When I entered the formula into Excel I got the message, “there is a problem with this formula.” In order to get Excel to make the calculation, the user must add parentheses to clarify the order of calculation. So yes, the formula as written is ambiguous and the person needs to clarify how the problem should be solved. In other words, don’t leave all of the decision making to a calculator.
there's a difference between how a computer calculator reads and how mathematics is done by hand by a person. When we are writing a division sign by hand, we use the traditional division sign ÷, but for a computer calculator we end up using the /..which can also be interpreted by a computer calculator as a division sign separating the numerator and the denominator..so for this question, it was written as 60 ÷ 5(7-4) which is the correct way to write this question for the answer to be 36. If you want the answer to be 4, it should be written by hand like this: 60 ÷ (5(7-4)). This has always been how mathematicians do math by hand. Using the / sign on computers, laptops, phones is what is causing the confusion. People are so used to seeing the / on digital devices, they think it's creating a fraction and start thinking numerator and denominator.
This problem is the reason why you should use fractions instead of the "divide"symbol: makes it completely unambiguous. Either the (7-5) is in the lower part of the fraction (denominator in English?), making the answer 6, or the (7-5) is completely outside of the fraction, making the answer 24.
There is no reason you can't switch the position of the 5 and the (7-5). Then you would have 60 in the numerator, and the 7-5 in the denominator. What next you would do is calculate 60 divided by 2, which is 30. Then you would multiply 30 times 5, to get 150.
Fractions are not interchangeable with division. 1 vinculum 2 is the fraction one-half 1 solidus 2 is the division 1 divided by 2 You cannot just replace one with the other willy nilly.
The kicker is the "divided by" operator in its presented form. (At school in Germany in the 70s we used : for division). This sign however suggests a fraction with 60 in the numerator and everything that follows the division sign, hence 5(7-5), in the denominator. That would be 6 then. In our school we were encouraged to express divisions in fractions because they are visually easier to resolve when they become large and contain many variables. It seems like the sequential PEMDAS convention is the generally accepted one mainly because of computers.
That symbol has meant divide from before the 70's. In true math, one does not use fractions...ever. One uses decimals. 1/2 in a math is indicated as 0.5 in order to be absolutely clear. It leads to less problems, and in programming it leads to a lot less problems.
Sorry but nothing about a division sign suggests a fraction at all. It sounds more like you were given a bad suggestion by someone trying to make things seem easier. This is also not a restriction from computers. They could just as easily have been programmed to solve it following the second pattern but they weren't because that has been wrong for more than a century now (predating computers).
@@R2BMusicCH Yes of course you can convert it to a fraction. The fact that you can convert it does not mean it's implied to be that at all. Your original conversion was wrong. The fact that you did it wrong does not mean it was implied that it should be that way. It just means you don't understand how to convert between those representations. A correct conversion would be 60/5*(7-5)
I’m 70, and I’m just thrilled to find out I still remember being taught this! And no, I’m not a math geek. I’m a little old lady who has stayed motivated to keep learning all my life!
The correct answer is to use proper consistent notation. You want the answer to be 24? 60 / 5 * (7-5) 60 / 5 * 2 12 * 2 24 You want the answer 6? 60 / (5 * (7-5)) 60 / (5 * 2) 60 / 10 6
So, everything you've written is correct but I'd like to add: 5*(7-5) vs 5(7-5) There isn't a clear-cut difference but I'd lite to think that the latter represents factorization whilst the other is normal multiplication. If this was the case 6 would be the correct answer. Considering how unclear the notation is you wouldn't know the difference but this would simplify your second calculation.
@@peckapuder The multiplication sign separates terms in the expression. The coefficient is part of the term. Order of operations applies to each separate term in the expression. What people are calling "implied multiplication" is simply using the number as a coefficient of the parenthetical expression as a term within the complete expression.
The author has completely misunderstood the issue. It's got nothing to do with any historical interpretation of ÷ as he claims at 2:09 and everything to do with the priority of implied multiplication, which he fails to even mention. In formulae, implied multiplication takes priority over division. For example, on the Casio website, it states "A radian is 1/2πr of the circumference of a circle." This is the standard definition and it does NOT, repeat NOT mean (1/2) * π * r. No, it means 1 / (2 * π * r). The implied multiplication is done BEFORE the division. And remember Casio makes calculators, so they should understand this point. The problem arises when folk blindly substitute numerical values into a formula and enter the result into a calculator. Calculators don't know the difference between implied and explicit multiplication, so the answer comes out wrong. So, returning to the original equation, 60÷5(7-5), the question I would ask before calculating the answer is "where did this come from?" If it is the result of blindly substituting values into a formula such as a/b(c-d), then the correct answer is probably 6 rather than 24. Also, you are more likely to see division represented by / rather than ÷ in such formulae, so the formula 1/2πr really means: 1 _____ 2 π r
When using Excel, all you need to do is be sure of your maths. I know your "parentheses syndrome" because I suffer from it, too. But the truth is that I'm just not good enough at intuitively simplifying fractions, so I force the program to jump through all the hoops I need to be sure I got it right ;)
I see the need to do that too to ensure I get the correct answer. Although it might be a bit more complicated, it is worth to to avoid later headaches.
Math and CS teacher here. I think everyone misses the most important part here: spacing. A common practice in CS is to use spaces to display precedence, so for example you would write a*b + c*d. It helps readability and can be really usefull for less known operators precedences like and/or. And also not all languages follow the exact same precedence rules, especially for bitwise operators. So in the ambiguous expression shown here, the modern precedence rules would give 24 but the spacing indicates that it's actually 6. For the same reason, when I see 1 / 2x, I tend to understand it as 1 / (2x).
@@ChespiritoChavo322 There's no "the result is ...", it's all about conventions. Don't take conventions as rules written in marble, they change over time, they change from a country to another, from a book to another, from a calculator to another, etc. We don't know the context of this expression, maybe it's from an old book for example, so we cannot know for sure that modern precedence rules apply. But the spacing clearly shows the intention, and that's something we can rely on.
@@ChespiritoChavo322 There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. That's what's causing the different answers. Division is used in either case.
And no one genuine has used the obelus symbol in the same expression as parenthetical multiplication. It just isn't done, except during these social media "math experiments" that offer no insight into how mathematics works. If anything, these problems just confuse math students (particularly young or inexperienced ones) trying to figure out order-of-operations rules in a realistic setting.
Not to mention that the ISO for mathematical notation has (for quite some time) said that the obelus should never be used for division. Math classes and mathematical exercises are not supposed to use the symbol, so teaching it is only a way to confuse younger students.
I asked my father who was an engineer for 45 years and literally helped build parts for the space program and the nuclear programs, and he said the answer is 6. He explained that there are 2 elements. 60 and 5(7-5), these values represent something and are not just numbers. So, there are only 2 expressions. The equation should be 60 / (5(7-5)). This shows how setting an equation up correctly is most important. Given the fact that these guys sent several capsules to the moon and back, I'm going to go with his answer.
I agree with your conclusion. I was taught that Mathematicians, Engineers and Physicists preferred where possible to rewrite an equation without the division ÷ sign to avoid ambiguity. If mathematical conventions are being changed to suit Calculators preferences, surely an honest person would consider that a very dangerous precedent. I am willing to be corrected.
@@malcolmbrewis5582 One other thing I did to test my dad's conclusion was I googled pictures of famous mathematic problems and equations. Secretly I was hoping to prove the old man wrong, lol. But I could not. I could not find a ÷ symbols on any of those blackboards. I took that to mean this issue of confusing how to write an equation had come up before, so to be clear and accurate, they did not use them. It makes since that they would not want to have their proofs interrupted in different ways. The same issue could easily surface in grammar as well by including or omitting punctuation like comma's.
Almost anyone who works in STEM or has higher education will give the answer of 6. Japanese calculators also give the answer of 6. Anyone who only did high school, American high school teachers, and newer American calculators, will give the answer of 24. Make of that what you will.
The reason I treat the answer as 6 is simple - if I see an equation like "x/2y = 1" then I don't think it should ever be interpreted to actually mean "xy/2 = 1", which is basically the same question. Nobody who ever said an equation like that would mean for it to be interpreted that way (unless they're deliberately trying to trick you), and having rules that make it function differently will only ever make things more convoluted than they need to be for no practical benefit. If you wanted to write 60/5(2) to mean you're dividing by 5, then instead write it the sane way as 60(2)/5 instead.
@@thetruth3828 Must be a quadratic, because the claim is, if you are old school it's 6. But if you are new school it's 24. Don't know how it can be either or either, as there must be an intended definite outcome.
@@onlythetruth883 You clearly don't know what a quadratic is--but without knowing what you're doing or saying you've accidentally hit on the problem with many of the arguments in this thread: the given task is to evaluate a simple ARITHMETIC EXPRESSION using the generally-accepted rules for doing that. Attempting to translate it to an ALGEBRAIC EXPRESSION and applying rules useful there, then translati9ng back to do the arithmetic, does not work. If it involves just numbers and arithmetic operators and brackets, so one could evaluate it on a calculator or calculator app, then use the usual rules for evaluating arithmetic expressions and don't try to remember your high-school algebra and misuse that very foggy recollection to confuse yourself and others.
@@waynebrehaut7183 Of course I was being sarcastic when I said must be a quadratic. And you did get the point-->. There is no point until the rules are firmly established.
I remember being taught that parenthesis was calculated first, multiplication came next, then division, then addition & lastly subtraction. This gave me 6.
That's what they taught for most in our country's basic education too, literal PEMDAS in strict order (as the letters). Only in college that both math and computer science professors agree on the real correct method. I'm mildly infuriated that they always teach children outdated or plainly wrong things like this (the four taste regions also comes to mind, so wrong)
@@mk_rexx well, pemdas (or pedmas as I know it) isnt wrong though. But the order of division or multiplication doesnt matter, and the order of addition and subtraction doesnt matter, as in both cases they are effectively the same operation. So everything in brackets first. Then all multiplication and division. Then all addition and subtraction.
according to my 1989 public USA education the answer is 6. 60÷5(7-5) = 60÷5(2) = And here is where the fight begins. Technically, according to the 1989 USA public education I received, the PARENTHESES still exist that this point, and therefore has to be resolved first by Order of Operations 60÷10 = 6 Parentheses (inside first, then anything dealing with the Parentheses), Exponent, multiply/divide, add/subtract. Even the distribution rule give the same answer 60÷5(7-5) = 60÷(35-25) = 60÷10 = 6
Your memory must be faulty or you had a bad teacher... I have at least 5 different math books from 1907 to the present and they all state the same thing... You evaluate what's (WITHIN) the grouping symbol not outside. And ALL multiplication and division can be evaluated equally from left to right..... When there are no (OPERATIONS INSIDE) the brackets/parentheses left to evaluate you can remove the parentheses and replace with an explicit multiplication sign or leave them to represent implicit multiplication and nothing more.... When you have a single value inside the parentheses that step is done... (7-5) is a parenthetical priority 5(2) is NOT a parenthetical priority and is exactly the same as 5*2 As for distribution, the whole point of distribution is to eliminate the need for parentheses by pulling what's inside to the outside not the other way around... Distribution requires that you multiply all the terms inside the parentheses with the TERM outside the parentheses. Terms are seoerated by addition and subtraction....60÷5 is one term to be multiplied by the two terms 7 and 5 60÷5(7-5)= 60÷5*7-60÷5*5= 12*7-12*5= 84-60= 24 60÷(5 (7-5))= 60÷(5*7-5*5)= 60÷(35-25)= 60÷10= 6 2+3+4+5 is 4 terms 10-9-8-7 is 4 terms 10÷2×6÷3 is 1 term 10÷2+5×3 is 2 terms I hope that helps you understand better....
Richard S Again, that is how I was taught and I noted when and the type of education. That's why I explained it the way I did. It was so everyone can see 1) the logic I used because 2) it was the logic I was taught by educators 3.) using math books they provided. So, with the correct answer being 24, you now have to ask the question; why are so many people like myself getting the answer 6? Because we were educated wrong!
@@ComputerGarageLLC unfortunately a lot of people swear that they were never taught to multiply and divide before they add and subtract. Are we to believe this as well?LOL I graduated in 1985 and was not taught in that manner. I have never seen a math book that supports your argument. I would be very interested in seeing a math book that supports your argument? It is very concerning that so many people do get this wrong considering that the order of operations supports 24 as well as the commutative property and distributive property support 24 and the multiplicative inverse of division supports 24 as well as the majority of online math engines and scientific calculators support 24. I guess this just goes to show that most people don't have to use math other than basic addition and subtraction on a regular basis. Thank you for your input. Have a great day
You are free to not believe me. That is your choice. But it was how I was taught through the public education system. Clearly I was taught wrong, and it appears that many others were taught wrong too. we, those who are wrong, are a reflection of what we were taught. And you are correct. a majority of people never use more than adding and subtracting most of their lives. Perfect example. Today a shirt cost $11.99, but tomorrow that shirt is on sale for 25% off. How much will you save by purchasing the shirt tomorrow? The answer that most people will give you......25%. Another example I use. Mary has $10, but she need 2 gallons of milk @ $1.98/gallon and at least $5 in fuel. Does Mary have enough money. Doesnt matter, as mary will go buy the 2 gallons of milks at the gas station, and tell the clerk to put the rest in fuel. So now, most of us never use more than very basic math most of our life. And you have a wonderful day also.
If you trust Texas Instruments' calculators, then the rule changed between 1993 and 1996. My TI-83Plus user's manual (page 1-24) says implied multiplication has the same priority as regular multiplication and division, so 1/2x is evaluated as (1/2)x, *but* the TI-82 gives a higher priority to implied multiplication so 1/2x is evaluated as 1/(2x). According to Wikipedia, the TI-82 was released in 1993 while the TI-83 in 1996. Modern TI-85Plus also has same precedence for implied and explicit multiplications, so they give answer 24. But modern Casio (at least my fx-CG50) work like old TI-82 and gives answer 6.
Don't key expressions unthinkingly, verbatim. Electronic calculators are not to be trusted that much. That is learned very early. The insertion of brackets is often needed. Rewriting with or without a fractional exponent can be useful. Sometimes, as in 1° 1', a "+" must be inserted to show addition. Juxtaposition can mean different things. 3pi indicates multiplication. 31 indicates the addition (of 3 × 10 plus 1 × 0). An electronic calculator frequently need to be told how to operate.
My Sharp EL-520W gives an answer of 6 for the expression "60÷5(7-5)", while it gives an answer of 24 for the expression "60÷5×(7-5)". This is also the way I was taught it in school. Implied multiplication with no operation symbol as in expressions like "xy or 3(5)" takes precedence over division indicated by the ÷ sign, while multiplication indicated with a × symbol has the same precedence as ÷, evaluated left to right. I didn't even realize this was controversial till I saw this mentioned in some of your videos. When did this other convention become popular?
Same. My first answer is 6 cause the first thing I do is multiple 5*(7-5) wich is be come (35-25) and decrease the number at parenthesis, so it will be 60÷10 and is 6. Sorry for my bad grammar...
This is interesting and the reason for the change is that in the old interpretation the division symbol was actually a fraction symbol. The point above the bar represented all of the equation to the left and the point below the bar represented all of the equation to the right. Now however the division symbol is simply that, a symbol to divide the order of operations to the left by the order of operations to the right. It's somewhat akin to English changing from archaic to modern English. The meaning of words has changed and if you keep up with the current meaning, you will understand what is being said. For example, If I said, your room is in shambles. Currently that would mean your room is a mess, however it would have meant that your room is in a meat market. What fun.
Prior to 1917 SOME text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses. So the ERROR was corrected post 1917... This was an ERROR brought about by the text book printing industry in regards to the misuse of the obelus. This is not why most people evaluate this expression incorrectly. They get the wrong answer 6 because they incorrectly believe that parenthetical implicit multiplication has priority over division.
I'm 59 years old, for what it's worth. I was taught the fractional representation method in school and it still makes sense to me. Draw a line and solve for the numerator, then the denominator, then divide. That is how it was done then. If it is incorrect then how did we ever get to the Moon? LOL
@@rrsharizam I don't use a CASIO calculator I use Wolfram Alpha a math engine and I dbl check with Mathway another math engine and if the two don't agree I find out why. But for basic arithmetic I only use them to validate my answer not to give me the answer... CASIO fx-82es will give 24 CASIO fx-570es will give 24 CASIO fx-50fh will give 24 CASIO fx-991es will give 24 CASIO fx-570ms will give 24 My response to anyone who says the answer is 6 is to evaluate 60a(7-5)=24......a =? Well a= 0.2 or 1/5 and the divisional reciprocal of 60*(1/5) is 60÷5 Soooo 60*(1/5)(7-5)=60÷5(7-5)=24
@@RS-fg5mf "will give" ??? So, you don't even use Casio, yet you say it will answer 24? I don't care whether the answer is 6 or 24. I just wanna say that Casio & Sharp answer 6. That's all
@@rrsharizam I have a pic of these model CASIOS giving the answet 9 to the expression 6÷2(1+2) So if it will give 9 to that expression it will give 24 to this expression....
Part of the issue is whether one considers the number parked outside the parentheses to be a common factor of the terms within the parentheses, or just another number in the sequence. I was taught that the number just outside the parentheses (in this case 5) is a part of the terms inside the parentheses ((a-b), with in this case a=7, b=5) unless separated by a multiplication sign. So (5a-5b) is the same as 5(a-b), but not the same as 5*(a-b). This would lead to a result of 6, which I would consider to be the proper result. Also, look at the division sign itself. The top dot is the stuff to the left, the bottom dot is the stuff to the right. Which would also yield 6. I learned back in the 1980s and 90s that you have to interpret equations for computers and calculators to get the proper results. So I would input the above equation as =60/(5(7-5)) when using a calculator or computer. Which would again yield 6.
It's honestly feeling like a bunch of people were the subject of teachers trying bad ideas in an attempt to make things easier. That's not what "common factor" means at all. And as someone has already pointed out having the * explicitly changes absolutely nothing. It wouldn't make sense to have it change anything.
“So I would input the above equation as 60/[5(7-5)].” You completely changed the equation the way you wrote it. You can’t just add an extra set of brackets in the middle of the equation. Had it been presented in that form, then yes, the answer would be 6. People are getting confused with what “brackets first” actually means. They think if they see brackets, that means everything touching the brackets gets done first. Brackets first means you solve the inside of the brackets first. Once you do that, the brackets part is done. 5(2) is 5X2 is 5*2. It doesn’t matter what form you use, they’re all the same thing. Since it’s now just a straight up multiplication and division equation because the brackets have been solved, you move from left to right. And the above commenter is correct that 5(a-b) is the exact same thing as 5*(a-b) is the exact same thing as 5a-5b. If a=4 and b=2 5(4-2)= 5(2) 5(2)=10 Also 5*4-5*2=20-10 20-10=10
Kudos! The expression on the RHS must be evaluated first before the division. What the RHS says is that there is a common factor of 5 and so the full expression on the RHS is 5(7-5) = 35-25 =10. And so the answer is 6. I don't care what Google says!
@@mohasat01 You also don't care how math works. That's not what a common factor is. And even if it were common factors is just an interesting fact of the numbers and has nothing to do with how or when you evaluate them. Per order of operations 60 / 5 must be evaluated before 5 * (7-5) because multiplication and division are to be evaluated left to right.
I prefer the ”special rule” version from 1917 I like writing my division as a fraction. That way there is no doubt as to what is numerator and denominator. The special rule seems to follow this process.
i dont get why the separate division and fraction, isn't 1 over 2 0.5? Isn't 1 divided by 2 0.5? Then why are they so FKN different when they are the SAME?!
@Anika Anjum That's why writing everything on a single line is ambiguous. The school I was taught is the division is a grouping operator so that everything to the right of it comes under the operator IE in the denominator. You were taught in a different school of thought. These different schools of thought are why equations need to be clearly written out.
Why would you use a calculator as the way to measure what interpretation to use. A calculator is just a computer and a computer only does what a human programmed it to do.
Why would you use a calculator for such a simple task? However when my daughter was 13 in 1989 I bought 13 candles at the local stationery shop. I gave the girl 13pence but she said I'd better check its correct & rang up 1penny 13 times. No it was an old till not computerised connected to stock control. She then said "Yes you are right 13 pence" & put out her hand for the money.
We couldn’t even use calculators in high school (they weren’t available in grade school) in an effort to prevent the inevitable, the DDOA (the dumbing down of America).
@Michael Stocker WRONG. There is absolutely nothing wrong with this expression except for the ignorance people have about parenthetical implicit multiplication.... The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 24
sorry for not knowing all the correct english terms So do I the paranthesis is broken down for easy of handeling and shopuld be multiplied as it stated 5(7-5) -> (35-25), of the five should be diveded down to a 1 by devidind all groups by 5 to clear it out (60 / 5(7-4) -> (60/5)/((5(7-5))/5 ---> 12/(1(7-5) --.> 12/(2) The 5(7-5) is a part of the paranthese operations and even in pedmas paranthese has priority
@@ronhan9 Wrong... 60/5(7-5) does NOT equal 60/(35-25) Easy handling is to simplify what is inside the parentheses. 5(2) is not a parenthetical priority and is exactly the same as 5×2... The TERM 60/5 is to be multiplied by the value of the parentheses 2 and the only correct answer is 24
Although we have modern PEMDAS to adjudicate how to interpret such expressions, this is really an inherent language flaw, as you pointed out mid video. It is rooted in the idea that you can omit the multiplication symbol between and number and an opening parenthesis. If you write it as 60 ÷ 5 * (7 - 5), you still need PEMDAS to interpret it, but it is much less tempting to get it wrong.
I read a while ago that it was 4% of Mathematicians who use it this way. The rest of the population didn't. Probably someone in a wee office somewhere decided.
I've used HP calculators with Reverse Polish notation from the start when they hit the market! In that system you start calculating the content of parenthesis and then go outward. With this logic, the result is definitely 6. During the whole time of my physics studies (that means dozens of textbooks in physics and applied mathematics), I haven't found a single case being confronted with any ambiguity of a mathematical term!!! If someone gives me such an ambiguous expression to calculate, I simply refuse to calculate! I will tell him to study mathematical semantics first! (This has already happened)
I memorized times tables in the mid 60's; PEMDAS wasn't a thing when I went to school; I never took physics or calculus, only went as far as trig; the answer I got is 6.
There's a reason why most math teachers have rarely used the '÷' symbol in decades. Almost every teacher will teach division in fraction form because the division symbol is very ambiguous. If written with a '/' or in fraction form, there would be no question what the right answer is. 60/5(7-5)=6 Reason is, everything multiplied on the right of the '/' is part of the denominator. Which is the reason most people are tripped up using the archaic '÷' symbol. The rules are slightly different. In order to get 24 with the '/', you would have to write it as: (60/5)(7-5) Easy. Thats why nobody who actually works with math uses '÷'. And in higher level math, such as calculus in fluid mechanics or thermodynamics, the order of operations is practically useless. You're stuck developing your own equations by following your units of measure to get from one place to another. No real need for PEMDAS when you have a force in Newtons or pounds, and you need to solve for pressure in kPa or psi. Or maybe you need max power output in Watts or horsepower. Then again, if it wasnt for archaic symbols used to confuse people who dont do math in this respect regularly, this channel would probably have died out long ago
WRONG.... Prior to the 1900's that's how the obelus ÷ was being misused. The solidus was never used in this manner 60÷5(7-5) and 60/5(7-5) are exactly the same and both equal 24 The solidus is NOT a grouping symbol only the vinculum (horizontal fraction bar) has grouping power.... 60 ------(7-5) = 60/5(7-5)=24 5 60 -------- = 60/(5 (7-5))=6 5(7-5) Extra brackets required to keep the grouping of operations together that the vinculum provided when written in a linear format with infix notation.... That is not why most people get this wrong. They incorrectly believe that implicit multiplication has priority over division. It doesn't...
Not necessarily, Richard. If it was written properly, the (7-5) is part of the numerator. So, without parentheses, you would have to write it 60(7-5)/5=24 Everything multiplied on the left of the slash is numerator, everything on the right is denominator. You're welcome to disagree. That's cool. However my college professor would mark my answer wrong if I wrote it 60/5(7-5)=24 As I said, nobody writes equations or mathematical phrases like this for good reason. There are simple programs to write and paste complex formula as they should appear, not like this with the intent to befuddle. Best of luck to you, bud.
@@boredbales12345 WRONG again. Multiplication is Commutative. 60÷5(7-5)= 60 (7-5)÷5= (7-5)÷5*60= 24 All 3 expressions are equal to 24.. Evaluate this equation 60a(7-5)=24...... a= ?
neither ÷ or / have the special treatment of taking a photo of content to the left, to the right and using the operation afterward. None of that is in the order of operations. In 60/5(7-5) the / is a division symbol, and the order of operations says 24... You must be thinking of the fraction slash but that requires (7-5) to be subscript, ⁶⁰/₅ₓ₍₇₋₅₎, to equal 6.
The correct ways to phrase the questions (depending on what you want to ask) would be: 60/[5(7-5)] for which the answer is 6. Or (60/5)(7-5) for which the answer is 24. The question as originally phrased makes no sense. The division sign is never used beyond grade school nowadays (it is not there even in a computer keyboard), but it was there in the question but without the multiplication sign. It was not only confusing but sloppy. One set of parentheses would have eliminated all ambiguity. Assuming the question was originally an algebra question for which you then substitute in the actual numbers, then "6" as the answer actually makes more sense.
@@Ok-th2gd 60÷5(7-5) can be changed to 60/5(7-5). From that 60 is the numerator and 5(7-5) is the denominator. 5(7-5) becomes 5(2) = 10 so 60/5(7-5) changes to 60/10 which is 6.
I was helping my 13 year old with his math homework 15 years ago and learned something that I was never taught in school. Not even in College. "Please Excuse My Dear Aunt Sally".
@@aligator7181 That's (3*47)-(1/4398473)+(10)-(8/33) = 150 + (25/33) = 150.7575757575 . . . Most/all decent calculators will get that without using ( )s
@@Chris_5318 Yes, but the trick is to get the order right. I have never used an expensive scientific calculator, I am assuming they probably sort out the order automatically?
@@haroldprice1030 Different, but almost identical, models from the same manufacture can give 6 or 24. The correct answer is the one found by using the same convention that the author used. We have not bee given tha info. However, the author would have to be crazy if he was expecting anyone to get 24.
You said : The "MODERN" interpretation. A lot of people, including myself, have been taught the one that gives 6 for result. I love math and was always at the top of my class. 24 would never have been the answer.
The issue is that the video author doesn't understand BODMAS correctly... "brackets" means you grab the brackets first and solve them themselves using BODMAS. So 60/5(7-5) the bracketed term is 5(7-5) which expands to 35-25 which makes 10. 60/10 = 6. Now, if you add a multiplication sign then it changes the precedence because you are actually changing the equation significantly. 5 * (7-5) the bracketed term becomes only (7-5) which is of course 2. A deliberate nuance used to create a video I think. Fair play.
I think everybody is missing the point. The fact is that a mathematical expression like this is derived to calculate an aswer to a problem in the real world. Before we can know which binary tree to follow, we have to know the real-world problem. What does 60 represent - it it people, who are being divided by ... what? We also need to know what the 7 and the 5 represent, and why they are bound trogether in the bracket. Mathematics is a tool - not an entity in itself.
Seems i was thought the 1917 version. My result was 6 too. Maybe you could do a follow up video on why the modern version is now used. What advantage does that interpretation bring?
In large part because expressions cannot be presented to computers by use of a divide bar that clearly shows what is in the numerator and what is in the denominator thereby showing grouping. Computer languages demand expressions all be in-line and there is no way to group subexpressions other than with explicit use of parenthesis.
it is manipulating the mathematics as they do it with everything this days. All depends who is calculating and for whom. If that was you assessed by tax office it would be 24 but if that tax would be calculated for Bill G. it would be 6. - 😏 the sentence when be written as a fraction with 60 on the top and the rest in the bottom and the result is obvious.
@@lubanskigornik282 And I just know if I buy Bitcoin, somewhere along the line my payout is going to use the New Math and end up dividing my payout by 24 instead of 6.
Correct!! But then I was in school in 1917!!! I think it is used to save space and characters in computer. 5(7-5) uses 1 less character than 5x(7-5). New Math!! You know, 2+2=5.
Left to right, what a nonsense. The fact there is no multiplication sign between bracket and the 5 is a clear indicator, that this is just one term, that the 5 and the bracket belong together, period. Anything else is sophism. 6 is the solution, period.
It's surprising how some modern calculators like CASIO, which are recommended by math teachers, also give 6 as the answer! (tested with models fx-82ES PLUS and fx-82SPXII Iberia)
Thanks for the info! CASIO's calculators were a thing for 6÷2(1+2) as well. I found one video, for example, that shows 9 on one calculator (fx-50FH) and 1 on another (fx-3650P), both which are marked in the video as "H.K.E.A.A. approved" (Hong Kong examinations and assessment authority). ua-cam.com/video/IXUBepvylQg/v-deo.html I would love to speak to someone at CASIO about this--would make for a great video!
it is something called syntax. it is not as much math as it is programming. it is the programming of how to READ math in a single line. LIKE A TRANSLATOR FOR THE CALCULATOR.(it works in binary data) you do not.
When writing an expression parser, you may want to capture the intent of the user input. As I mentioned in another comment, the expression 1/2a is most likely meant to be interpreted as 1/(2*a), not (1/2)*a. The intent is generally to raise the precedence of implied multiplication above that of explicit division.
also it could be written as 60/(5(7-5))=6 or (60/5)(7-5)=24 to be less ambiguous. I've done a significant amount of coding over the years and I like the use of parenthesis to reduce confusion.
24 isn't ambiguous from the get-go, though. To get a different answer just assumes grouping around 5(7-5) which does not exist in the original problem.
I've done a lot of coding over the years and I hate it when people overuse parenthesis trying to reduce confusion because it just makes the statements harder to read. Now I have to parse a bunch of parenthesis to figure out if you actually changed the PEMDAS order at all with them only to find out you didn't, you just wasted my time.
"also it could be written as 60/(5(7-5))=6" No, it couldn't. That would be wrong for this problem. There is no ambiguity. Extra () are not required when the operations are already in order. You CAN use them, but the problem has clear meaning without them
The expression typed into my Casio calculator exactly as shown returns the result 6. Which is exactly what I calculated as I was always taught that if there was no operator between a number and an expression in parentheses, then they were linked and to be calculated together. I.e. 5(7-5) = 10
Exactly so. And his sentence is not ambiguous. The verb saw separates the subject (I) from the direct object (man) and any modifiers of the object (binoculars). So if you wanted to say you saw the man by using binoculars, the binoculars would have to modify the verb saw.
As someone who mainly learned math through programming, I'm sometimes disadvantaged by a lack of theory and long-hand methods. I struggle with deciphering mathematical notion in order to translate it into something I'm working on, mainly because of all of the implied, rather than explicit operators and evaluation order of written mathematical notation. While there is an underlying default evaluation order in programming, you can explicate everything to the order you want. The result of 60 / 5 x (7 - 5) would be evaluated as (60 / 5) x (7 - 5) = 24. If you meant something different, you'd explicitly say so with parentheses and operators: 60 / ( 5 x (7 - 5)) = 6. The difference between / and ÷ would I guess be one of which programming language you are using. ÷ may be a valid in programming, but it's not a common keyboard character, so I don't know, because I've never used it.
Also depends on what math you're doing... Algebra, geometry and calculus always order of operations is always parenthesis 1st, then multiplication, division, addition and then subtraction... That's the basic order of operations for any higher math except for programmers because the computer is doing the math not the programmer! Discreet Mathematics, is what computer programmers learn... Euler circuits, TSP, Fibonacci numbers, etc.
@@victorglaviano: Not discreet mathematics, discrete mathematics (i.e. noncontinuous math, vs. being unobtrusive, since context matters). At some point, for society to function, there have to be rules / standards that are agreed on, or there would be complete chaos (as per the Tower of Babel meme). I was pleasantly surprised to see how Microsoft Excel (V 2010, from Office 2010, which works fine for my private use at home), handled it. When asked to evaluate the equation: 60 / 5 (7 - 5), it forces you to clarify what you mean, instead of just giving an answer. By default, it asks if you mean: 60 / 5 * (7-5), and if you agree, it gives 24. It also gives you the option to "correct" the formula (i..e make it nonambiguous) yourself, by updating it some other way. Remarkably good behavior for typical application software, likely brought about over time by so many people using highly competitive spreadsheets for critical applications, and FORCING companies to eliminate ambiguity would be my semi-educated guess, as someone who spent a career in application and then system programming at IBM on mainframes. For example, my CPA uses spreadsheets constantly for taxes. Can you imagine the chaos if they misinterpreted ambiguous formulas? Best just not to allow them.
As a programmer you should know that some of the languages are using polish notation where execution is from right to left. 60/5(7-2) would be equal 6 and 60/5×7-2 would also be equal to 6 and 60/5×-7 2 would be equal to 6 as well
@@Grim_Reaper_from_Hell - I have no idea what you are saying. Every language I have ever worked with evaluates right to left; It's not unusual. Also, I just compiled your first 2 operations and the printed results are as follows: 60 / 5 * (7-2) = 60 60 / 5 * 7-2 = 82 Your third operation would return an error as you have two operators next to each other: 60/5×-7 2 If you meant 7-2, then this is identical to the second operation you described, and the result would be 82. If you meant 60 / 5 * -7 * 2, the result would be -168.
when I was in school we were taught to distribute the 5 to the numbers in the parenthesis first. thereby the resulting answer would be 6. At some point in time we changed the way we did math in order to confuse our children...I mean make math easier lol. I still enjoy the videos; keeps the mind working.
@@kevinsanderson4112 as written i would think the 5 was factored out 60/((5× 7) -(5×5))i know some teachers who teach it this way and my calc class was like that so my immediate thought was 60/10 =6
I also came to the historical way, although I think part of it for me was how I viewed the question. I saw it similar to 60/5x where x is (7-5) being 2.
The reason I came up with 6 was the fact I was taught that the order of operations was in the actual order of the letters. Parenthesis first then exponents, Math then Division, Addition then subtraction. VERY EYE-OPENING AND EDUCATIONAL. GREAT VIDEO!!!
That is how I also learned it. I was taught to remember - (P)lease (E)xcuse (M)y (D)ear (A)unt (S)ally. (P)arenthesis, (E)xponents, (M)ultiplication, (D)ivision, (A)ddition, and (S)ubtraction. Please note...I went to a public school. LOL!
Although it wasn't mentioned in the video, the reason multiplication/division are not given a specific importance is because they are the same operation, so you perform them in the order as written. Division is really just multiplying by a fraction. Ex: 60÷5 = 60 x (1/5). The same holds true for addition/subtraction. Subtraction is really just adding a negative number. Ex: 23 - 8 = 23 + (-8) If you change all division operations to the equivalent multiplication operation, and then multiply straight across, you would see the answer will always be 24 to the equation presented in this video.
Y'all learned wrong or were taught wrong. The correct translation of the acronym is "...Multiplication AND Division..." (equal rank performed left to right), "...Addition AND Subtraction.. " (equal rank performed left to right).
I remember being taught that when there is an “understood” multiplication because no “x” sign is there, then this calculation would be done before the preceding division sign. The 5 and the solution to the calculation in the parentheses are linked together, like the expression 5y are linked. If y=2, then 5y=10. Then divide what is on the other side of the division sign by 10. If they wanted me to do the division before the multiplication, they would have used a multiplication symbol in place between the 5 and the parentheses.
correct because the 5 is the coefficient of the parentheses. whenever you have a parentheses, you have a coefficient, and whenever you have a coefficient, you have to utilize the distributive property.
Ditto. No times sign between the 5 and the 2, just parentheses, was to be calculated first with how I was taught. I see it both ways but unless the order of operations changed in the last 25 years and it was not made public knowledge, then my math teachers would tell me I’m wrong to give 24 as the answer.
All of these type example are due to someone writing mathematical statements in the most confusing way; in REAL mathematics, physics and computer programming we choose the write mathematical statements so as to prevent confusion. These example-makers lift a few excerpts from journal (or written text) articles where one is forced to use only a single line of text space; however most likely elsewhere equations are presented in an correct format.
@Chris Travers when I typed it into my ti-83 I got 24. And that was after I solved it without a calculator. Anything touching but not in parenthesis only means multiplication nothing else.
When I see a number right next to a parentheses I interpret it as a factored number, represented as one unit. So 5(7-5) is really a factored expression of number 35 minus the number 25. I'm just saying, ambiguity must be interpreted in context instead of immediately concluding that a number outside of a parentheses is synonymous to a simple multiplication symbol as 5 x (7-5) un which case your final answer of the full equation would be 24. However if you see the 5(7-5) as a factored expression of 35 minus 25 then the answer of the full equation would be 6.
The problem is the symbol ÷ that is used. It causes ambiguity and should not be used. If a problem is not precisely stated, in terms of math the only correct answer should be that the problem is ambiguous
Many commenters seem to dislike the symbol ÷. @presh can you weigh in on this? I personally like the symbol and wish it could be defined as follows : the line in the centre shows that there will be division. The dot above represents a placeholder for the numerator, the term immediately left (preceding) and the dot below representing the denominator, the term immediately following the ÷ symbol. If this sounds strange, just look at these symbols and I think there is enough precedent: x÷y x%(special case where the numerator is always 100) and x/y. If this is always the interpretation, and it remains consistent, then it implies 60 over the rest, or the historical usage, or the tree on the right. Perhaps the historical usage of ÷ was as I suggest? This removes any ambiguity and also preserves order of operations. In an era where each character of text printed increased the cost a shorthand like ÷ would have been valuable...
Nope, removal of the multiplication symbol does not change the order of operations. It's just shorthand that has been adopted into common usage (more likely laziness in removing the sybol).
Yes, and 6 is the correct answer worldwide. Let me summarize the positions as I see them: > for folks who are followers of the PEMDAS philosophy and believe such things as x/3x is equal to x squared divided by 3 the answer is 24. > for folks like me who believe that PEMDAS is BS and screwing up the teaching of math in America and believe in such things as x/3x = 1/3 the answer is 6. Now I do recognize that this is America and one is free to choose, but from my viewpoint it does appear that the PEMDAS philosophy falls into the category of metaphysics; - - - you know, that abstract theory with no basis in reality.
See, when you get to "60 / 5(2)", to my mind, the 5(2) is an outer parenthesis+bracket expression (which should be evaluated after the inner parentheses+bracket) and should be evaluated before the typical multiplication+division. I was taught pre-PEMDAS, however, but I think that approach clarifies a lot of these "ambiguous" problems.
No in that case it would be [5(7-5)]. That’s where you would use inner brackets and then outer and that’s where you’d multiply by 5 before moving left to right from the beginning. In this case there are no outer brackets so once you’ve solved what’s inside of them, you move left to right from the beginning. 5(2) is the same as 5X2.
PEMDAS has been around for centuries, modern PEMDAS has been around for more than a century (as the video showed), so no you weren't taught pre-PEMDAS. You were just unfortunately taught wrong.
@@trickortrump3292 This is why I get dinged when I write essays in school - I genuinely use *too* many parentheses and think parenthetically. Not so much that I'm particularly proficient in LISP, though.
I was taught that 5(7-5) implies another parentheses because the multiplication symbol is omitted. So that the true form of that problem would be 60 ÷ (5(7-5)) and the answer would be 6. However if the problem is written as 60 ÷ 5 x (7-5) then the answer would be 24.
And what natural, empirical, universal law says that's wrong? Its not something we can prove with science. Its only what we all agree on it. And the fact that 3 videos exist on this channel is evidence that we don't all agree on it.
If we are being correct we should always write all parentheses e.g. ((2(x+3)(x-3))-1) but as you can imagine that gets messy real quick especially when writing by hand. That's why we omit some of the parentheses (at least where I am from) to make it cleaner and easier to read ((2(x+3)(x-3))-1) = 2(x+3)(x-3) -1
You calculated correctly. For some reason these kids are wanting to do the multiplication on the right before the division on the left, madmen all of them. It’s easy to see that if you take 60 / 5 (7-5) you start with the parenthesis 60 / 5 (2) So you have 60 / 5 x 2 If you do math incorrectly and do the multiplication on the right first, you get a sum of 6, but anyone who passed 5th grade math knows you go from left to right 12 x 2 is the final product before solution
The answer is "Don't use the flipping ÷ symbol", because there's no way to be certain whether the one who wrote the problem means (60/5)(7-5) or 60/(5(7-5)). The former would be 12(2)=24, the latter would be 60/10=6. People tend to see implied multiplication (5(7-5) as opposed to 5×(7-5)) as including the multiplier 5 as part of the term, so I imagine most would say the latter, but strictly speaking it makes no difference to PEMDAS. And my scientific calculator app actually gives a warning about the ambiguity of the problem as written and offers to let me change the "operand grouping" setting, giving the example of 1÷2π to demonstrate what changing the setting does. It does default to 24, however.
Before I watched, I knew the video would conclude with 24, but when I was in school the correct answer was 6. Why? When you have any number just outside a parentheses (without a separate multiplication sign) you multiply to what was in parentheses prior to any other operation. Same is true with a variable like x or y. So 12 / 3x when x is 2 is 2 (as opposed to 8). My school taught me to presume 3x is in parentheses as though it were virtually (3x) as far as order of operations. This is also true of 60 / 5(7 - 5). There is no difference between that equation and 60 / (5(7 - 5)) which in turn is 6. To get the answer 24 the equation should have been written 60 / 5 * (7 - 5). It makes sense to me in a way. If I have a 12 dollar buy for 3 packages of x toys and each package of x had 2 toys and I wanted to know the price per toy I’d write the equation as 12 / 3x = 2 per toy. However this video says that the correct equation to get the proper answer is 12 / (3x) = 2. This is also true of exponentials. At my school the answer to the following equation 60 / 5 (7 - 5) ^ 2 would = 3. Or to be very precise could be written as 60 / ( 5 ((7 - 5) ^ 2)) I’m assuming this video would say the correct answer is 48 I’m not saying the correct mathematical solution is my way or not, just saying this was how it was taught to me in high school.
Yes, both are valid since it is simply using ambiguous notation. It's terrible writing. Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
I have always struggled with maths having left school at age 14 but I am passionate in trying to figure out any maths problem. Don't often get the correct answer but I enjoy trying. Thanks for the exercise. Blessings.
There's disagreement on this answer because the notation is very sloppy. If you are uncertain about this particular math question, you might actually be better at math than you think. Your math might be very good when the notation is clear.
@Judy Cherry Yes but when you are dealing with modern Neo Marxism like we are now there never is any correct answer. You know in their warped minds 2+2 can equal 5. Every thing is fluid, You know like the Genders are. God help us if we don't take the World back from the Satanic Globalists.
All that changed was notation not the rules of maths themselves which is what a lot of people think is changing. It's like using Sin²x to mean (Sinx)² or using Roman numerals, MCMXIX, instead of Arabic numerals, 1919. Both are valid notations and using one over another doesn't break any rules or axioms etc. The problem with the question here is it isn't written to modern international standards, the ISOs. If it was written properly then everyone would agree on just 6 or just 24.
@@bingcherry2008 Oh there should be just 1 answer. You are right about what. What about the question 16/8/2? Or "What is 10 divided by 5 multiplied by 2?". They are ambiguous also, just like the one in the video, without more information to clarify what the person writing it meant by what they wrote. If someone was writing an academic paper and wrote 60÷5(2) they mean 6. If a programmer wrote a book on how to learn Python and wrote 60÷5(2) for an example they mean 24. The issue is the notation is ambiguous now. That's why we have international standards, to bridge the gap. With 60÷(5(2)) everyone agrees on 6. With 60÷5×(2) everyone agrees on 24. One of those is what the person writing the question meant but we will never know which. Until we do, both are valid.
5(2) counts the same as (2) ex x. the modern method generates 60/5x1(7-5) . the main problem with the modern method is the improper disposing of (). as soon as you make it 5x1() your disposing it the same as 5() without changing the number inside. no matter what () must be removed before proceeding even if exponents or multiplication takes place. this was know as completing operations between signs in the 90's. an example would be 2 x abc= vs. 2 x a x b x c . abc is a complete expression of 1 number to mutipply 2 by. no it may turn out a shortcut is it's all multiplied together so order doesn't mean much. but only if the shortcut doesn't ater the answer. in this case the shortcut 5x1(2) alters the correct answer so you have to follow the 5(2) = 10. since rewriteing 5(2) as 5x1(2) allows the interpation 60/5 x2 this is a basic half step to proper order of exponents and such. probably one of the most disturbing parts of teaching maths in a system
Quite literally elementary school math equation that people are arguing over?? This is disturbing... This should take one seconds to figure out that the answer is 24. Math really wasn't a lot of people's cup of tea.
Early on after scientific calculators became popular in doing this type of equation, math teachers told us not to use a calculator because it would give the wrong answer. When learning how to solve complex equations written in fraction form, the math teachers taught us to do the math above and below the line separately, then do the division. Engineers and physicists will use the old school method, which is called juxtaposition. This method accounts for the equation written in fraction form. The divided sign or a "/" use in the equation is just syntax. It replaces the horizontal line in fraction form. When written in one line using the arithmetic symbols and parentheses, some of these symbols are implied. So, when converting an equation from fraction to line form if the person writing the equation doesn't include a parentheses or bracket after the division symbol according the to PEDMAS, it changes the equation and the answer given. However, the rule for converting the equation from the signal line expression is to put everything left of the division symbols in the numerator and everything right of the division symbol in the numerator. This indicates that there is an implied bracket, or parentheses, in the equation. Which method really is correct? Having worked in the engineering field where my calculations had to have the correct answer to make what we were designing to work, I used the juxtaposition method and always got the correct answer. When using a calculator, I inserted the implied parentheses in the calculation. It is my opinion that in order of operations, multiplication should take presidence over division. I challenge a math teacher to prove which is the correct method to use on an ambiguous written equation.
I was on the math team in school and was taught that either side of the / was implied parenthesis and the ÷ was not used at all. That was in the 90s so my memory might be wrong now, but i think you are right.
I don't remember being taught that and would argue against it, because then you get to pretty iffy territory. That seems like a special, jargon-like usage convention: If everything is always of that form in some field, it makes sense to omit superfluous parentheses for readability, but it is problematic for general usage. Of course, a lot of it is *visual:* Are you using/imagining a large slash extending a character height above and below the rest of the expression with room around it or a small one packed tightly in one of multiple separated addition terms? To me, however, it is obvious that you can't just break an expression at a point, where there is no operator to break at (that matches the implied operation) for the sake of binding a part of that grouping to some another operator (with the same or lower preference). If you do that, you are just willy-nilly chopping the term in half at a completely unmarked place. The purpose of notation isn't to mislead. I've used examples of 3x/xy and xy/3x elsewhere.
The minute you start "implying" something that isn't there in math you're wrong. We could certainly have decided that multiplication has some precedence over division, but that would require us to change how we write our equations. That's the point of all of this. The rules have to be fixed in order to do math at all. In theory we could make order of operations anything we wanted to. What we choose dictates how we construct the equations though. And some ways make creating equations much more complicated than others. The simple fact is there is nothing ambiguous about this equation. You just can't invent things that aren't from the established rules for how math is to be evaluated and then complain when you get a different answer than the one the writer of the equation wrote it to produce.
@@Cdaragorn huh? we were taught that multply goes before division and what ever number is outside of the brackets it getting multplied by the inside number then division comes. you cant just change math and thing youre going to get the right result.
@@keenanvanaalst9865 My entire comment was explaining why you can't just change math so you're right. The problem is multiply doesn't go before divide and it hasn't for more than 100 years. Multiply and divide are equal in the order of operations. You do them together. I'm sorry if you were taught wrong. Seems like a lot of people were given that misconception.
I got 6, whenever I see a ➗ I automatically turn that in a fraction /. So I simplify the top and the bottom independently before finishing the division.
and that's why you never use ➗ after the 4 grade... except when you want to make a semi trap video. Math is supposed to be clear, not interpreted. If you would have seen the correct fraction you whould have given the correct answer. It's not a matter of age, they just made this purposefully confusing. You won't find an engineer use this kind of writing.
@@grigturcescu6190 what this demonstrates is that people can't follow a few simple rules and that they need to be hand held all the way to the correct answer... When you actually understand and apply the Order of Operations and the various properties and axioms of math you get the ONLY correct answer 9 It doesn't help that on average 70% of adults incorrectly believe that 5+2×10=70.... You have people under educated who fail to understand the Order of Operations AND yoy have people who are over educated and try to make more out of a basic 4th grade arithmetic expression than it is...
Not arguing but talking from my schooling 50 years ago. I got 6. I subtracted 5 from 7 resulting in 60 / 5 * 2. I then multiplied five by two. Then divided 60 by 10 = 6. I just typed into excel =60/5(7-5). Excel insisted in inserting the multiplication symbol in between 5 & (7-2). It then produced the answer 24. I think excel did the problem in the following order, 7 - 5 = 2, 60 / 5 = 12, 12 * 2 = 24. Clearly I should now treat multiplication an division problems as equal actions and operate from the left. Fun presentation.
That's what most people remember but what you forget is the TERM outside the parentheses is multiplied by the value of the parentheses not just the factor next to it. TERMS are seperated by addition and subtraction not multiplication or division. 60÷5 is one TERM attached to and multiplied with the value of the parentheses 2... The correct answer is 24 60÷5(7-5)= 24 60÷(5(7-5))=6 60+5(7-5)= 60+5×2= 60+10=70 60-5(7-5)= 60-5×2= 60-10= 50
@@JJJJ-hp9oz there is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to group and give priority to operations INSIDE the parentheses and nothing more. 5(2) is not a parenthetical priority and is exactly the same as 5×2 You then demonstrate the Distributive Property incorrectly. The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses. 60÷5(7-5)= 60÷5*7-60÷5*5 parentheses eliminated 12*7-12*5= 84-60= 24 60÷(5(7-5))= 60÷(5*7-5*5) inner parentheses removed 60÷(35-25)= 60÷10= 6 60÷5(7-5) does NOT equal 60÷(35-25)
@@JJJJ-hp9oz LMAO... The Order of Operations were formally established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's... New Math is an excuse for people who fail to understand the basic rules of math... The correct answer is and always has been 24 not 6 You FAIL to understand what constitutes a TERM and you FAIL to understand that when written in an inline format only the number to the right of the obelus is in the denominator unless WITHIN a grouping symbol...
I'm not even that old and I was thought by all my math teachers that you would solve the multiplication next to parenthesis first regardless of from left to right, so I came up with 6. Blame my math teachers.
So I went directly to comments -> and found Alpharex Rex! You are my kinda guy 🙋 Saved me from even Watching the video. Clearly we made it this far in life, paying bills, so there must be Alternative Math that also works 😉
I'm 27 and I use this equation all of the time. Whenever I get stopped for drunk driving, the officer always asks, "Miss, how many beers have you had?" I always say, "60÷5(7-5)" While he stands there tryna figure it out, I slip out of the cuffs and steal his patrol car. Voila!
It's an equation for time travel. You'll be able to go back in time and get those software programmers to fix this problem first. Then we don't have to waste our time on problems like these.
The correct answer is 6. Not only do the brackets represent multiplication, they also come first in the order of operations. Thus, not only is it 5x2, but the 2 is in brackets, and so must be solved first.
I inserted 60÷5(7-5)= into an Excel spreadsheet. Excel accepted it as a text but did not recognize it as a formula that needed to be solved. I had to change the entry to =60/5*(7-5) in order for Excel to accept it as a formula and solved it with an answer of 24. However, if I changed the entry to =60/(5*(7-5)), Excel accepted it as a formula and solved it with an answer of 6.
thats how math works... in PEDMAS, when a multiplication and division are on the same line and you must choose between either one, you will go from left to right
Well, yes. All formulas in excel require you to start with = so it knows it's a formula to be solved. Also adding another set of parentheses will change the order you solve it. The point of PEDMAS/BODMAS is to standardise the way we solve formulas, and adding more parentheses to an equation will change the order in which you solve it
It IS written as a fraction. The rub comes in grouping what's in the denominator. The form 5(2) is really 5x2 which is not priority bound by PEMDAS. You are using parentheses to disambiguate to your preference. If you don't do that and evaluate by PEMDAS you get 24.
@@garymartin9777 That's not necessarily true. Remember in PEMDAS, multiplication comes _before_ division in order of precedence (M before D). So it's 60 over 5(7-5) = 60/10 = 6. Notice I did parentheses, THEN multiplication, THEN division. That's how it should always work. What the video got wrong is that they interpreted M and D to be on the same precedence level. They are not. M is above D, and that's why it's 6.
You are not correct, it would be (60/5) would be the fraction with a *2 after, just as it is written. Don't try to change the equation please. Multiplication and division have the SAME priority, so whichever comes first is first.
I'm not going to rely on a calculator's "judgement" on what is ambiguous. The calculator is merely following rules programmed by a human that could have interpreted an ambiguous statement one way or another.
There's nothing ambiguous there. It's plain and simple, unless you were born in 1910 or something. Rules change, so people need to adapt and forget the old ones.
@@BypassOne I agree the problem is not ambiguous but I'm merely pointing out that a calculator result is not proof of the answer to the problem but merely the result of human programming, which is not infallible.
@@percyfaith11 Human programming that is based on mathematical rules. Calculators were invented to easen and speed up calculations, exactly because people tend to forget them. So, believe me, the expression is not ambiguous just because YOU forgot the rules.
I think you have to work it as 5(2) then divide that into 60 because it's connected to the parenthesis. Work the entire parenthesis first, not just what's in the inside.. just like an exponent on the parenthesis. 6 is my final answer
Fun fact, all of the Marx brothers loved go to Alabama to shoot elephants for their tusks! Why Alabama, you ask? Cuz everyone know that in Alabama, the Tuscaloosa.
I think the ambiguity stems from the omission of the operator symbol when using parentheses, making the expression "5(7-5)" appear as though it should be viewed as a unit. If the expression was "60 / 5 x (7-5)" instead, any ambiguity would be lost (imo).
Yep, that's pretty much it. The academic interpretation of 60÷5(2) is 60÷(5(2)). It's just the notation used for years and is in academic papers. Multiplication by juxtaposition was given higher priority so less brackets were needed. Feynman, for example, used this, and other, common shorthand notations. That's all they are, shorthand. Writing a÷b(c) is now bad writing because of how popular programming has become. Programming views 60÷5(2) as 60÷5×(2) which means there is no juxtaposition so no ambiguity like you said. Moral of the story is, the question in the video is flawed which you have noticed. Many, many other people in the comments are oblivious to this and just argue the answer is only 6 or only 24 but are missing the real problem completely.
@@GanonTEK I'm not sure your correct about the programming thing. I was taking the 4 maths (Alg, pre-Calc, Geom, Alg II) back in the 80's before programming was a big thing, and the books we used in my poor little country school were not the newest and greatest. And I knew the answer was 24 based on my learnings back then (before your programming theory) Not saying there weren't programming and computers in the 80's, I had a C64 and programmed in Basic. But our books were probably from the 70's and it wasn't a "popular programming" thing that taught me to know the answer was 24.
A number or variable directly in front (or behind) of a quantity is considered part of the expression. The parenthesis are not just symbolic of a multiplication. They are also symbolic of the Distribution Property. The 5 in front of the quantity (7-5) should be read as 5 distributed (multiplied) to 7 and to -5 or the sum of the latter two. So 5(7-5) is equivalent to (35-25), one unit and can't be treated as separate. As another said earlier... 60÷5x would not be solved as 60÷5 and then the dividend multiplied by x. Also, substituting an x for any member in the equation 60÷5(7-5)=6 will give you the correct number back...i.e. X÷5(7-5)=6 will lead to X=60. 60÷X(7-5)=6 will lead to X=5. Etc. This will not be the case if you say the answer is 24. 60÷5(7-5) does not equal 24.
The answer is 24. 60/5*2 = 12*2 = 24. Multiplication and division have the same priority, so when no parentheses are present, perform the operations as they are encountered from left to right.
3:00 Exactly. As a programmer, I would consider that expression poorly written. The fact that a compiler can evaluate it unambiguously doesn't change that. It's always better to use parentheses to make the meaning clear.
@@NetAndyCz nope. None of the languages I have ever programmed in will accept it as is, they all require an explicit multiply operaror. And they would all produce the same answer: 24. Expression evaluators are different of course.
It is more how math programs and calculators parse it, anyway no one (or almost no one) argues about the order of operation for explicit multiplication.
So a majority of people believe 5+2×10=70 is this because the expression is ambiguous or because they don't understand the rules?? Did we have to use parentheses to make it clear 5+(2×10) or should these people have to learn the rules?? If you follow the Order of Operations as they are intended to be followed there is no ambiguity in 60÷5(7-5) there however are a lot of people who do not understand the rules and require crutches in order to evaluate the expression correctly...
I would have thought 6, but this is why I avoid using multiplication and division symbols, and instead use parenthesis and fractions. As long as the faction lines (especially if it's fractions within fractions) are correctly sized, there's no ambiguity.
60 -----(7-5)=60÷5(7-5) 5 60 ---------- = 60÷(5(7-5)) 5(7-5) A vinculum (horizontal fraction bar) is a grouping symbol and groups operations within the denominator and when written in a inline infix format extra brackets are required to maintain the grouping of operations within the denominator... Remove the grouping power of the vinculum, replace it with the grouping power of another grouping symbol i.e parentheses...
@RS-fg5mf The first equation I'd write 60 ---- (7-5) = 60÷5*(7-5) 5 While 60 ------- = 60÷5(7-5) 5(7-5) Because I learned that a missing multiplication sign indicates a closer connection and so is seen as if it being in parentheses with the following term. So 60÷5x is not equal to 60÷5*x
It is 6 using the Distributive Law of Mathematics. There IS NO modern interpretation outside of America. A student at Oxford, Cambridge, Sydney, Fudon etc DO NOT depart from the law! There is no ambiguity either!
@@garyquinlan4075 To be fair, Order of Operations is useful for... oh, about three weeks during pre-algebra - until you start learning the properties of addition and multiplication. The problem is, some people rely on it and apply it far beyond its pedagogical purpose. No mathematician, scientist (to include economists), or engineer should ever reference it.
The fact that this continues to come up is evidence that there are two very different interpretations that have been taught to various people depending on when and where they were taught; and because this is the Internet, people are more than happy to boldly proclaim the other side to be wrong. FWIW, I was taught in school that the 5(7-5) is resolved completely before the division. The answer seems really to be more explicit, brackets are cheap.
Think about this. This situation and debate about the correct way to solve an equation has come up before, I am sure. Engineers and physicists need their proofs to be interrupted accurately for peer review. There is no room for misunderstanding. I googled pictures of famous equations and I found no ÷ signs. They don't use them. Maybe we should abandon them entirely.
"I was taught in school that the 5(7-5) is resolved completely before the division. " You were taught wrong. That isn't a valid interpretation. That is just you having been given false information. There is nothing in math saying to include the 5 as part of the () Sorry
@MrGreensweightHist actually there is. Saying 6÷5 is the same thing as 6/5. The division symbol replaces a fraction, quite literally showing this to a trig professor and an a math calc professor, both have said 5(7-5) is the denominator. There is a reason they don't use the division symbol anymore and just use fractions.
@@Dogsparkster I am sorry you have bad teachers. "There is a reason they don't use the division symbol anymore and just use fractions." The division symbol IS a fraction bar 3÷4 is 3/4 is ¾ The reason ÷ isn't used anymore is simply because / is one line while ÷ is a line and two dots. / is faster to write. that's the ONLY reason it changed. X however, became * because X is too easy to confuse with the variable x. using X instead of * can cause confusion. Using ÷ instead of / alters nothing.
@@MrGreensweightHist yes that is the type of cocksure reply I expect from UA-cam comments, thank you. Juxtaposition having higher precedence than explicit multiplication or division is a long accepted notational convention that doesn’t appear to be universally accepted because it contradicts the sacred PEMDAS rule children are taught in elementary school, hence these internet controversies that continue to spring up. Since we aren’t in 1890 and trying to minimize characters when printing equations in books, we can all just be more clear for everybody’s sake and use more brackets.
All through school and into industry I have never seen anyone write an equation using the division symbol when writing by hand. It’s always a horizontal or forward slash line, and to make it clear it will be a horizontal line with the numerator over the denominator. The devision symbol used daily is somewhat recent and comes from spreadsheets and coding where one is forced to write on a single line. So to get over the confusion lots of parentheses are needed on that single line or these debates occur because the equation is unclear to people thinking in terms of math equations written on paper.
"I have never seen anyone write an equation using the division symbol when writing by hand. It’s always a horizontal or forward slash line," They all mean the same thing. Honestly does not matter which one you use. 60÷5(7-5) is 60/5(7-5) is 60 ----- (7-5) 5 All the same problem.
These types of problems are a waste of time. There is a reason nobody doing actual math uses the ÷ symbol. Hell, it's not even on your computer keyboard. A proper division bar ---------- would eliminate any ambiguity of whether the multiplication should take place before or after division.
are your examples complex math symbols or what? I see a black and white smiley face, followed by 20 rectangles with ? in it, followed by E, T and C. i.imgur.com/vWjbBlW.png
@ groszak1: Looks like your browser, which appears to be designed for the Polish language, isn't able to render emojis. The characters that Brandon typed was a series of smiley faces and other emojis. I assume he was using a mobile device. (And the "ETC." is for "etc." [should be lower case], an abbreviation for the Latin phrase "et cetera", which means "and so forth" .) Anyway, markgriz is right. Nobody in real life ever uses the ÷ symbol.
The first one is a regular smiley face, not an emoji. I intentionally deleted the annoying colored emoji fonts from my Android tablet. And please don't criticize the division symbol.
Yeah the whole video is just getting people to argue about well defined methods of solving fractions,,,,the cardinal rule is, multiply and divide before you add and subtract but you must solve the maths in the brackets first(---)
*WRONG! The 5(2) must be solved first to remove the ( ) from the equation. The answer is 6.* What you are doing is INCORRECTLY changing the ( ) to a X between the integers...and THAT is your flaw. The two are NOT the same, even though we treat them the same way when we get to them..the difference is WHEN we getto them. You are "getting to them" too late. *The parenthesis must be addressed BEFORE multiplication/divisions are addressed left to right.* #PEMDASnotDEPMAS
Ah Math, in the 1970s, I was taught that the parentheses rule also includes any 'assumed' multiplication, thus the 5 gets multiplied by the (2) first before any other expressed multiplication and division. Hard to do it differently.
There were some groups that tried to push for implicit multiplication to have higher precedence in the 70's and that's part of the problem. Unfortunately, giving implicit multiplication a higher priority breaks the Order of Operations. The Commutative Property. The Distributive Property and the Multiplicative inverse of division... Solve these two expressions... 60A(7-5)=24....A=? 60A(7-5)=6....A=? I will explain later when you have answered...
@thegrandfinale2 5a is a coefficient/variable bond. Real numbers do not have coefficients. 5(2) The 5 is not a coefficient of 2 and is nothing more than implicit multiplication. 5a does not equal 5(2) 5a would equal (5*2) when you replace the variable with a real number you break the coefficient/variable bond and parentheses are required to maintain that bond. Parentheses only give priority to (OPERATIONS WITHIN) the grouping symbol not outside.... 60÷5(7-2)=24 60÷(5 (7-2))=6
@thegrandfinale2 the historical usage was due primarily to text book publishers pre 1900's that were trying to save page space by using the obelus like a vinculum. The printing technology at the time was cumbersome and using a vinculum took up more vertical space than was acceptable to the publishers so they pushed the misuse of the obelus even though it was in direct contradiction to the Order of Operations... As for the two equations. 60A(7-5)=24..... A=0.2 or 1/5 60*(1/5)(7-5)=24 What is the divisional reciprocal of 60*(1/5)....WELL that would be 60÷5....So. 60÷5(7-5)=24 60A(7-5)=6....A=.05 or 1/20 60*(1/20)(7-5)=6 What is the divisional reciprocal of 60*(1/20) WELL that would be 60÷20....So 60÷20(7-5)=6 If 60÷5(7-2)=6 then 60a(7-2)=6 a should be equal to 1/5 but it isn't. I hope that helps.
@thegrandfinale2 you are still missing the point. A variable because it can be any number or value is in and of itself a symbol of agragation... x = 5+5 that does not mean you can write 10÷5+5 you have to write 10÷(5+5) When you have x= 2*3 and you have 6÷2x You do not write 6÷2*3 or 6÷2(3) you would write 6÷(2*3) There really is no reason other than laziness to use implicit multiplication in a basic arithmetic expression.... One reason implicit multiplication exists is due to the possibility of confusion between the multiplication sign x and the variable x in an algebraic expression.... 2x(1+2) does this mean 2 times (1+2) or does it mean 2x times (1+2)... In any case I have at least 6 different math books from 1907 to the present and they all clearly state that you evaluate what's WITHIN the grouping symbol and that ALL multiplication and division can be evaluated equally from left to right.... Show me a math book that states something to the contrary... Also.... 60a(7-5)=6 Why would a not equal 1/5 ?? 60÷5 = 60*(1/5) 60*(1/5)(7-5)=?? 60*(1/5)÷(1 (7-5))=?? 60÷5÷(1 (7-5))=?? All multiplication can be changed to division by the reciprocal. All division can be changed to multiplication by the reciprocal. Do you believe that implicit multiplication should override the Order of Operations and invalidate the Commutative Property. The Distributive Property and the Multiplicative inverse of division??
@thegrandfinale2 I agree with you on two points... The notation confuses people, but is this the fault of the notation or the lack of knowledge many people have about math?? Primarily, they do not follow the Order of Operations and choose to see 5(2) as a grouping. BUT math books for more than a Century have clearly stated that you evaluate what's INSIDE the grouping symbol and ALL multiplication and division are to be evaluated from left to right. I also agree that there is not a valid reason to use implicit multiplication in an arithmetic expression except for laziness . Use an explicit symbol like it's meant to be used and use parentheses the way they were meant to be used.... I have friends who believe that variables should not get to break the rules and believe that 6a÷2a=3a^2 and not 3÷a AND I argue with them as well. There is plenty of text book evidence that demonstrates 6a÷2a= 6a÷(2a) and not (6a÷2)a .... You break the coefficient/variable bond by seperation. 6a÷2a ≠ 6a÷2*a or 6a÷2(a) as there would be no need to put parentheses around the a unless you intended to show seperation of the 2a . Do you agree that 60a(7-5) equals 60÷(1/a)(7-5) ?? If not how do you resolve the conflict between the coefficient/variable bond 60a and the implicit multiplication of a(7-5) Is it (60a)(7-5) or 60(a (7-5))??
My answer was 6. That is how I understood order of operations. I was probably taught that exception to the rule or decided it through my own intuition. Look at the large amount of space between "60," the division symbol, and the rest of the expression (I stopped the video at the four minute mark, in case you mentioned this). Then notice how the "5" is hugging the parentheses. I would be inclined to calculate physically closer, i.e. closer on the page, operations before more spaced out ones. I would venture a guess that when writing math textbooks and the like, it is standard practice to punctuate in ways people will find intuitive. The division symbol itself, resembling a fraction, also implies the order that results in 6.
WRONG... Parenthetical implicit multiplication does not have priority over division. In fact the TERM not just the factor outside the parentheses is attached to the parentheses. TERMS are seperated by addition and subtraction not multiplication or division. 60÷5 is one TERM attached to and multiplied by the value of the parentheses 2. The correct answer is 24 A(B+C)= AB+AC where A is equal to the TERM not just the factor outside the parentheses. A is the monomial factor outside the parentheses to be multiplied by the value of the binomial factors inside the parentheses or to be Distributed across the two TERMS inside the parentheses that makeup the binomial.... A=60÷5 B=7 C= -5 60÷5(7-5)= 60÷5*7-60÷5*5= 12*7-12*5= 84-60= 24
I don't know what you're writing all that for. I did watch enough of the video to get to that part. Maybe you missed the part about the "exception to the rule."
@@MoonJung82 the video doesn't explain that it isn't an exception to the rule. Prior to 1917 some textbook printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space for the printing methods at that time and was more costly to print. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math so the ERROR was corrected post 1917.... Get it?? ERROR Would you have evaluated 60/5(7-5) any differently... using a solidus rather than an obelus??
@@RS-fg5mf So you're saying the video is wrong? 2:05 "Historically, this division symbol had a special meaning when you wrote it in text..." Go yell at him if you're sure of that. I believe I was taught that special case--that it was not ideal but allowable. By the time we reached high school, most of us had been weaned off of using ÷. You may also find it interesting that the video title reads: "60÷5(7-5) = ?" and in the video itself, it's a little different: "60 ÷ 5(7-5) = ?" (spacing) Writer's intent matters and I wonder if the guy making this video added those spaces with that in mind. "Would you have evaluated 60/5(7-5) any differently... using a solidus rather than an obelus??" Yeah, I would have said you forgot parentheses, either (60/5)(7-5) or 60/(5(7-5)) And obviously, those added parentheses would also clear up the confusion of the original expression. This video does not primarily present a math problem but a communication problem. I responded in kind and your attempts to correct me are missing the point.
I sometimes refer to myself as a recovering math head (I aced the math ACT test and was second in my high school in the MAA competition at the age of 16) but I am not recovering all that well. When I see this mistake being shown as the correct answer I cringe. Evaluate the numerator, evaluate the denominator, then divide.
_ One cannot use the calculator as the basis for the correct way to proceed. _ It depends on the rule one establishes for evaluating mathematical expressions. _ Engineering, physics and other fields of mathematics do not generally use PEDMAS as the procedure for evaluation of mathematical expressions. They establish their own rules of precedence, usually stating that multiplication ALWAYS precedes division. _ If all these professions accept PEMDAS as their new rule then it will become common, but I do not believe that it is so currently.
This is not a real math problem, this is a problem about how we understand the order of operations and using parentheses when uncertain: So: 60/5*(7-5) can be: 60/(5*(7-5))=6 or (60/5)*(7-5)=24
I got six because the 2 was still in parentheses. The confusion lies in when a number is next to variable in parentheses it implies multiplication. But because of how it's written, one could make that mistake of multiplying 5*2 first because according to order of operations, parentheses come first. So therefore I wont fault people who come up with 6 or 24.
@@anthony420181 multiplication was before division though since it's the M also if its from left to right then why did the parenthesis get done first on the right (even though i know why)
@@anthony420181 The ambiguity is that there is no agreed upon convention on whether multiplication by juxtaposition implies grouping also or not. I.e. does 5(2) = (5×2) or 5×2? Both are widely taught and used. 60/(5×2) gives 6. 60/5×2 gives 24. It's simply a poorly written expression that no one would ever write now. It doesn't follow modern international standards like ISO-80000-1 which mentions this case of division on one line with multiplication or division directly after and that brackets are needed to remove the ambiguity.
@@anthony420181 Many scientific calculators give 6 also. It's very common to find one that gives multiplication by juxtaposition higher priority. Especially with some companies. I've a brand new Casio myself from last year that does it for example that's marketed for schools and colleges. Microsoft Math gives both answers on screen at the same time (one as the answer and one in the box you typed in after it converts it to a fraction notation but they don't match). I think it is unintentional but highlights the ambiguity. I've seen photos of other calculators with the same equation in both and two different answers for the output. If you search "The PEMDAS Paradox" you can get an article and academic paper by a PhD student on this very ambiguity which has the photo I've seen included. If you search "Ambiguous PEMDAS Harvard" you get an article and a separate short document on the ambiguity also with references and documents. Wolfram Alpha's article on the Solidus is another place that mentions this ambiguity. If you search PEJMDAS from The How and Why of Mathematics on UA-cam you get videos, with references, which talk about the interpretation that gives multiplication by juxtaposition higher priority. There is alot of evidence out there backing up my claims. I wouldn't say it otherwise. The question in the video is written poorly. For any reasonablely written question, most likely all calculators, online and offline, agree on the same answer as well as people who learned PEMDAS or PEJMDAS.
It wasn't mentioned as over in America they seem to teach that multiplication by juxtaposition does not imply grouping so to them this is generally the taught method. You also need a bracket there with what you wrote: 60/(35-25) and 60/35-25 are not the same answer when written on one line. However, it seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping though, or it may not, so the notation is ambiguous making both answers valid. It depends on context (academic or programming). Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it. The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation. Wolfram Alpha's Solidus article mentions this ambiguity also. Microsoft Math gives both answers. Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus. Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274) "If an arithmetic or algebraic term contains both ÷ and ×, there is at present no agreement as to which sign shall be used first..." It then goes on to say that brackets should be used to "avoid ambiguity in such cases" "The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue) "Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous and says, "shame on that person for writing an ambiguous expression". "Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division". "Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause. Other credible sources are: - The PEMDAS Paradox (a paper by a PhD student on this ambiguity) - The Failure of PEMDAS (the writer has a PhD in maths) - Harvard Math Ambiguity (Cajori's book above is talked about here) - Berkeley Arithmetic Operations Ambiguity - PopularMechanics Viral Ambiguity (AMS's statement is here) - Slate Maths Ambiguity - Education Week Maths Ambiguity - The Math Doctors - Implicit Multiplication - YSU Viral Question (Highly decorated maths professor says it's ambiguous) - hmmdaily viral maths (Another maths professor says it's ambiguous) The volume of evidence highly suggests it's ambiguous.
That is because they dont actually solve the equation, they input it into a calculator from left to right without any thought to how calculators operate. this gives them an answer of 24 which they then seek ways to justify the answer given by the calculator solving 60/5*1(7-5)=x instead of solving 60/5(7-5)=x.
The problem I have with this kind of problem is that it is not really math. It’s grammar. Just write the darn expression in unambiguous way so we can do actual math. We have more interesting concepts to learn in geometry, trig, calculus, etc.
@@UniversalS757 Don’t worry. Math grammar has correct answers too, but math grammar is different than math concepts. I have been an engineer working in the real world for 30 years. Not once have I debated stuff like this on the job. I will just put parentheses in there and keep it moving. This is a discussion that may be fun, but it needs to stay on academia.
you're close. it's about order of operations. first parentheses second multiplication and division are at the same operation so left to right. so... 60/5=12 * 2 = 24
Ronald Dunn 11 hours ago Division is just multiplication of fractions. Rewrite the expression using this idea and there is no ambiguity. 60 x (1/5) x 2 = 24 (advanced students should understand this)
All the math we have is not meant for real life. It exercises and trains your brain to make it sharp so that your brain works instantaneously and perfectly to find a solution to your real life problems and also to help in your decision making..!!!!
The correct answer is 6. If the equation is solved BASED ON THE WAY IT IS WRITTEN, which defines the way the mathematical operations should be executed the answer is 6. Looking at the written problem there are no brackets ( ) surrounding the 60/5 therefore it is not done first. The brackets determine the order of operations and if they are not there on the left side then that division operation is not done first. Respect the rules of: order of operations.
From the first calculation the 5 was followed by 2 in brackets, and so that has to be resolved first. This gives 10, which divided into 60 gives you 6, which is the correct answer.
WRONG. 5(2) is not a bracketed priority and is exactly the same as 5×2 Brackets only group and give priority to operations INSIDE the symbol not outside the symbol... The correct answer is 24
@@paulbiddlecombe3279 don't blame the expression for your failure to understand it correctly.... 60 boxes are delivered equally to 5 locations. Each box contains 7 winter coats and 5 of those coats are childrens coats. How many adult coats did each location receive?? 60÷5(7-5)= 24 adult coats. 60 adult coats are delivered equally to 5 locations. Each location had 7 people waiting for coats. 5 of these people are children... How many coats could each adult receive?? 60÷(5(7-5))= 6 coats per adult. 60÷5(7-5) EQUALS 60÷5×2
@@paulbiddlecombe3279 the only people confused are the people that do not understand basic arithmetic. The symbols used to represent each operation changes nothing. I really do not understand why so many people don’t understand the answer is 24
Paul, You are correct. They are wrong. I am a chemist and in chemistry we would get 6. Here's why: The expression written as 5(7-5) is to be treated as a single expression because it shows a relationship between the 5 and the 7 and the 5 and the 5. This expression IS TO BE TREATED AS A SINGLE UNIT because the parenthesis is used to create the relationship. So 5(7-5) = (5x7)-(5x5) = (35)-(25)=10. The rewrite of the expression would be 60/5(7-5) = 60/10 = 6 The 5(7-5) is to be treated as a single expression. It is NOT (60/5) x 2. And it is NOT 60÷35-25 either. It's 60/5(7-5)
I love the emphasis that 24 is the answer according to the modern interpretation of rules of operation. It really drives home the point that order of operation is a convention rather than a fundamental law of mathematics. It's important to keep to the convention to avoid difficulties communicating with others. But in theory, all mathematics should be possible if addition/subtraction came before multiplication/division. We would just write things differently (and I suspect would be forced to use much more parentheses.)
This is why the Order of Operations and the various properties and axioms of math were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses when dealing with inline infix notation.... As for Addition having priority over Multiplication, if that were the rule it would work but it wouldn't make much sense since Multiplication is shorthand notation for repeated Addition... As it stands the Order of Operations and the various properties (LAWS) and axioms of math work logically and consistently across the board...
@@RS-fg5mf I was thinking more along the lines of we have 2 sets of eggs. Example: One is 6x2, the other is 12x8 Writing it out as 6x2+12x8 without multiplication as a higher priority than addition, we would need (6x2)+(12x8) In order for addition to make sense, we would need a setup more like: Example 2: We have an L shaped array of eggs container that is 3 rows by 2 columns. We then add 2 more rows and 5 columns. How many eggs can the container hold? Writing it out as described would be 3+2x2+5. In this case though, you need to solve addition first. So parentheses are required for our current order of operations (3+2)x(2+5), but would not be required if addition was higher priority. In the real world though, example 1 is far more common than example 2, which is why we developed the multiplication priority. That's my best guess anyway. Your guess about multiplication being shorthand for repeated multiplication may also be correct.
@@plentyofpaper I understand what you're saying. What I'm saying is it wouldn't make logical sense to do it that way since Multiplication is shorthand for repeated addition... (5×3)+(4×2) would still equal 3+3+3+3+3+2+2+2+2
The mistake is that once the operations INSIDE the parenthesis are done, they should be replaced by the correct operand. 2(3) is the lazy way to say 2x3.
The thing that I don't understand that they're doing in the "modern" interpretation of this problem is that they are just dropping the parentheses after calculating (7-5), so 5(2) becomes just a regular multiplication, not a parentheses calculation now?
@@towmlvb3423 5(7-5) is like a single sentence therefore the answer is 6. If it was written like 60/5*(7-5) then that will be different. Like seriously is it that hard to understand that the way u write it will determine the answer?
Solve Parenthesis means to remove the parenthesis from the equation first. Before left to right the parenthesis marks need to have been solved (removed from the equation). You can't move on to the next step in either PEMDAS or BODMAS until the parenthesis have been solved and thus removed. 60/5(2) still contains parenthesis (brackets) and it would be outside of BOTH methods to continue while the presence of a parenthesis/brackets still exist. Regardless of the fact that it is a multiplication it is still a bracketed expression. 60/5(7-2) and 60/5*(7-2) are not the same expression. This is also why it is not ambiguous.
I was taught calculations are done in a specific order. Given Inside parentheses is always done first. 1. Multiplication 2. Division 3. Addition 4. Subtraction. So my answer is 6, & that's what my teacher, who was never wrong, would accept.
NO... his teacher was right. And the way he described the order of operations is correct. The YT content creator also stated the order correctly - but then... for totally INEXPLICABLE reasons, what he entered in Google was a DIFFERENT EQUATION. Sheesh!
If you view 5(7-5) as 5 being factored out of (35-25) you end up with the second binary tree. This is what we were still taught in the 60's and 70s, the parenthetical surrounding 5(7-5) is inferred, and this is NOT the same as 5*(7-5) where the parenthetical is not inferred.
I feel like I was lied to my whole life. In school, the way I was taught, it would be 6. The parentheses isn't solved till the exponent is factored in, then a multiplication that is before it. That was the way they taught it in the 80's at my school.
If I remember correctly when I was in junior high at the end of the 60s the part of the equation in parentheses was solved first then the rest of the equation.
I was taught this in the 60's ; it was 24. Correct order of operations: ua-cam.com/users/shortsMaPZGyudFzo 60/5(7-5); expression A=60/5=12; first operation and step. B=(7-5)=2; second step, group, do together. A(B)=12(2)=24; final step. Or use Distributive multiplication A(7)-A(5)=84-60=24; parenthesis content not done first or together. Video : ua-cam.com/video/y9h1oqv21Vs/v-deo.html, great message at end. ___ BIDMAS Brackets refers to any part of the equation that is in brackets. These should always be complete first. Indices simply means to the power of. For example, 3² or 5³. Division and Multiplication: Starting from the left, work these out in the order that they appear in the equation. If multiplication appears first you should complete this before division. Addition and Subtraction: Also start from the left and work these out in the order that they appear in the equation. If subtraction appears before addition, you should complete this first.
The way they taught us in school in the 80's was Bedmas; Brackets, Exponents, Division and Multiplication, Addition and Subtraction. And left to right. It seems there were a lot of schools teaching different ways of interpreting the order.
Good to know I haven't lost any of my math skills after all these years. I was wrong then and I'm wrong now.
This video is wrong. The anwer is 6.
The correct answer is 24.
60÷5(7-5)=12(7-5)=12(2)=24. You are still correct, the other person is wrong.
You weren't wrong, he is.
@@CarlMCole you are wrong, including about your false claim of genius.
I'll stick to the old way. So 24
When I entered the formula into Excel I got the message, “there is a problem with this formula.” In order to get Excel to make the calculation, the user must add parentheses to clarify the order of calculation. So yes, the formula as written is ambiguous and the person needs to clarify how the problem should be solved. In other words, don’t leave all of the decision making to a calculator.
there's a difference between how a computer calculator reads and how mathematics is done by hand by a person. When we are writing a division sign by hand, we use the traditional division sign ÷, but for a computer calculator we end up using the /..which can also be interpreted by a computer calculator as a division sign separating the numerator and the denominator..so for this question, it was written as 60 ÷ 5(7-4) which is the correct way to write this question for the answer to be 36. If you want the answer to be 4, it should be written by hand like this: 60 ÷ (5(7-4)). This has always been how mathematicians do math by hand. Using the / sign on computers, laptops, phones is what is causing the confusion. People are so used to seeing the / on digital devices, they think it's creating a fraction and start thinking numerator and denominator.
This problem is the reason why you should use fractions instead of the "divide"symbol: makes it completely unambiguous. Either the (7-5) is in the lower part of the fraction (denominator in English?), making the answer 6, or the (7-5) is completely outside of the fraction, making the answer 24.
you would need to write as 60:(5(7-5)) to make the whole thing the lower part of fraction. Otherwise, modern way to calculating will give u 24.
agreed entirely. written with a division symbol introduces ambiguity
There is no reason you can't switch the position of the 5 and the (7-5). Then you would have 60 in the numerator, and the 7-5 in the denominator. What next you would do is calculate 60 divided by 2, which is 30. Then you would multiply 30 times 5, to get 150.
Fractions are not interchangeable with division.
1 vinculum 2 is the fraction one-half
1 solidus 2 is the division 1 divided by 2
You cannot just replace one with the other willy nilly.
@@sinub801
60
-----------
5(7-5)
The kicker is the "divided by" operator in its presented form. (At school in Germany in the 70s we used : for division).
This sign however suggests a fraction with 60 in the numerator and everything that follows the division sign, hence 5(7-5), in the denominator. That would be 6 then.
In our school we were encouraged to express divisions in fractions because they are visually easier to resolve when they become large and contain many variables.
It seems like the sequential PEMDAS convention is the generally accepted one mainly because of computers.
That symbol has meant divide from before the 70's.
In true math, one does not use fractions...ever. One uses decimals. 1/2 in a math is indicated as 0.5 in order to be absolutely clear. It leads to less problems, and in programming it leads to a lot less problems.
Sorry but nothing about a division sign suggests a fraction at all. It sounds more like you were given a bad suggestion by someone trying to make things seem easier.
This is also not a restriction from computers. They could just as easily have been programmed to solve it following the second pattern but they weren't because that has been wrong for more than a century now (predating computers).
@@Nempo13 What do you mean, no fractions ever? How do you write x/y in decimals?
@@Cdaragorn That's not true. A division 5÷3 (or 5:3 as we did in my school) can be written as a fraction 5/3 or in words five over three.
@@R2BMusicCH Yes of course you can convert it to a fraction. The fact that you can convert it does not mean it's implied to be that at all.
Your original conversion was wrong. The fact that you did it wrong does not mean it was implied that it should be that way. It just means you don't understand how to convert between those representations.
A correct conversion would be 60/5*(7-5)
People aren't passionate about mathematics - they're passionate about arguing.
I'll dispute that. Passionately.
Buckhorn Cortez NO, I’m not!
I know I'm passionate about arguing.💗 But I'm always drawn to problems that require solving.
Passionate about correcting people that i know are incorrect*
I am passionate about truth
I’m 70, and I’m just thrilled to find out I still remember being taught this! And no, I’m not a math geek. I’m a little old lady who has stayed motivated to keep learning all my life!
So true Lynn. We may never need this particular equation in our everyday life, but it's nice to know the method anyway.
Hi Lynn, you and me, both.
I'm 66 and went to one of top 20 High Schools and the then top Accounting and business University in the nation. The answer is 6.
You are awesome!!!
Ditto. Great isn't it!
The correct answer is to use proper consistent notation.
You want the answer to be 24?
60 / 5 * (7-5)
60 / 5 * 2
12 * 2
24
You want the answer 6?
60 / (5 * (7-5))
60 / (5 * 2)
60 / 10
6
Exactly. Notation is key
24 not 12 but yes ^^
@@thairorecordsamv1040 lol yes. Ill fix that
So, everything you've written is correct but I'd like to add:
5*(7-5) vs 5(7-5)
There isn't a clear-cut difference but I'd lite to think that the latter represents factorization whilst the other is normal multiplication. If this was the case 6 would be the correct answer. Considering how unclear the notation is you wouldn't know the difference but this would simplify your second calculation.
@@peckapuder The multiplication sign separates terms in the expression. The coefficient is part of the term. Order of operations applies to each separate term in the expression. What people are calling "implied multiplication" is simply using the number as a coefficient of the parenthetical expression as a term within the complete expression.
The author has completely misunderstood the issue. It's got nothing to do with any historical interpretation of ÷ as he claims at 2:09 and everything to do with the priority of implied multiplication, which he fails to even mention. In formulae, implied multiplication takes priority over division. For example, on the Casio website, it states "A radian is 1/2πr of the circumference of a circle." This is the standard definition and it does NOT, repeat NOT mean (1/2) * π * r. No, it means 1 / (2 * π * r). The implied multiplication is done BEFORE the division. And remember Casio makes calculators, so they should understand this point.
The problem arises when folk blindly substitute numerical values into a formula and enter the result into a calculator. Calculators don't know the difference between implied and explicit multiplication, so the answer comes out wrong. So, returning to the original equation, 60÷5(7-5), the question I would ask before calculating the answer is "where did this come from?" If it is the result of blindly substituting values into a formula such as a/b(c-d), then the correct answer is probably 6 rather than 24.
Also, you are more likely to see division represented by / rather than ÷ in such formulae, so the formula 1/2πr really means:
1
_____
2 π r
When using excel, i always “over use parentheses “ to force excel to evaluate exactly what i want. I can’t afford surprises.
I so the same thing.
When using Excel, all you need to do is be sure of your maths. I know your "parentheses syndrome" because I suffer from it, too. But the truth is that I'm just not good enough at intuitively simplifying fractions, so I force the program to jump through all the hoops I need to be sure I got it right ;)
I see the need to do that too to ensure I get the correct answer. Although it might be a bit more complicated, it is worth to to avoid later headaches.
Plug this into Excel =60/5*(7-5) answer =24
ABSOLUTELY agree.
Math and CS teacher here. I think everyone misses the most important part here: spacing. A common practice in CS is to use spaces to display precedence, so for example you would write a*b + c*d. It helps readability and can be really usefull for less known operators precedences like and/or. And also not all languages follow the exact same precedence rules, especially for bitwise operators. So in the ambiguous expression shown here, the modern precedence rules would give 24 but the spacing indicates that it's actually 6. For the same reason, when I see 1 / 2x, I tend to understand it as 1 / (2x).
you can add all the spaces you want. The result is still 24..
60 / 5 (7-5)
= 60 * 1/5 * (7-5)
@@ChespiritoChavo322 There's no "the result is ...", it's all about conventions. Don't take conventions as rules written in marble, they change over time, they change from a country to another, from a book to another, from a calculator to another, etc. We don't know the context of this expression, maybe it's from an old book for example, so we cannot know for sure that modern precedence rules apply. But the spacing clearly shows the intention, and that's something we can rely on.
@@pierreardouin6441 i didn't follow any rule. Just used the formal definition of division.
@@ChespiritoChavo322 There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not.
That's what's causing the different answers. Division is used in either case.
Why in the hell would you put parentheses around 2x when it did not have parentheses to begin with???? Spacing or no spacing, the answer is 24.
÷ is terrible notation
True. I learned in 5th class to use fractions and no terrible Division Symbol ...
And no one genuine has used the obelus symbol in the same expression as parenthetical multiplication. It just isn't done, except during these social media "math experiments" that offer no insight into how mathematics works. If anything, these problems just confuse math students (particularly young or inexperienced ones) trying to figure out order-of-operations rules in a realistic setting.
inline division signs deserve a painful death.
Not to mention that the ISO for mathematical notation has (for quite some time) said that the obelus should never be used for division. Math classes and mathematical exercises are not supposed to use the symbol, so teaching it is only a way to confuse younger students.
So ÷ =! /... plus I hated it because it sometimes looks like a minus sign if your dots are too small or a plus sign if your dots are too big
I asked my father who was an engineer for 45 years and literally helped build parts for the space program and the nuclear programs, and he said the answer is 6. He explained that there are 2 elements. 60 and 5(7-5), these values represent something and are not just numbers. So, there are only 2 expressions. The equation should be 60 / (5(7-5)). This shows how setting an equation up correctly is most important. Given the fact that these guys sent several capsules to the moon and back, I'm going to go with his answer.
I agree with your conclusion. I was taught that Mathematicians, Engineers and Physicists preferred where possible to rewrite an equation without the division ÷ sign to avoid ambiguity.
If mathematical conventions are being changed to suit Calculators preferences, surely an honest person would consider that a very dangerous precedent.
I am willing to be corrected.
@@malcolmbrewis5582 One other thing I did to test my dad's conclusion was I googled pictures of famous mathematic problems and equations. Secretly I was hoping to prove the old man wrong, lol. But I could not.
I could not find a ÷ symbols on any of those blackboards. I took that to mean this issue of confusing how to write an equation had come up before, so to be clear and accurate, they did not use them. It makes since that they would not want to have their proofs interrupted in different ways. The same issue could easily surface in grammar as well by including or omitting punctuation like comma's.
In physics I haven't seen a single time the ÷ symbol being used, it's always a fraction, like V=∆s/∆t
Almost anyone who works in STEM or has higher education will give the answer of 6. Japanese calculators also give the answer of 6. Anyone who only did high school, American high school teachers, and newer American calculators, will give the answer of 24. Make of that what you will.
The reason I treat the answer as 6 is simple - if I see an equation like "x/2y = 1" then I don't think it should ever be interpreted to actually mean "xy/2 = 1", which is basically the same question. Nobody who ever said an equation like that would mean for it to be interpreted that way (unless they're deliberately trying to trick you), and having rules that make it function differently will only ever make things more convoluted than they need to be for no practical benefit.
If you wanted to write 60/5(2) to mean you're dividing by 5, then instead write it the sane way as 60(2)/5 instead.
If I owe you, my calculation is 6.
If you owe me my calculation is 24.
Yeah its 6. Isnt it?? I dont want to watch the whole thing. It should be 6.
@@thetruth3828
Must be a quadratic, because the claim is, if you are old school it's 6.
But if you are new school it's 24.
Don't know how it can be either or either, as there must be an intended definite outcome.
Good answer! Ha ha!
@@onlythetruth883 You clearly don't know what a quadratic is--but without knowing what you're doing or saying you've accidentally hit on the problem with many of the arguments in this thread: the given task is to evaluate a simple ARITHMETIC EXPRESSION using the generally-accepted rules for doing that.
Attempting to translate it to an ALGEBRAIC EXPRESSION and applying rules useful there, then translati9ng back to do the arithmetic, does not work.
If it involves just numbers and arithmetic operators and brackets, so one could evaluate it on a calculator or calculator app, then use the usual rules for evaluating arithmetic expressions and don't try to remember your high-school algebra and misuse that very foggy recollection to confuse yourself and others.
@@waynebrehaut7183
Of course I was being sarcastic when I said must be a quadratic. And you did get the point-->. There is no point until the rules are firmly established.
I remember being taught that parenthesis was calculated first, multiplication came next, then division, then addition & lastly subtraction. This gave me 6.
I graduated from high school in 1987 and that's the way I was taught. The answer would be 6.
Peace.
That's what they taught for most in our country's basic education too, literal PEMDAS in strict order (as the letters). Only in college that both math and computer science professors agree on the real correct method. I'm mildly infuriated that they always teach children outdated or plainly wrong things like this (the four taste regions also comes to mind, so wrong)
BODMAS????
@@davidevans8858 B-Braces/Brackets, O-Orders
@@mk_rexx well, pemdas (or pedmas as I know it) isnt wrong though. But the order of division or multiplication doesnt matter, and the order of addition and subtraction doesnt matter, as in both cases they are effectively the same operation. So everything in brackets first. Then all multiplication and division. Then all addition and subtraction.
according to my 1989 public USA education the answer is 6.
60÷5(7-5) =
60÷5(2) = And here is where the fight begins. Technically, according to the 1989 USA public education I received, the
PARENTHESES still exist that this point, and therefore has to be resolved first by Order of Operations
60÷10 =
6
Parentheses (inside first, then anything dealing with the Parentheses), Exponent, multiply/divide, add/subtract.
Even the distribution rule give the same answer
60÷5(7-5) =
60÷(35-25) =
60÷10 =
6
Your memory must be faulty or you had a bad teacher... I have at least 5 different math books from 1907 to the present and they all state the same thing... You evaluate what's (WITHIN) the grouping symbol not outside. And ALL multiplication and division can be evaluated equally from left to right..... When there are no (OPERATIONS INSIDE) the brackets/parentheses left to evaluate you can remove the parentheses and replace with an explicit multiplication sign or leave them to represent implicit multiplication and nothing more....
When you have a single value inside the parentheses that step is done... (7-5) is a parenthetical priority 5(2) is NOT a parenthetical priority and is exactly the same as 5*2
As for distribution, the whole point of distribution is to eliminate the need for parentheses by pulling what's inside to the outside not the other way around... Distribution requires that you multiply all the terms inside the parentheses with the TERM outside the parentheses. Terms are seoerated by addition and subtraction....60÷5 is one term to be multiplied by the two terms 7 and 5
60÷5(7-5)=
60÷5*7-60÷5*5=
12*7-12*5=
84-60=
24
60÷(5 (7-5))=
60÷(5*7-5*5)=
60÷(35-25)=
60÷10=
6
2+3+4+5 is 4 terms
10-9-8-7 is 4 terms
10÷2×6÷3 is 1 term
10÷2+5×3 is 2 terms
I hope that helps you understand better....
Richard S Again, that is how I was taught and I noted when and the type of education. That's why I explained it the way I did. It was so everyone can see 1) the logic I used because 2) it was the logic I was taught by educators 3.) using math books they provided.
So, with the correct answer being 24, you now have to ask the question; why are so many people like myself getting the answer 6?
Because we were educated wrong!
@@ComputerGarageLLC unfortunately a lot of people swear that they were never taught to multiply and divide before they add and subtract. Are we to believe this as well?LOL
I graduated in 1985 and was not taught in that manner. I have never seen a math book that supports your argument. I would be very interested in seeing a math book that supports your argument?
It is very concerning that so many people do get this wrong considering that the order of operations supports 24 as well as the commutative property and distributive property support 24 and the multiplicative inverse of division supports 24 as well as the majority of online math engines and scientific calculators support 24.
I guess this just goes to show that most people don't have to use math other than basic addition and subtraction on a regular basis.
Thank you for your input. Have a great day
You are free to not believe me. That is your choice. But it was how I was taught through the public education system. Clearly I was taught wrong, and it appears that many others were taught wrong too. we, those who are wrong, are a reflection of what we were taught.
And you are correct. a majority of people never use more than adding and subtracting most of their lives. Perfect example. Today a shirt cost $11.99, but tomorrow that shirt is on sale for 25% off. How much will you save by purchasing the shirt tomorrow? The answer that most people will give you......25%.
Another example I use. Mary has $10, but she need 2 gallons of milk @ $1.98/gallon and at least $5 in fuel. Does Mary have enough money. Doesnt matter, as mary will go buy the 2 gallons of milks at the gas station, and tell the clerk to put the rest in fuel.
So now, most of us never use more than very basic math most of our life.
And you have a wonderful day also.
that's probably faulty education as Richard S said. 5(2) is a multiplication, and you can't split 60÷5 in half with distribution of lower priority.
If you trust Texas Instruments' calculators, then the rule changed between 1993 and 1996. My TI-83Plus user's manual (page 1-24) says implied multiplication has the same priority as regular multiplication and division, so 1/2x is evaluated as (1/2)x, *but* the TI-82 gives a higher priority to implied multiplication so 1/2x is evaluated as 1/(2x). According to Wikipedia, the TI-82 was released in 1993 while the TI-83 in 1996.
Modern TI-85Plus also has same precedence for implied and explicit multiplications, so they give answer 24. But modern Casio (at least my fx-CG50) work like old TI-82 and gives answer 6.
Don't key expressions unthinkingly, verbatim. Electronic calculators are not to be trusted that much. That is learned very early. The insertion of brackets is often needed. Rewriting with or without a fractional exponent can be useful. Sometimes, as in 1° 1', a "+" must be inserted to show addition. Juxtaposition can mean different things. 3pi indicates multiplication. 31 indicates the addition (of 3 × 10 plus 1 × 0). An electronic calculator frequently need to be told how to operate.
My Sharp EL-520W gives an answer of 6 for the expression "60÷5(7-5)", while it gives an answer of 24 for the expression "60÷5×(7-5)". This is also the way I was taught it in school. Implied multiplication with no operation symbol as in expressions like "xy or 3(5)" takes precedence over division indicated by the ÷ sign, while multiplication indicated with a × symbol has the same precedence as ÷, evaluated left to right. I didn't even realize this was controversial till I saw this mentioned in some of your videos. When did this other convention become popular?
Exactly. No one in their right mind would evaluate 1/xy as y/x.
I'm not sure it's the convention becoming popular so much as a simplified set of rules being widely taught in some places.
Goodness, you're right. The syntax changes depending on whether you use / or ÷
Same. My first answer is 6 cause the first thing I do is multiple 5*(7-5) wich is be come (35-25) and decrease the number at parenthesis, so it will be 60÷10 and is 6. Sorry for my bad grammar...
Same for me on Sharp EL-531W
This is interesting and the reason for the change is that in the old interpretation the division symbol was actually a fraction symbol. The point above the bar represented all of the equation to the left and the point below the bar represented all of the equation to the right. Now however the division symbol is simply that, a symbol to divide the order of operations to the left by the order of operations to the right.
It's somewhat akin to English changing from archaic to modern English. The meaning of words has changed and if you keep up with the current meaning, you will understand what is being said.
For example, If I said, your room is in shambles. Currently that would mean your room is a mess, however it would have meant that your room is in a meat market. What fun.
Prior to 1917 SOME text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses. So the ERROR was corrected post 1917...
This was an ERROR brought about by the text book printing industry in regards to the misuse of the obelus. This is not why most people evaluate this expression incorrectly. They get the wrong answer 6 because they incorrectly believe that parenthetical implicit multiplication has priority over division.
Im not sure if y'all do it too but here in italy we use : without the fraction symbol
I'm 59 years old, for what it's worth. I was taught the fractional representation method in school and it still makes sense to me. Draw a line and solve for the numerator, then the denominator, then divide. That is how it was done then. If it is incorrect then how did we ever get to the Moon? LOL
@@BeerIndependence4All fractions and divisions are 2 different things
@@SmashingCapital how?
Scientific calculators (Casio & Sharp) give answer *6.*
The rest answer 24.
Pick your side.
And even half of Casio calculators give the correct answer 24
@@RS-fg5mf
I test it on fx-570EX
What do you use?
@@rrsharizam I don't use a CASIO calculator I use Wolfram Alpha a math engine and I dbl check with Mathway another math engine and if the two don't agree I find out why. But for basic arithmetic I only use them to validate my answer not to give me the answer...
CASIO fx-82es will give 24
CASIO fx-570es will give 24
CASIO fx-50fh will give 24
CASIO fx-991es will give 24
CASIO fx-570ms will give 24
My response to anyone who says the answer is 6 is to evaluate 60a(7-5)=24......a =?
Well a= 0.2 or 1/5 and the divisional reciprocal of 60*(1/5) is 60÷5
Soooo
60*(1/5)(7-5)=60÷5(7-5)=24
@@RS-fg5mf
"will give" ???
So, you don't even use Casio, yet you say it will answer 24?
I don't care whether the answer is 6 or 24.
I just wanna say that Casio & Sharp answer 6.
That's all
@@rrsharizam I have a pic of these model CASIOS giving the answet 9 to the expression 6÷2(1+2) So if it will give 9 to that expression it will give 24 to this expression....
Part of the issue is whether one considers the number parked outside the parentheses to be a common factor of the terms within the parentheses, or just another number in the sequence. I was taught that the number just outside the parentheses (in this case 5) is a part of the terms inside the parentheses ((a-b), with in this case a=7, b=5) unless separated by a multiplication sign. So (5a-5b) is the same as 5(a-b), but not the same as 5*(a-b). This would lead to a result of 6, which I would consider to be the proper result. Also, look at the division sign itself. The top dot is the stuff to the left, the bottom dot is the stuff to the right. Which would also yield 6.
I learned back in the 1980s and 90s that you have to interpret equations for computers and calculators to get the proper results. So I would input the above equation as =60/(5(7-5)) when using a calculator or computer. Which would again yield 6.
It's honestly feeling like a bunch of people were the subject of teachers trying bad ideas in an attempt to make things easier.
That's not what "common factor" means at all. And as someone has already pointed out having the * explicitly changes absolutely nothing. It wouldn't make sense to have it change anything.
“So I would input the above equation as 60/[5(7-5)].”
You completely changed the equation the way you wrote it. You can’t just add an extra set of brackets in the middle of the equation. Had it been presented in that form, then yes, the answer would be 6. People are getting confused with what “brackets first” actually means. They think if they see brackets, that means everything touching the brackets gets done first. Brackets first means you solve the inside of the brackets first. Once you do that, the brackets part is done. 5(2) is 5X2 is 5*2. It doesn’t matter what form you use, they’re all the same thing. Since it’s now just a straight up multiplication and division equation because the brackets have been solved, you move from left to right.
And the above commenter is correct that 5(a-b) is the exact same thing as 5*(a-b) is the exact same thing as 5a-5b.
If a=4 and b=2
5(4-2)= 5(2)
5(2)=10
Also
5*4-5*2=20-10
20-10=10
I put this in a calculator on a computer and it came out 24 so you're wrong
Kudos! The expression on the RHS must be evaluated first before the division. What the RHS says is that there is a common factor of 5 and so the full expression on the RHS is 5(7-5) = 35-25 =10. And so the answer is 6. I don't care what Google says!
@@mohasat01 You also don't care how math works. That's not what a common factor is. And even if it were common factors is just an interesting fact of the numbers and has nothing to do with how or when you evaluate them.
Per order of operations 60 / 5 must be evaluated before 5 * (7-5) because multiplication and division are to be evaluated left to right.
I prefer the ”special rule” version from 1917
I like writing my division as a fraction. That way there is no doubt as to what is numerator and denominator.
The special rule seems to follow this process.
I you use Google UA-cam to post this then you should stick to Google way of evaluating math expression. Google is the best guide.
You can just write it as a fraction and not division
i dont get why the separate division and fraction, isn't 1 over 2 0.5? Isn't 1 divided by 2 0.5? Then why are they so FKN different when they are the SAME?!
true, that's why don't use parentheses for multiplication in these situations, use * or the dot instead
@Anika Anjum That's why writing everything on a single line is ambiguous. The school I was taught is the division is a grouping operator so that everything to the right of it comes under the operator IE in the denominator. You were taught in a different school of thought. These different schools of thought are why equations need to be clearly written out.
Why would you use a calculator as the way to measure what interpretation to use. A calculator is just a computer and a computer only does what a human programmed it to do.
Read my comment above....I think you will agree with me...
Because a calculator follows rules laid out by its human programmers instead of the unqualified presumptions of youtubers
Why would you use a calculator for such a simple task? However when my daughter was 13 in 1989 I bought 13 candles at the local stationery shop. I gave the girl 13pence but she said I'd better check its correct & rang up 1penny 13 times. No it was an old till not computerised connected to stock control. She then said "Yes you are right 13 pence" & put out her hand for the money.
We couldn’t even use calculators in high school (they weren’t available in grade school) in an effort to prevent the inevitable, the DDOA (the dumbing down of America).
but computers are programmed to do math same way as us...7-5=2. 60/5=12. 12/2=6. In that order.
This shows that I am getting old, I came up with the answer of 6
One can get forgetful with age but plenty of young people fail to get the correct answer as well. The correct answer is 24
@Michael Stocker WRONG. There is absolutely nothing wrong with this expression except for the ignorance people have about parenthetical implicit multiplication.... The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 24
sorry for not knowing all the correct english terms
So do I the paranthesis is broken down for easy of handeling and shopuld be multiplied as it stated 5(7-5) -> (35-25), of the five should be diveded down to a 1 by devidind all groups by 5 to clear it out (60 / 5(7-4) -> (60/5)/((5(7-5))/5 ---> 12/(1(7-5) --.> 12/(2)
The 5(7-5) is a part of the paranthese operations and even in pedmas paranthese has priority
@@RS-fg5mf This has nothing to do with being forgetful and everything to do with what method an individual is taught on precedence.
@@ronhan9 Wrong...
60/5(7-5) does NOT equal 60/(35-25)
Easy handling is to simplify what is inside the parentheses. 5(2) is not a parenthetical priority and is exactly the same as 5×2...
The TERM 60/5 is to be multiplied by the value of the parentheses 2 and the only correct answer is 24
Although we have modern PEMDAS to adjudicate how to interpret such expressions, this is really an inherent language flaw, as you pointed out mid video. It is rooted in the idea that you can omit the multiplication symbol between and number and an opening parenthesis. If you write it as 60 ÷ 5 * (7 - 5), you still need PEMDAS to interpret it, but it is much less tempting to get it wrong.
I am old as dirt.
I always distinguished a difference between
N*(a-b) and N(a-b)
With N(a-b)
== (N(a-b))==(N*f(x))
Just my shorthand.
"Why would they change Math?? Math is Math!!!"
Well said Bob/Mr. Incredible , well said
Math did not change in this case. Writing and glyph interpretation changed.
Oh Contraire, math must now be expected to include critical race theory.
Math developed from the human ability to conceptualize, there is no inherent law of nature behind math.
I read a while ago that it was 4% of Mathematicians who use it this way. The rest of the population didn't. Probably someone in a wee office somewhere decided.
Not in "1984"... "He" is "she" and "She" is "He"... or whatever they say it is...
I've used HP calculators with Reverse Polish notation from the start when they hit the market! In that system you start calculating the content of parenthesis and then go outward. With this logic, the result is definitely 6. During the whole time of my physics studies (that means dozens of textbooks in physics and applied mathematics), I haven't found a single case being confronted with any ambiguity of a mathematical term!!! If someone gives me such an ambiguous expression to calculate, I simply refuse to calculate! I will tell him to study mathematical semantics first! (This has already happened)
yeah, but our text books were kick ass.
notated,indexed and bibliographied with special symbols etc.
RPN rules!
The answer for us is 6.
I memorized times tables in the mid 60's; PEMDAS wasn't a thing when I went to school; I never took physics or calculus, only went as far as trig; the answer I got is 6.
I went to school in the 70s and 80s. Was always taught the method that gives the answer 6.
According to this 12x÷6x = 2x²
You don't see that often...
(I would go for the ambiguous)
There's a reason why most math teachers have rarely used the '÷' symbol in decades. Almost every teacher will teach division in fraction form because the division symbol is very ambiguous. If written with a '/' or in fraction form, there would be no question what the right answer is.
60/5(7-5)=6
Reason is, everything multiplied on the right of the '/' is part of the denominator. Which is the reason most people are tripped up using the archaic '÷' symbol. The rules are slightly different. In order to get 24 with the '/', you would have to write it as:
(60/5)(7-5)
Easy. Thats why nobody who actually works with math uses '÷'. And in higher level math, such as calculus in fluid mechanics or thermodynamics, the order of operations is practically useless. You're stuck developing your own equations by following your units of measure to get from one place to another. No real need for PEMDAS when you have a force in Newtons or pounds, and you need to solve for pressure in kPa or psi. Or maybe you need max power output in Watts or horsepower. Then again, if it wasnt for archaic symbols used to confuse people who dont do math in this respect regularly, this channel would probably have died out long ago
WRONG.... Prior to the 1900's that's how the obelus ÷ was being misused. The solidus was never used in this manner
60÷5(7-5) and 60/5(7-5) are exactly the same and both equal 24
The solidus is NOT a grouping symbol only the vinculum (horizontal fraction bar) has grouping power....
60
------(7-5) = 60/5(7-5)=24
5
60
-------- = 60/(5 (7-5))=6
5(7-5)
Extra brackets required to keep the grouping of operations together that the vinculum provided when written in a linear format with infix notation....
That is not why most people get this wrong. They incorrectly believe that implicit multiplication has priority over division. It doesn't...
Not necessarily, Richard. If it was written properly, the (7-5) is part of the numerator. So, without parentheses, you would have to write it
60(7-5)/5=24
Everything multiplied on the left of the slash is numerator, everything on the right is denominator. You're welcome to disagree. That's cool. However my college professor would mark my answer wrong if I wrote it
60/5(7-5)=24
As I said, nobody writes equations or mathematical phrases like this for good reason. There are simple programs to write and paste complex formula as they should appear, not like this with the intent to befuddle. Best of luck to you, bud.
@@boredbales12345 WRONG again. Multiplication is Commutative.
60÷5(7-5)=
60 (7-5)÷5=
(7-5)÷5*60=
24
All 3 expressions are equal to 24..
Evaluate this equation
60a(7-5)=24...... a= ?
@@boredbales12345Your professor would be wrong for counting 24 wrong... LMAO
neither ÷ or / have the special treatment of taking a photo of content to the left, to the right and using the operation afterward. None of that is in the order of operations.
In 60/5(7-5) the / is a division symbol, and the order of operations says 24...
You must be thinking of the fraction slash but that requires (7-5) to be subscript, ⁶⁰/₅ₓ₍₇₋₅₎, to equal 6.
The correct ways to phrase the questions (depending on what you want to ask) would be:
60/[5(7-5)] for which the answer is 6.
Or (60/5)(7-5) for which the answer is 24.
The question as originally phrased makes no sense. The division sign is never used beyond grade school nowadays (it is not there even in a computer keyboard), but it was there in the question but without the multiplication sign. It was not only confusing but sloppy. One set of parentheses would have eliminated all ambiguity.
Assuming the question was originally an algebra question for which you then substitute in the actual numbers, then "6" as the answer actually makes more sense.
the division symbol is above the "8" on the numerical side of my computer keyboard... I do however agree
That's why we use fractions instead of the division symbol
I agree. It completely avoids the issue
That division symbol is called an obelus. Just fyi.
? Thats division still.
@@Ok-th2gd of course it's division, but using the fraction instead of the obelus it eliminates confusion like from this problem
@@Ok-th2gd 60÷5(7-5) can be changed to 60/5(7-5). From that 60 is the numerator and 5(7-5) is the denominator. 5(7-5) becomes 5(2) = 10 so 60/5(7-5) changes to 60/10 which is 6.
I was helping my 13 year old with his math homework 15 years ago and learned something that I was never taught in school. Not even in College. "Please Excuse My Dear Aunt Sally".
My 7th grade math teacher used her name in it: "Pretty Please, Mrs. Dovers Always Says".
try to apply it to : 3*47-1/4398473+10-8/33 without parenthesis
@@aligator7181 That's (3*47)-(1/4398473)+(10)-(8/33) = 150 + (25/33) = 150.7575757575 . . .
Most/all decent calculators will get that without using ( )s
@@Chris_5318
Yes, but the trick is to get the order right. I have never used an expensive scientific calculator, I am assuming they probably sort out the order automatically?
@@haroldprice1030 Different, but almost identical, models from the same manufacture can give 6 or 24.
The correct answer is the one found by using the same convention that the author used. We have not bee given tha info. However, the author would have to be crazy if he was expecting anyone to get 24.
You said : The "MODERN" interpretation. A lot of people, including myself, have been taught the one that gives 6 for result. I love math and was always at the top of my class. 24 would never have been the answer.
6 is absolutely the correct answer. 24 is result of a different equation. The video is wrong.
i think my math training also results in 6. so we just changed the definitions. no right or wrong.
ABSOLUTELY......me too !!!
The issue is that the video author doesn't understand BODMAS correctly... "brackets" means you grab the brackets first and solve them themselves using BODMAS. So 60/5(7-5) the bracketed term is 5(7-5) which expands to 35-25 which makes 10. 60/10 = 6.
Now, if you add a multiplication sign then it changes the precedence because you are actually changing the equation significantly. 5 * (7-5) the bracketed term becomes only (7-5) which is of course 2.
A deliberate nuance used to create a video I think. Fair play.
You were taught incorrectly.
I think everybody is missing the point. The fact is that a mathematical expression like this is derived to calculate an aswer to a problem in the real world. Before we can know which binary tree to follow, we have to know the real-world problem.
What does 60 represent - it it people, who are being divided by ... what?
We also need to know what the 7 and the 5 represent, and why they are bound trogether in the bracket.
Mathematics is a tool - not an entity in itself.
Seems i was thought the 1917 version. My result was 6 too. Maybe you could do a follow up video on why the modern version is now used. What advantage does that interpretation bring?
In large part because expressions cannot be presented to computers by use of a divide bar that clearly shows what is in the numerator and what is in the denominator thereby showing grouping. Computer languages demand expressions all be in-line and there is no way to group subexpressions other than with explicit use of parenthesis.
it is manipulating the mathematics as they do it with everything this days. All depends who is calculating and for whom. If that was you assessed by tax office it would be 24 but if that tax would be calculated for Bill G. it would be 6. - 😏
the sentence when be written as a fraction with 60 on the top and the rest in the bottom and the result is obvious.
@@lubanskigornik282 And I just know if I buy Bitcoin, somewhere along the line my payout is going to use the New Math and end up dividing my payout by 24 instead of 6.
Correct!! But then I was in school in 1917!!! I think it is used to save space and characters in computer. 5(7-5) uses 1 less character than 5x(7-5). New Math!! You know, 2+2=5.
Left to right, what a nonsense. The fact there is no multiplication sign between bracket and the 5 is a clear indicator, that this is just one term, that the 5 and the bracket belong together, period. Anything else is sophism. 6 is the solution, period.
It's surprising how some modern calculators like CASIO, which are recommended by math teachers, also give 6 as the answer! (tested with models fx-82ES PLUS and fx-82SPXII Iberia)
Thanks for the info! CASIO's calculators were a thing for 6÷2(1+2) as well. I found one video, for example, that shows 9 on one calculator (fx-50FH) and 1 on another (fx-3650P), both which are marked in the video as "H.K.E.A.A. approved" (Hong Kong examinations and assessment authority). ua-cam.com/video/IXUBepvylQg/v-deo.html
I would love to speak to someone at CASIO about this--would make for a great video!
SHARP Scientific Calculator EL-531LH , gives 6 as well
The calculators tend to put in a bracket (in this case before the 5 and at the end) before displaying the result.
it is something called syntax. it is not as much math as it is programming. it is the programming of how to READ math in a single line. LIKE A TRANSLATOR FOR THE CALCULATOR.(it works in binary data) you do not.
When writing an expression parser, you may want to capture the intent of the user input. As I mentioned in another comment, the expression 1/2a is most likely meant to be interpreted as 1/(2*a), not (1/2)*a. The intent is generally to raise the precedence of implied multiplication above that of explicit division.
also it could be written as 60/(5(7-5))=6 or (60/5)(7-5)=24 to be less ambiguous. I've done a significant amount of coding over the years and I like the use of parenthesis to reduce confusion.
24 isn't ambiguous from the get-go, though. To get a different answer just assumes grouping around 5(7-5) which does not exist in the original problem.
I've done a lot of coding over the years and I hate it when people overuse parenthesis trying to reduce confusion because it just makes the statements harder to read. Now I have to parse a bunch of parenthesis to figure out if you actually changed the PEMDAS order at all with them only to find out you didn't, you just wasted my time.
A fan as well. Being explicit is the way to go.
"also it could be written as 60/(5(7-5))=6"
No, it couldn't.
That would be wrong for this problem.
There is no ambiguity.
Extra () are not required when the operations are already in order.
You CAN use them, but the problem has clear meaning without them
It definitely doesn't make it harder to read, quite the opposite imo
The expression typed into my Casio calculator exactly as shown returns the result 6. Which is exactly what I calculated as I was always taught that if there was no operator between a number and an expression in parentheses, then they were linked and to be calculated together. I.e. 5(7-5) = 10
Old Casio calculators do not handle the order of operations correctly.
@@Harmonic14 My TI-85 also states 6. I was always of the school of thought that when in doubt, use more parenthesis.
Exactly so. And his sentence is not ambiguous. The verb saw separates the subject (I) from the direct object (man) and any modifiers of the object (binoculars). So if you wanted to say you saw the man by using binoculars, the binoculars would have to modify the verb saw.
So did I. And it confirmed my "old fashioned way" to interpret that unclear calculation. It is from 1981.
@@wacholder5690 The order of operations has the answer
24.
As someone who mainly learned math through programming, I'm sometimes disadvantaged by a lack of theory and long-hand methods. I struggle with deciphering mathematical notion in order to translate it into something I'm working on, mainly because of all of the implied, rather than explicit operators and evaluation order of written mathematical notation. While there is an underlying default evaluation order in programming, you can explicate everything to the order you want. The result of 60 / 5 x (7 - 5) would be evaluated as (60 / 5) x (7 - 5) = 24. If you meant something different, you'd explicitly say so with parentheses and operators: 60 / ( 5 x (7 - 5)) = 6. The difference between / and ÷ would I guess be one of which programming language you are using. ÷ may be a valid in programming, but it's not a common keyboard character, so I don't know, because I've never used it.
in many programming languages, * and / are used for multiplication and division respectively.
Also depends on what math you're doing... Algebra, geometry and calculus always order of operations is always parenthesis 1st, then multiplication, division, addition and then subtraction... That's the basic order of operations for any higher math except for programmers because the computer is doing the math not the programmer! Discreet Mathematics, is what computer programmers learn... Euler circuits, TSP, Fibonacci numbers, etc.
@@victorglaviano: Not discreet mathematics, discrete mathematics (i.e. noncontinuous math, vs. being unobtrusive, since context matters).
At some point, for society to function, there have to be rules / standards that are agreed on, or there would be complete chaos (as per the Tower of Babel meme).
I was pleasantly surprised to see how Microsoft Excel (V 2010, from Office 2010, which works fine for my private use at home), handled it.
When asked to evaluate the equation: 60 / 5 (7 - 5), it forces you to clarify what you mean, instead of just giving an answer. By default, it asks if you mean: 60 / 5 * (7-5), and if you agree, it gives 24. It also gives you the option to "correct" the formula (i..e make it nonambiguous) yourself, by updating it some other way.
Remarkably good behavior for typical application software, likely brought about over time by so many people using highly competitive spreadsheets for critical applications, and FORCING companies to eliminate ambiguity would be my semi-educated guess, as someone who spent a career in application and then system programming at IBM on mainframes.
For example, my CPA uses spreadsheets constantly for taxes. Can you imagine the chaos if they misinterpreted ambiguous formulas? Best just not to allow them.
As a programmer you should know that some of the languages are using polish notation where execution is from right to left. 60/5(7-2) would be equal 6 and 60/5×7-2 would also be equal to 6 and 60/5×-7 2 would be equal to 6 as well
@@Grim_Reaper_from_Hell - I have no idea what you are saying. Every language I have ever worked with evaluates right to left; It's not unusual.
Also, I just compiled your first 2 operations and the printed results are as follows:
60 / 5 * (7-2) = 60
60 / 5 * 7-2 = 82
Your third operation would return an error as you have two operators next to each other: 60/5×-7 2
If you meant 7-2, then this is identical to the second operation you described, and the result would be 82.
If you meant 60 / 5 * -7 * 2, the result would be -168.
Ok, I figured out.the equation…
It’s (9min video)+(wrong answer)+(huge comments engagement)+(3,000,000 views) = $6000
You have the winning answer
The guy even _said_ at the beginning of the video why he was making it. Cha-ching!
Good for him and I am glad that he is uncovering something that is making us say 🤔
@@sandragrant327 I agree, it’s quite an undertaking to make math controversial
Yep, telling people that 5(2) is (5+5) .. first order operation or 5(2) is a scalar .. second order operation, wouldn't have given him my 2 cents.
when I was in school we were taught to distribute the 5 to the numbers in the parenthesis first. thereby the resulting answer would be 6. At some point in time we changed the way we did math in order to confuse our children...I mean make math easier lol. I still enjoy the videos; keeps the mind working.
I was taught In school to do it the “historical way” because it’s still in parentheses so you multiply it first
vikpun XD thats not how it works. You do whats IN the parenthese first not mulitplying or dividing the parenthesis
@@kevinsanderson4112 as written i would think the 5 was factored out 60/((5× 7) -(5×5))i know some teachers who teach it this way and my calc class was like that so my immediate thought was 60/10 =6
@@jmanwild87 Your expression correct and that is the way I learn maths. How 24 become the unambiguous answer.
right????
I also came to the historical way, although I think part of it for me was how I viewed the question. I saw it similar to 60/5x where x is (7-5) being 2.
The reason I came up with 6 was the fact I was taught that the order of operations was in the actual order of the letters. Parenthesis first then exponents, Math then Division, Addition then subtraction.
VERY EYE-OPENING AND EDUCATIONAL. GREAT VIDEO!!!
That is how I also learned it. I was taught to remember - (P)lease (E)xcuse (M)y (D)ear (A)unt (S)ally. (P)arenthesis, (E)xponents, (M)ultiplication, (D)ivision, (A)ddition, and (S)ubtraction. Please note...I went to a public school. LOL!
Although it wasn't mentioned in the video, the reason multiplication/division are not given a specific importance is because they are the same operation, so you perform them in the order as written.
Division is really just multiplying by a fraction. Ex: 60÷5 = 60 x (1/5). The same holds true for addition/subtraction. Subtraction is really just adding a negative number. Ex: 23 - 8 = 23 + (-8)
If you change all division operations to the equivalent multiplication operation, and then multiply straight across, you would see the answer will always be 24 to the equation presented in this video.
No you’re right it’s 6, cause multiplication is before division. The creator is just trying to cause division
Y'all learned wrong or were taught wrong. The correct translation of the acronym is "...Multiplication AND Division..." (equal rank performed left to right), "...Addition AND Subtraction.. " (equal rank performed left to right).
@@Paul-yb8pf no. Multiplication and division are equal. You solve left to right.
I remember being taught that when there is an “understood” multiplication because no “x” sign is there, then this calculation would be done before the preceding division sign. The 5 and the solution to the calculation in the parentheses are linked together, like the expression 5y are linked. If y=2, then 5y=10. Then divide what is on the other side of the division sign by 10. If they wanted me to do the division before the multiplication, they would have used a multiplication symbol in place between the 5 and the parentheses.
I wasn’t taught this but I’ve always followed it as it seems more intuitive
correct because the 5 is the coefficient of the parentheses. whenever you have a parentheses, you have a coefficient, and whenever you have a coefficient, you have to utilize the distributive property.
Ditto. No times sign between the 5 and the 2, just parentheses, was to be calculated first with how I was taught. I see it both ways but unless the order of operations changed in the last 25 years and it was not made public knowledge, then my math teachers would tell me I’m wrong to give 24 as the answer.
I'm old school (65) and we were taught the same thing. 6
agreed
All of these type example are due to someone writing mathematical statements in the most confusing way; in REAL mathematics, physics and computer programming we choose the write mathematical statements so as to prevent confusion. These example-makers lift a few excerpts from journal (or written text) articles where one is forced to use only a single line of text space; however most likely elsewhere equations are presented in an correct format.
I was always taught that anything that touches the parentheses / brackets was next after evaluating what was inside the parentheses/ brackets.
That's how I was taught to calculate too and I absolutely stand by it even if the rocket crashes. :-P
Anything touching them simply implies multiplication if they wanted it to be the 5*2 first then they should have done this
60*(5(7-5))
@@starlordz6111 True.
@Chris Travers when I typed it into my ti-83 I got 24. And that was after I solved it without a calculator. Anything touching but not in parenthesis only means multiplication nothing else.
In "60 ÷ 5(2)", the bracket "(2)" has higher precedence than " ÷ ".
When I see a number right next to a parentheses I interpret it as a factored number, represented as one unit. So 5(7-5) is really a factored expression of number 35 minus the number 25.
I'm just saying, ambiguity must be interpreted in context instead of immediately concluding that a number outside of a parentheses is synonymous to a simple multiplication symbol as 5 x (7-5) un which case your final answer of the full equation would be 24. However if you see the 5(7-5) as a factored expression of 35 minus 25 then the answer of the full equation would be 6.
You have a good point!
The problem is the symbol ÷ that is used. It causes ambiguity and should not be used. If a problem is not precisely stated, in terms of math the only correct answer should be that the problem is ambiguous
Many commenters seem to dislike the symbol ÷. @presh can you weigh in on this? I personally like the symbol and wish it could be defined as follows : the line in the centre shows that there will be division. The dot above represents a placeholder for the numerator, the term immediately left (preceding) and the dot below representing the denominator, the term immediately following the ÷ symbol.
If this sounds strange, just look at these symbols and I think there is enough precedent: x÷y x%(special case where the numerator is always 100) and x/y. If this is always the interpretation, and it remains consistent, then it implies 60 over the rest, or the historical usage, or the tree on the right. Perhaps the historical usage of ÷ was as I suggest? This removes any ambiguity and also preserves order of operations.
In an era where each character of text printed increased the cost a shorthand like ÷ would have been valuable...
Nope, removal of the multiplication symbol does not change the order of operations. It's just shorthand that has been adopted into common usage (more likely laziness in removing the sybol).
@@rickyhall7514
There was never any "multiplication symbol" in the equation so we cannot "remove" what's not present.
Please elaborate your analysis.
Based on the education I received in the USSR in the 1980's the answer is 6.
I guess we now know why your economy and government crumbled when someone over there found MTV.
Your teaching was correct and equivalent to the teaching in the US in the 1970s, which is when I did my primary education.
Yes, and 6 is the correct answer worldwide.
Let me summarize the positions as I see them:
> for folks who are followers of the PEMDAS philosophy and believe such things as
x/3x is equal to x squared divided by 3
the answer is 24.
> for folks like me who believe that PEMDAS is BS and screwing up the teaching of math in America and believe in such things as
x/3x = 1/3
the answer is 6.
Now I do recognize that this is America and one is free to choose, but
from my viewpoint it does appear that the PEMDAS philosophy falls into the category of metaphysics; - - - you know, that abstract theory with no basis in reality.
@@Borvo1 its A/BC = AB/C?
True or False?
See, when you get to "60 / 5(2)", to my mind, the 5(2) is an outer parenthesis+bracket expression (which should be evaluated after the inner parentheses+bracket) and should be evaluated before the typical multiplication+division. I was taught pre-PEMDAS, however, but I think that approach clarifies a lot of these "ambiguous" problems.
No in that case it would be [5(7-5)]. That’s where you would use inner brackets and then outer and that’s where you’d multiply by 5 before moving left to right from the beginning. In this case there are no outer brackets so once you’ve solved what’s inside of them, you move left to right from the beginning. 5(2) is the same as 5X2.
PEMDAS has been around for centuries, modern PEMDAS has been around for more than a century (as the video showed), so no you weren't taught pre-PEMDAS. You were just unfortunately taught wrong.
PEMDAS is not used by mathematicians, physicists or engineers:
ua-cam.com/video/lLCDca6dYpA/v-deo.htmlsi=Rzfnvk4hUtqL6ZVq
@@trickortrump3292 This is why I get dinged when I write essays in school - I genuinely use *too* many parentheses and think parenthetically. Not so much that I'm particularly proficient in LISP, though.
@@trickortrump3292 No, the outer brackets are implied and aren't necessary. 5 is the coefficient of 2, so 5x2 must be done first.
I was taught that 5(7-5) implies another parentheses because the multiplication symbol is omitted. So that the true form of that problem would be 60 ÷ (5(7-5)) and the answer would be 6. However if the problem is written as 60 ÷ 5 x (7-5) then the answer would be 24.
You were taught wrong. Parenthesis are never implied. If it’s not in parentheses don’t add one.
@@uhohhotdog it's implied by the distributive property
Dann Clark no. That’s not how math works. It goes left to right. You’re going right to left.
And what natural, empirical, universal law says that's wrong? Its not something we can prove with science. Its only what we all agree on it. And the fact that 3 videos exist on this channel is evidence that we don't all agree on it.
If we are being correct we should always write all parentheses e.g. ((2(x+3)(x-3))-1) but as you can imagine that gets messy real quick especially when writing by hand. That's why we omit some of the parentheses (at least where I am from) to make it cleaner and easier to read ((2(x+3)(x-3))-1) = 2(x+3)(x-3) -1
I’m 53 years old and calculated 24 as the answer due to the way I was taught mathematics at school.
You calculated correctly.
For some reason these kids are wanting to do the multiplication on the right before the division on the left, madmen all of them.
It’s easy to see that if you take
60 / 5 (7-5) you start with the parenthesis
60 / 5 (2)
So you have 60 / 5 x 2
If you do math incorrectly and do the multiplication on the right first, you get a sum of 6, but anyone who passed 5th grade math knows you go from left to right
12 x 2 is the final product before solution
Same here Im 56.
Same here ans is 24. I'm 42 btw
I’m 58. I get 24.
I’m 55... I got 24
Sorry, you’re all wrong. The answer is 42. The answer to everything is 42.
So the new code for you being wrong is 42. LOL
There is an old Ford Prefect out there somewhere with a reg plate . ANS 42. How cool to own that huh!
This guy really knows were his towel is!
Hey you seen my guide to the galaxy?
@@DanielMartinez-hk2dc , I didn't see you're guide to the universe. Did you leave it in your pocket? 😈🤣😆😂
The answer is "Don't use the flipping ÷ symbol", because there's no way to be certain whether the one who wrote the problem means (60/5)(7-5) or 60/(5(7-5)). The former would be 12(2)=24, the latter would be 60/10=6.
People tend to see implied multiplication (5(7-5) as opposed to 5×(7-5)) as including the multiplier 5 as part of the term, so I imagine most would say the latter, but strictly speaking it makes no difference to PEMDAS.
And my scientific calculator app actually gives a warning about the ambiguity of the problem as written and offers to let me change the "operand grouping" setting, giving the example of 1÷2π to demonstrate what changing the setting does. It does default to 24, however.
Before I watched, I knew the video would conclude with 24, but when I was in school the correct answer was 6.
Why? When you have any number just outside a parentheses (without a separate multiplication sign) you multiply to what was in parentheses prior to any other operation. Same is true with a variable like x or y. So 12 / 3x when x is 2 is 2 (as opposed to 8).
My school taught me to presume 3x is in parentheses as though it were virtually (3x) as far as order of operations.
This is also true of 60 / 5(7 - 5). There is no difference between that equation and 60 / (5(7 - 5)) which in turn is 6. To get the answer 24 the equation should have been written 60 / 5 * (7 - 5).
It makes sense to me in a way. If I have a 12 dollar buy for 3 packages of x toys and each package of x had 2 toys and I wanted to know the price per toy I’d write the equation as 12 / 3x = 2 per toy. However this video says that the correct equation to get the proper answer is 12 / (3x) = 2.
This is also true of exponentials. At my school the answer to the following equation 60 / 5 (7 - 5) ^ 2 would = 3. Or to be very precise could be written as 60 / ( 5 ((7 - 5) ^ 2))
I’m assuming this video would say the correct answer is 48
I’m not saying the correct mathematical solution is my way or not, just saying this was how it was taught to me in high school.
Yes, both are valid since it is simply using ambiguous notation.
It's terrible writing.
Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
Yep..I got 6..
Left school in 1962
Yes, to me it is 6.
Agreed 👍
I got 6 as well, I must be old? 😂😢🎉
I have always struggled with maths having left school at age 14 but I am passionate in trying to figure out any maths problem. Don't often get the correct answer but I enjoy trying. Thanks for the exercise. Blessings.
There's disagreement on this answer because the notation is very sloppy.
If you are uncertain about this particular math question, you might actually be better at math than you think. Your math might be very good when the notation is clear.
Maureen Mallett.....it's math. No plural
@@joc8092 Depends on where you live.
In England, it is pronounced Maths plural.
@@MrGreensweightHist k, I stand corrected
@@joc8092 I only know from watching Doctor who, and thinking, "That sounds so bizarre" until i got used to it :D
The problem is in the “modern interpretation”. How do we justify changing math when it completely changes the answer? It makes no sense to me at all.
@Judy Cherry Yes but when you are dealing with modern Neo Marxism like we are now there never is any correct answer. You know in their warped minds 2+2 can equal 5. Every thing is fluid, You know like the Genders are. God help us if we don't take the World back from the Satanic Globalists.
All that changed was notation not the rules of maths themselves which is what a lot of people think is changing.
It's like using Sin²x to mean (Sinx)² or using Roman numerals, MCMXIX, instead of Arabic numerals, 1919.
Both are valid notations and using one over another doesn't break any rules or axioms etc.
The problem with the question here is it isn't written to modern international standards, the ISOs.
If it was written properly then everyone would agree on just 6 or just 24.
@@GanonTEK I hear you, but in the end, there should only be one correct answer, not two. Math used to always be an absolute. My answer is, and was, 6.
@@bingcherry2008 Oh there should be just 1 answer. You are right about what.
What about the question 16/8/2?
Or "What is 10 divided by 5 multiplied by 2?".
They are ambiguous also, just like the one in the video, without more information to clarify what the person writing it meant by what they wrote.
If someone was writing an academic paper and wrote 60÷5(2) they mean 6.
If a programmer wrote a book on how to learn Python and wrote 60÷5(2) for an example they mean 24.
The issue is the notation is ambiguous now. That's why we have international standards, to bridge the gap.
With 60÷(5(2)) everyone agrees on 6. With 60÷5×(2) everyone agrees on 24. One of those is what the person writing the question meant but we will never know which.
Until we do, both are valid.
See my reply from 5 months ago. Look above. All is explained.
5(2) counts the same as (2) ex x. the modern method generates 60/5x1(7-5) . the main problem with the modern method is the improper disposing of (). as soon as you make it 5x1() your disposing it the same as 5() without changing the number inside. no matter what () must be removed before proceeding even if exponents or multiplication takes place. this was know as completing operations between signs in the 90's.
an example would be 2 x abc= vs. 2 x a x b x c . abc is a complete expression of 1 number to mutipply 2 by. no it may turn out a shortcut is it's all multiplied together so order doesn't mean much. but only if the shortcut doesn't ater the answer.
in this case the shortcut 5x1(2) alters the correct answer so you have to follow the 5(2) = 10. since rewriteing 5(2) as 5x1(2) allows the interpation 60/5 x2 this is a basic half step to proper order of exponents and such.
probably one of the most disturbing parts of teaching maths in a system
The fact it takes 9 minutes to describe a math problem, is a problem🤷🏼♂️
@@polarblue7468 No, I think the man with binoculars saw him...
@@blackcosmos No, but I stayed at a Holiday Inn Express last week and it took me like 3 seconds to complete in my head😂
No, it makes it cool.
I was doing this stuff in Grade 7. People are making this much harder than it actually is.
Quite literally elementary school math equation that people are arguing over?? This is disturbing... This should take one seconds to figure out that the answer is 24. Math really wasn't a lot of people's cup of tea.
Early on after scientific calculators became popular in doing this type of equation, math teachers told us not to use a calculator because it would give the wrong answer. When learning how to solve complex equations written in fraction form, the math teachers taught us to do the math above and below the line separately, then do the division. Engineers and physicists will use the old school method, which is called juxtaposition. This method accounts for the equation written in fraction form. The divided sign or a "/" use in the equation is just syntax. It replaces the horizontal line in fraction form. When written in one line using the arithmetic symbols and parentheses, some of these symbols are implied. So, when converting an equation from fraction to line form if the person writing the equation doesn't include a parentheses or bracket after the division symbol according the to PEDMAS, it changes the equation and the answer given. However, the rule for converting the equation from the signal line expression is to put everything left of the division symbols in the numerator and everything right of the division symbol in the numerator. This indicates that there is an implied bracket, or parentheses, in the equation. Which method really is correct? Having worked in the engineering field where my calculations had to have the correct answer to make what we were designing to work, I used the juxtaposition method and always got the correct answer. When using a calculator, I inserted the implied parentheses in the calculation. It is my opinion that in order of operations, multiplication should take presidence over division. I challenge a math teacher to prove which is the correct method to use on an ambiguous written equation.
I was on the math team in school and was taught that either side of the / was implied parenthesis and the ÷ was not used at all. That was in the 90s so my memory might be wrong now, but i think you are right.
I don't remember being taught that and would argue against it, because then you get to pretty iffy territory. That seems like a special, jargon-like usage convention: If everything is always of that form in some field, it makes sense to omit superfluous parentheses for readability, but it is problematic for general usage. Of course, a lot of it is *visual:* Are you using/imagining a large slash extending a character height above and below the rest of the expression with room around it or a small one packed tightly in one of multiple separated addition terms?
To me, however, it is obvious that you can't just break an expression at a point, where there is no operator to break at (that matches the implied operation) for the sake of binding a part of that grouping to some another operator (with the same or lower preference). If you do that, you are just willy-nilly chopping the term in half at a completely unmarked place. The purpose of notation isn't to mislead.
I've used examples of 3x/xy and xy/3x elsewhere.
The minute you start "implying" something that isn't there in math you're wrong.
We could certainly have decided that multiplication has some precedence over division, but that would require us to change how we write our equations.
That's the point of all of this. The rules have to be fixed in order to do math at all. In theory we could make order of operations anything we wanted to. What we choose dictates how we construct the equations though. And some ways make creating equations much more complicated than others.
The simple fact is there is nothing ambiguous about this equation. You just can't invent things that aren't from the established rules for how math is to be evaluated and then complain when you get a different answer than the one the writer of the equation wrote it to produce.
@@Cdaragorn huh? we were taught that multply goes before division and what ever number is outside of the brackets it getting multplied by the inside number then division comes. you cant just change math and thing youre going to get the right result.
@@keenanvanaalst9865 My entire comment was explaining why you can't just change math so you're right. The problem is multiply doesn't go before divide and it hasn't for more than 100 years. Multiply and divide are equal in the order of operations. You do them together.
I'm sorry if you were taught wrong. Seems like a lot of people were given that misconception.
I got 6, whenever I see a ➗ I automatically turn that in a fraction /. So I simplify the top and the bottom independently before finishing the division.
You failed to turn it into the correct fraction.
60
-------(7-5) ÷ 60÷5(7-5)= 24
5
60
---------- = 60÷(5(7-5))= 6
5(7-5)
@@RS-fg5mf that was exactly what I did! Thank you for explaining!
@@KrogTharr so you understand now that the correct answer is 24, right?
and that's why you never use ➗ after the 4 grade... except when you want to make a semi trap video. Math is supposed to be clear, not interpreted. If you would have seen the correct fraction you whould have given the correct answer. It's not a matter of age, they just made this purposefully confusing. You won't find an engineer use this kind of writing.
@@grigturcescu6190 what this demonstrates is that people can't follow a few simple rules and that they need to be hand held all the way to the correct answer...
When you actually understand and apply the Order of Operations and the various properties and axioms of math you get the ONLY correct answer 9
It doesn't help that on average 70% of adults incorrectly believe that 5+2×10=70....
You have people under educated who fail to understand the Order of Operations AND yoy have people who are over educated and try to make more out of a basic 4th grade arithmetic expression than it is...
Not arguing but talking from my schooling 50 years ago. I got 6. I subtracted 5 from 7 resulting in 60 / 5 * 2. I then multiplied five by two. Then divided 60 by 10 = 6.
I just typed into excel =60/5(7-5). Excel insisted in inserting the multiplication symbol in between 5 & (7-2). It then produced the answer 24.
I think excel did the problem in the following order, 7 - 5 = 2, 60 / 5 = 12, 12 * 2 = 24. Clearly I should now treat multiplication an division problems as equal actions and operate from the left.
Fun presentation.
I was taught that the 5 before the parentheses would multiply what was inside so 7-5 is 2 times 5 equals 10 divided by 60 equals 6
That's what most people remember but what you forget is the TERM outside the parentheses is multiplied by the value of the parentheses not just the factor next to it.
TERMS are seperated by addition and subtraction not multiplication or division. 60÷5 is one TERM attached to and multiplied with the value of the parentheses 2... The correct answer is 24
60÷5(7-5)= 24
60÷(5(7-5))=6
60+5(7-5)= 60+5×2= 60+10=70
60-5(7-5)= 60-5×2= 60-10= 50
True. 6
@@stephenkinyanjui477 WRONG. The correct answer is 24 not 6
@@JJJJ-hp9oz there is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to group and give priority to operations INSIDE the parentheses and nothing more.
5(2) is not a parenthetical priority and is exactly the same as 5×2
You then demonstrate the Distributive Property incorrectly. The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses.
60÷5(7-5)=
60÷5*7-60÷5*5 parentheses eliminated
12*7-12*5=
84-60=
24
60÷(5(7-5))=
60÷(5*7-5*5) inner parentheses removed
60÷(35-25)=
60÷10=
6
60÷5(7-5) does NOT equal 60÷(35-25)
@@JJJJ-hp9oz LMAO... The Order of Operations were formally established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's... New Math is an excuse for people who fail to understand the basic rules of math... The correct answer is and always has been 24 not 6
You FAIL to understand what constitutes a TERM and you FAIL to understand that when written in an inline format only the number to the right of the obelus is in the denominator unless WITHIN a grouping symbol...
I'm not even that old and I was thought by all my math teachers that you would solve the multiplication next to parenthesis first regardless of from left to right, so I came up with 6. Blame my math teachers.
So I went directly to comments -> and found Alpharex Rex! You are my kinda guy 🙋 Saved me from even Watching the video. Clearly we made it this far in life, paying bills, so there must be Alternative Math that also works 😉
Me also
I am right there with you. I remember doing parens by distributing the 5 to multiply it by the numbers in the parens, so 35-25=10, so 60/10 was 6.
Me too. I was taught that 5(2) was a single term and should be simplified. Changing math rules is self destructive.
I made it 6 too .
50 years old and still waiting to encounter a problem like this in real life.
Right.
I'm wondering what units would be used for an equation like this?
I'm 27 and I use this equation all of the time.
Whenever I get stopped for drunk driving, the officer always asks, "Miss, how many beers have you had?"
I always say, "60÷5(7-5)"
While he stands there tryna figure it out, I slip out of the cuffs and steal his patrol car.
Voila!
Watch more youtube and you will get more of these. ;)
I guess you never saw the the man with binoculars
It's an equation for time travel. You'll be able to go back in time and get those software programmers to fix this problem first. Then we don't have to waste our time on problems like these.
The correct answer is 6. Not only do the brackets represent multiplication, they also come first in the order of operations. Thus, not only is it 5x2, but the 2 is in brackets, and so must be solved first.
You are incorrect.
What is INDIDE the brackets comes first.
Not multiplication to what is OUTSIDE of them.
I inserted 60÷5(7-5)= into an Excel spreadsheet. Excel accepted it as a text but did not recognize it as a formula that needed to be solved. I had to change the entry to =60/5*(7-5) in order for Excel to accept it as a formula and solved it with an answer of 24. However, if I changed the entry to =60/(5*(7-5)), Excel accepted it as a formula and solved it with an answer of 6.
Yes thats how formulas in excel work
thats how math works... in PEDMAS, when a multiplication and division are on the same line and you must choose between either one, you will go from left to right
yeah 6 is the right answer to that
Well, yes. All formulas in excel require you to start with = so it knows it's a formula to be solved. Also adding another set of parentheses will change the order you solve it. The point of PEDMAS/BODMAS is to standardise the way we solve formulas, and adding more parentheses to an equation will change the order in which you solve it
@@julesssssssss You must evaluate the terms first. The OP is wrong in his treatment here.
If that was rewritten as a fraction it would be 60/(5(7-5)), not (60/5)(7-5).
Inside brackets first was my first clue. Because it was not my money that the problem is about, I didn't have 50%-100% interest to solve it.
Wolfram re-writes it as 60/5 as a fraction, times (7-5). 24.
It IS written as a fraction. The rub comes in grouping what's in the denominator. The form 5(2) is really 5x2 which is not priority bound by PEMDAS. You are using parentheses to disambiguate to your preference. If you don't do that and evaluate by PEMDAS you get 24.
@@garymartin9777 That's not necessarily true. Remember in PEMDAS, multiplication comes _before_ division in order of precedence (M before D). So it's 60 over 5(7-5) = 60/10 = 6. Notice I did parentheses, THEN multiplication, THEN division. That's how it should always work.
What the video got wrong is that they interpreted M and D to be on the same precedence level. They are not. M is above D, and that's why it's 6.
You are not correct, it would be (60/5) would be the fraction with a *2 after, just as it is written. Don't try to change the equation please. Multiplication and division have the SAME priority, so whichever comes first is first.
I'm not going to rely on a calculator's "judgement" on what is ambiguous. The calculator is merely following rules programmed by a human that could have interpreted an ambiguous statement one way or another.
There's nothing ambiguous there. It's plain and simple, unless you were born in 1910 or something. Rules change, so people need to adapt and forget the old ones.
Flat earther
@@BypassOne I agree the problem is not ambiguous but I'm merely pointing out that a calculator result is not proof of the answer to the problem but merely the result of human programming, which is not infallible.
@@wildasiandude432 Recognizing the fallibility of humans and technology is not the same as Luddism.
@@percyfaith11 Human programming that is based on mathematical rules. Calculators were invented to easen and speed up calculations, exactly because people tend to forget them. So, believe me, the expression is not ambiguous just because YOU forgot the rules.
"It used to be 6, but now it's 24."
No!
lol 😂
The correct answer is 6. Multiplication is done before division.
@@Slowburn726 Multiplication and division are the same level. Addition and subtraction are the same level. The mnemonic should really be PE(MD)(AS).
@@briant7265 I think you mean PEM/D(A-S)
@@Amblin80s PƏM/D(A-S). Stacked exponents are evaluated right to left.
x^y^z = x^(y^z).
I think you have to work it as 5(2) then divide that into 60 because it's connected to the parenthesis. Work the entire parenthesis first, not just what's in the inside.. just like an exponent on the parenthesis. 6 is my final answer
That's how I was taught.
Agreed 100%. In the order of operations, multiplication comes before division and 5x2 has to be calculated before looking to the left to divide.
BOMDAS so correct answer is 6 ie 60/10
Actually using the distributive factor it would be 60/35-25. We were taught to use Pemdas in exact order not equal precedence.
It’s still 6
Groucho Marx - "One morning I shot an elephant in my pajamas. How he got in my pajamas, I don't know."
autophyte,
Do you think, he possibly bored through with his erect trunk.
And did it, sorry, did he survive?
@@onlythetruth883 who? the elephant or Groucho?
@@NatandGeorge
Unfortunately it seems as if they are one and the same. That's why I was wondering about its fate. Sorry, his fate.
Fun fact, all of the Marx brothers loved go to Alabama to shoot elephants for their tusks! Why Alabama, you ask?
Cuz everyone know that in Alabama, the Tuscaloosa.
@@lightningmacqueen4097 Boom -tsssshhhh
I think the ambiguity stems from the omission of the operator symbol when using parentheses, making the expression "5(7-5)" appear as though it should be viewed as a unit. If the expression was "60 / 5 x (7-5)" instead, any ambiguity would be lost (imo).
Yep, that's pretty much it.
The academic interpretation of 60÷5(2) is 60÷(5(2)). It's just the notation used for years and is in academic papers. Multiplication by juxtaposition was given higher priority so less brackets were needed. Feynman, for example, used this, and other, common shorthand notations. That's all they are, shorthand.
Writing a÷b(c) is now bad writing because of how popular programming has become. Programming views 60÷5(2) as 60÷5×(2) which means there is no juxtaposition so no ambiguity like you said.
Moral of the story is, the question in the video is flawed which you have noticed. Many, many other people in the comments are oblivious to this and just argue the answer is only 6 or only 24 but are missing the real problem completely.
.exactly!
Yes, I agree. We would never treat 60 / 5x as 60/5 * x in my algebra classes.
@@GanonTEK I'm not sure your correct about the programming thing. I was taking the 4 maths (Alg, pre-Calc, Geom, Alg II) back in the 80's before programming was a big thing, and the books we used in my poor little country school were not the newest and greatest. And I knew the answer was 24 based on my learnings back then (before your programming theory) Not saying there weren't programming and computers in the 80's, I had a C64 and programmed in Basic. But our books were probably from the 70's and it wasn't a "popular programming" thing that taught me to know the answer was 24.
A number or variable directly in front (or behind) of a quantity is considered part of the expression. The parenthesis are not just symbolic of a multiplication. They are also symbolic of the Distribution Property. The 5 in front of the quantity (7-5) should be read as 5 distributed (multiplied) to 7 and to -5 or the sum of the latter two. So 5(7-5) is equivalent to (35-25), one unit and can't be treated as separate. As another said earlier... 60÷5x would not be solved as 60÷5 and then the dividend multiplied by x. Also, substituting an x for any member in the equation 60÷5(7-5)=6 will give you the correct number back...i.e. X÷5(7-5)=6 will lead to X=60.
60÷X(7-5)=6 will lead to X=5. Etc. This will not be the case if you say the answer is 24. 60÷5(7-5) does not equal 24.
The answer is 24. 60/5*2 = 12*2 = 24. Multiplication and division have the same priority, so when no parentheses are present, perform the operations as they are encountered from left to right.
wrong
3:00 Exactly. As a programmer, I would consider that expression poorly written. The fact that a compiler can evaluate it unambiguously doesn't change that. It's always better to use parentheses to make the meaning clear.
But different compliers interpret it unambiguously differently!
@@NetAndyCz nope. None of the languages I have ever programmed in will accept it as is, they all require an explicit multiply operaror. And they would all produce the same answer: 24. Expression evaluators are different of course.
It is more how math programs and calculators parse it, anyway no one (or almost no one) argues about the order of operation for explicit multiplication.
TheMonk72: Try FORTH, it uses RPN and is totally unambiguous and relies on no order of precedence:
60 5 7 5 - * ÷ = 6
60 5 ÷ 7 5 - * = 24
So a majority of people believe 5+2×10=70 is this because the expression is ambiguous or because they don't understand the rules?? Did we have to use parentheses to make it clear 5+(2×10) or should these people have to learn the rules??
If you follow the Order of Operations as they are intended to be followed there is no ambiguity in 60÷5(7-5) there however are a lot of people who do not understand the rules and require crutches in order to evaluate the expression correctly...
I would have thought 6, but this is why I avoid using multiplication and division symbols, and instead use parenthesis and fractions. As long as the faction lines (especially if it's fractions within fractions) are correctly sized, there's no ambiguity.
60
-----(7-5)=60÷5(7-5)
5
60
---------- = 60÷(5(7-5))
5(7-5)
A vinculum (horizontal fraction bar) is a grouping symbol and groups operations within the denominator and when written in a inline infix format extra brackets are required to maintain the grouping of operations within the denominator... Remove the grouping power of the vinculum, replace it with the grouping power of another grouping symbol i.e parentheses...
@RS-fg5mf
The first equation I'd write
60
---- (7-5) = 60÷5*(7-5)
5
While
60
------- = 60÷5(7-5)
5(7-5)
Because I learned that a missing multiplication sign indicates a closer connection and so is seen as if it being in parentheses with the following term. So 60÷5x is not equal to 60÷5*x
I agree
It is 6 using the Distributive Law of Mathematics. There IS NO modern interpretation outside of America. A student at Oxford, Cambridge, Sydney, Fudon etc DO NOT depart from the law!
There is no ambiguity either!
@@garyquinlan4075 To be fair, Order of Operations is useful for... oh, about three weeks during pre-algebra - until you start learning the properties of addition and multiplication. The problem is, some people rely on it and apply it far beyond its pedagogical purpose. No mathematician, scientist (to include economists), or engineer should ever reference it.
I’m 65 and calculated I’d rather have another beer ......
Case study!!!?
A good calculation.
@@toddruthig4048 : Spot on !
@@kurtfrancis4621 : Cheers 🥃
This is the first honest answer a young person (65 I know) has given. I will also have a cooling ale.
The fact that this continues to come up is evidence that there are two very different interpretations that have been taught to various people depending on when and where they were taught; and because this is the Internet, people are more than happy to boldly proclaim the other side to be wrong. FWIW, I was taught in school that the 5(7-5) is resolved completely before the division.
The answer seems really to be more explicit, brackets are cheap.
Think about this. This situation and debate about the correct way to solve an equation has come up before, I am sure. Engineers and physicists need their proofs to be interrupted accurately for peer review. There is no room for misunderstanding. I googled pictures of famous equations and I found no ÷ signs. They don't use them. Maybe we should abandon them entirely.
"I was taught in school that the 5(7-5) is resolved completely before the division. "
You were taught wrong.
That isn't a valid interpretation.
That is just you having been given false information.
There is nothing in math saying to include the 5 as part of the ()
Sorry
@MrGreensweightHist actually there is. Saying 6÷5 is the same thing as 6/5. The division symbol replaces a fraction, quite literally showing this to a trig professor and an a math calc professor, both have said 5(7-5) is the denominator. There is a reason they don't use the division symbol anymore and just use fractions.
@@Dogsparkster I am sorry you have bad teachers.
"There is a reason they don't use the division symbol anymore and just use fractions."
The division symbol IS a fraction bar
3÷4 is 3/4 is ¾
The reason ÷ isn't used anymore is simply because / is one line while ÷ is a line and two dots.
/ is faster to write.
that's the ONLY reason it changed.
X however, became * because X is too easy to confuse with the variable x.
using X instead of * can cause confusion.
Using ÷ instead of / alters nothing.
@@MrGreensweightHist yes that is the type of cocksure reply I expect from UA-cam comments, thank you. Juxtaposition having higher precedence than explicit multiplication or division is a long accepted notational convention that doesn’t appear to be universally accepted because it contradicts the sacred PEMDAS rule children are taught in elementary school, hence these internet controversies that continue to spring up. Since we aren’t in 1890 and trying to minimize characters when printing equations in books, we can all just be more clear for everybody’s sake and use more brackets.
All through school and into industry I have never seen anyone write an equation using the division symbol when writing by hand. It’s always a horizontal or forward slash line, and to make it clear it will be a horizontal line with the numerator over the denominator. The devision symbol used daily is somewhat recent and comes from spreadsheets and coding where one is forced to write on a single line. So to get over the confusion lots of parentheses are needed on that single line or these debates occur because the equation is unclear to people thinking in terms of math equations written on paper.
Correct, especially when doing algebra!
"I have never seen anyone write an equation using the division symbol when writing by hand. It’s always a horizontal or forward slash line,"
They all mean the same thing.
Honestly does not matter which one you use.
60÷5(7-5)
is
60/5(7-5)
is
60
----- (7-5)
5
All the same problem.
These types of problems are a waste of time.
There is a reason nobody doing actual math uses the ÷ symbol. Hell, it's not even on your computer keyboard.
A proper division bar ---------- would eliminate any ambiguity of whether the multiplication should take place before or after division.
and neither are ×, ½, ² or ³ on the keyboard
and neither is this: ☺😊😀😁😂😃😄😅😆😇😈😉😯😐😑😕😠😬😡😢😴ETC.
are your examples complex math symbols or what? I see a black and white smiley face, followed by 20 rectangles with ? in it, followed by E, T and C. i.imgur.com/vWjbBlW.png
@ groszak1: Looks like your browser, which appears to be designed for the Polish language, isn't able to render emojis. The characters that Brandon typed was a series of smiley faces and other emojis. I assume he was using a mobile device.
(And the "ETC." is for "etc." [should be lower case], an abbreviation for the Latin phrase "et cetera", which means "and so forth" .)
Anyway, markgriz is right. Nobody in real life ever uses the ÷ symbol.
The first one is a regular smiley face, not an emoji. I intentionally deleted the annoying colored emoji fonts from my Android tablet. And please don't criticize the division symbol.
Whole idea behind this video to create controversy which will eventually lead to more people watching this video, helping this smartass channel.
exactly! he's just trolling us now. I doubt this problem was ever even on facebook.
LOL exactly
I KNOW!
On one hand, it is, but on the other hand, it isn't.
Yeah the whole video is just getting people to argue about well defined methods of solving fractions,,,,the cardinal rule is, multiply and divide before you add and subtract but you must solve the maths in the brackets first(---)
*WRONG! The 5(2) must be solved first to remove the ( ) from the equation. The answer is 6.* What you are doing is INCORRECTLY changing the ( ) to a X between the integers...and THAT is your flaw. The two are NOT the same, even though we treat them the same way when we get to them..the difference is WHEN we getto them. You are "getting to them" too late. *The parenthesis must be addressed BEFORE multiplication/divisions are addressed left to right.* #PEMDASnotDEPMAS
Ah Math, in the 1970s, I was taught that the parentheses rule also includes any 'assumed' multiplication, thus the 5 gets multiplied by the (2) first before any other expressed multiplication and division. Hard to do it differently.
There were some groups that tried to push for implicit multiplication to have higher precedence in the 70's and that's part of the problem.
Unfortunately, giving implicit multiplication a higher priority breaks the Order of Operations. The Commutative Property. The Distributive Property and the Multiplicative inverse of division...
Solve these two expressions...
60A(7-5)=24....A=?
60A(7-5)=6....A=?
I will explain later when you have answered...
@thegrandfinale2 5a is a coefficient/variable bond. Real numbers do not have coefficients. 5(2) The 5 is not a coefficient of 2 and is nothing more than implicit multiplication.
5a does not equal 5(2)
5a would equal (5*2) when you replace the variable with a real number you break the coefficient/variable bond and parentheses are required to maintain that bond.
Parentheses only give priority to (OPERATIONS WITHIN) the grouping symbol not outside....
60÷5(7-2)=24
60÷(5 (7-2))=6
@thegrandfinale2 the historical usage was due primarily to text book publishers pre 1900's that were trying to save page space by using the obelus like a vinculum. The printing technology at the time was cumbersome and using a vinculum took up more vertical space than was acceptable to the publishers so they pushed the misuse of the obelus even though it was in direct contradiction to the Order of Operations...
As for the two equations.
60A(7-5)=24..... A=0.2 or 1/5
60*(1/5)(7-5)=24
What is the divisional reciprocal of 60*(1/5)....WELL that would be 60÷5....So. 60÷5(7-5)=24
60A(7-5)=6....A=.05 or 1/20
60*(1/20)(7-5)=6
What is the divisional reciprocal of 60*(1/20) WELL that would be 60÷20....So 60÷20(7-5)=6
If 60÷5(7-2)=6 then 60a(7-2)=6
a should be equal to 1/5 but it isn't.
I hope that helps.
@thegrandfinale2 you are still missing the point. A variable because it can be any number or value is in and of itself a symbol of agragation... x = 5+5 that does not mean you can write 10÷5+5 you have to write 10÷(5+5)
When you have x= 2*3 and you have 6÷2x You do not write 6÷2*3 or 6÷2(3) you would write 6÷(2*3)
There really is no reason other than laziness to use implicit multiplication in a basic arithmetic expression.... One reason implicit multiplication exists is due to the possibility of confusion between the multiplication sign x and the variable x in an algebraic expression.... 2x(1+2) does this mean 2 times (1+2) or does it mean 2x times (1+2)...
In any case I have at least 6 different math books from 1907 to the present and they all clearly state that you evaluate what's WITHIN the grouping symbol and that ALL multiplication and division can be evaluated equally from left to right.... Show me a math book that states something to the contrary...
Also.... 60a(7-5)=6 Why would a not equal 1/5 ?? 60÷5 = 60*(1/5)
60*(1/5)(7-5)=??
60*(1/5)÷(1 (7-5))=??
60÷5÷(1 (7-5))=??
All multiplication can be changed to division by the reciprocal.
All division can be changed to multiplication by the reciprocal.
Do you believe that implicit multiplication should override the Order of Operations and invalidate the Commutative Property. The Distributive Property and the Multiplicative inverse of division??
@thegrandfinale2 I agree with you on two points... The notation confuses people, but is this the fault of the notation or the lack of knowledge many people have about math?? Primarily, they do not follow the Order of Operations and choose to see 5(2) as a grouping. BUT math books for more than a Century have clearly stated that you evaluate what's INSIDE the grouping symbol and ALL multiplication and division are to be evaluated from left to right.
I also agree that there is not a valid reason to use implicit multiplication in an arithmetic expression except for laziness . Use an explicit symbol like it's meant to be used and use parentheses the way they were meant to be used....
I have friends who believe that variables should not get to break the rules and believe that 6a÷2a=3a^2 and not 3÷a AND I argue with them as well. There is plenty of text book evidence that demonstrates 6a÷2a= 6a÷(2a) and not (6a÷2)a .... You break the coefficient/variable bond by seperation. 6a÷2a ≠ 6a÷2*a or 6a÷2(a) as there would be no need to put parentheses around the a unless you intended to show seperation of the 2a .
Do you agree that 60a(7-5) equals 60÷(1/a)(7-5) ?? If not how do you resolve the conflict between the coefficient/variable bond 60a and the implicit multiplication of a(7-5)
Is it (60a)(7-5) or 60(a (7-5))??
My answer was 6. That is how I understood order of operations. I was probably taught that exception to the rule or decided it through my own intuition. Look at the large amount of space between "60," the division symbol, and the rest of the expression (I stopped the video at the four minute mark, in case you mentioned this). Then notice how the "5" is hugging the parentheses. I would be inclined to calculate physically closer, i.e. closer on the page, operations before more spaced out ones. I would venture a guess that when writing math textbooks and the like, it is standard practice to punctuate in ways people will find intuitive.
The division symbol itself, resembling a fraction, also implies the order that results in 6.
WRONG... Parenthetical implicit multiplication does not have priority over division. In fact the TERM not just the factor outside the parentheses is attached to the parentheses. TERMS are seperated by addition and subtraction not multiplication or division. 60÷5 is one TERM attached to and multiplied by the value of the parentheses 2. The correct answer is 24
A(B+C)= AB+AC where A is equal to the TERM not just the factor outside the parentheses.
A is the monomial factor outside the parentheses to be multiplied by the value of the binomial factors inside the parentheses or to be Distributed across the two TERMS inside the parentheses that makeup the binomial....
A=60÷5
B=7
C= -5
60÷5(7-5)=
60÷5*7-60÷5*5=
12*7-12*5=
84-60=
24
I don't know what you're writing all that for. I did watch enough of the video to get to that part. Maybe you missed the part about the "exception to the rule."
@@MoonJung82 the video doesn't explain that it isn't an exception to the rule. Prior to 1917 some textbook printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space for the printing methods at that time and was more costly to print. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math so the ERROR was corrected post 1917.... Get it?? ERROR
Would you have evaluated 60/5(7-5) any differently... using a solidus rather than an obelus??
@@RS-fg5mf So you're saying the video is wrong? 2:05 "Historically, this division symbol had a special meaning when you wrote it in text..."
Go yell at him if you're sure of that. I believe I was taught that special case--that it was not ideal but allowable. By the time we reached high school, most of us had been weaned off of using ÷.
You may also find it interesting that the video title reads:
"60÷5(7-5) = ?"
and in the video itself, it's a little different:
"60 ÷ 5(7-5) = ?" (spacing)
Writer's intent matters and I wonder if the guy making this video added those spaces with that in mind.
"Would you have evaluated 60/5(7-5) any differently... using a solidus rather than an obelus??"
Yeah, I would have said you forgot parentheses, either (60/5)(7-5) or 60/(5(7-5))
And obviously, those added parentheses would also clear up the confusion of the original expression.
This video does not primarily present a math problem but a communication problem. I responded in kind and your attempts to correct me are missing the point.
Well said.
If UA-cam recommended this to you, it knows too much about you.
Probably
I mean, not necessarily. I never answer these questions because I know they're made to start arguments
The list of people with the wrong answer will help scammers rip these people off.
I feel called out.
I sometimes refer to myself as a recovering math head (I aced the math ACT test and was second in my high school in the MAA competition at the age of 16) but I am not recovering all that well. When I see this mistake being shown as the correct answer I cringe. Evaluate the numerator, evaluate the denominator, then divide.
_ One cannot use the calculator as the basis for the correct way to proceed.
_ It depends on the rule one establishes for evaluating mathematical expressions.
_ Engineering, physics and other fields of mathematics do not generally use PEDMAS as the procedure for evaluation of mathematical expressions. They establish their own rules of precedence, usually stating that multiplication ALWAYS precedes division.
_ If all these professions accept PEMDAS as their new rule then it will become common, but I do not believe that it is so currently.
This is not a real math problem, this is a problem about how we understand the order of operations and using parentheses when uncertain:
So:
60/5*(7-5) can be:
60/(5*(7-5))=6
or
(60/5)*(7-5)=24
Why are you putting parenthesis around 60 and 5 though
Basically they added that because the quotient of 60 and 5 and the difference of 7 and 5 is the one to be multiplied
Adrian Goia i think its 60 divided by 52
See this makes sure that it’s not 24, because if you write this sum as a fraction of 60/ 5(2) you simplify it down to 12/2 which is 6
Alexander Gandy Or, depending on how you would prefer to do it, you could do 60/5=12 and 12(2)=24
I got six because the 2 was still in parentheses. The confusion lies in when a number is next to variable in parentheses it implies multiplication. But because of how it's written, one could make that mistake of multiplying 5*2 first because according to order of operations, parentheses come first. So therefore I wont fault people who come up with 6 or 24.
@@anthony420181 but pemdas would still have multiply first so it would be 6
@@anthony420181 right to left? Doesnt that make it 60÷10?
@@anthony420181 multiplication was before division though since it's the M also if its from left to right then why did the parenthesis get done first on the right (even though i know why)
@@anthony420181 The ambiguity is that there is no agreed upon convention on whether multiplication by juxtaposition implies grouping also or not.
I.e. does 5(2) = (5×2) or 5×2?
Both are widely taught and used.
60/(5×2) gives 6.
60/5×2 gives 24.
It's simply a poorly written expression that no one would ever write now. It doesn't follow modern international standards like ISO-80000-1 which mentions this case of division on one line with multiplication or division directly after and that brackets are needed to remove the ambiguity.
@@anthony420181 Many scientific calculators give 6 also. It's very common to find one that gives multiplication by juxtaposition higher priority. Especially with some companies.
I've a brand new Casio myself from last year that does it for example that's marketed for schools and colleges.
Microsoft Math gives both answers on screen at the same time (one as the answer and one in the box you typed in after it converts it to a fraction notation but they don't match).
I think it is unintentional but highlights the ambiguity.
I've seen photos of other calculators with the same equation in both and two different answers for the output. If you search "The PEMDAS Paradox" you can get an article and academic paper by a PhD student on this very ambiguity which has the photo I've seen included.
If you search "Ambiguous PEMDAS Harvard" you get an article and a separate short document on the ambiguity also with references and documents.
Wolfram Alpha's article on the Solidus is another place that mentions this ambiguity.
If you search PEJMDAS from The How and Why of Mathematics on UA-cam you get videos, with references, which talk about the interpretation that gives multiplication by juxtaposition higher priority.
There is alot of evidence out there backing up my claims. I wouldn't say it otherwise.
The question in the video is written poorly. For any reasonablely written question, most likely all calculators, online and offline, agree on the same answer as well as people who learned PEMDAS or PEJMDAS.
If written as a fraction, and expand the brackets you get 60/35-25, which is also equal to 6.
I’m surprised that this method wasn’t mentioned.
It wasn't mentioned as over in America they seem to teach that multiplication by juxtaposition does not imply grouping so to them this is generally the taught method.
You also need a bracket there with what you wrote:
60/(35-25) and 60/35-25 are not the same answer when written on one line.
However, it seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping though, or it may not, so the notation is ambiguous making both answers valid. It depends on context (academic or programming).
Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it.
The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation.
Wolfram Alpha's Solidus article mentions this ambiguity also.
Microsoft Math gives both answers.
Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus.
Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274)
"If an arithmetic or algebraic term contains both ÷ and ×, there is at present no agreement as to which sign shall be used first..."
It then goes on to say that brackets should be used to "avoid ambiguity in such cases"
"The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue)
"Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous and says, "shame on that person for writing an ambiguous expression".
"Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division".
"Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause.
Other credible sources are:
- The PEMDAS Paradox (a paper by a PhD student on this ambiguity)
- The Failure of PEMDAS (the writer has a PhD in maths)
- Harvard Math Ambiguity (Cajori's book above is talked about here)
- Berkeley Arithmetic Operations Ambiguity
- PopularMechanics Viral Ambiguity (AMS's statement is here)
- Slate Maths Ambiguity
- Education Week Maths Ambiguity
- The Math Doctors - Implicit Multiplication
- YSU Viral Question (Highly decorated maths professor says it's ambiguous)
- hmmdaily viral maths (Another maths professor says it's ambiguous)
The volume of evidence highly suggests it's ambiguous.
But now you must follow the order of operations. 60/35=1.7143 then subtract 25 giving -23.2857.
That is because they dont actually solve the equation, they input it into a calculator from left to right without any thought to how calculators operate. this gives them an answer of 24 which they then seek ways to justify the answer given by the calculator solving 60/5*1(7-5)=x instead of solving 60/5(7-5)=x.
That correct.
@@BabySuzunaTechnically, solving left to right would still equal 24. 60/5=12. So 12(7-5) is distributed as 12*7=84 minus 12*5=60. 84-60=24.
The problem I have with this kind of problem is that it is not really math. It’s grammar. Just write the darn expression in unambiguous way so we can do actual math. We have more interesting concepts to learn in geometry, trig, calculus, etc.
100%
I wholeheartedly agree
@@jakemccoy Yea, I agree.
@@UniversalS757 Don’t worry. Math grammar has correct answers too, but math grammar is different than math concepts. I have been an engineer working in the real world for 30 years. Not once have I debated stuff like this on the job. I will just put parentheses in there and keep it moving. This is a discussion that may be fun, but it needs to stay on academia.
@@jakemccoy ok
I think I was taught (a long time ago) to do inside the brackets first and then anything outside the bracket to the bracket answer next.
that is the way i was taught. todays calculators do not allow that method of calculation. i believe the new method to be wrong
That is how I learned it too
you're close. it's about order of operations.
first parentheses
second multiplication and division are at the same operation so left to right.
so... 60/5=12 * 2 = 24
@@arizona_anime_fan The correct answer is 6
it feels like that was tattooed on my brain cells.
I was taught GEMDAS. G stands for groups to encompass both parentheses and brackets. I took advanced math throughout school and came up with 6.
Ronald Dunn
11 hours ago
Division is just multiplication of fractions. Rewrite the expression using this idea and there is no ambiguity.
60 x (1/5) x 2 = 24 (advanced students should understand this)
FAIL!
@@gizzyguzzi there has to be that one person...
"I took advanced math throughout school" lmao
You had to divide the 60 by 5 after you subtracted the 7 and 5
Me: when will I use this in real life
Math teacher: 13 years later on UA-cam
BRILLIAN
All the math we have is not meant for real life. It exercises and trains your brain to make it sharp so that your brain works instantaneously and perfectly to find a solution to your real life problems and also to help in your decision making..!!!!
😂
LOL!
@@RajaBabu-oe4be This was a JOKE......relax.....
The correct answer is 6. If the equation is solved BASED ON THE WAY IT IS WRITTEN, which defines the way the mathematical operations should be executed the answer is 6. Looking at the written problem there are no brackets ( ) surrounding the 60/5 therefore it is not done first. The brackets determine the order of operations and if they are not there on the left side then that division operation is not done first. Respect the rules of: order of operations.
From the first calculation the 5 was followed by 2 in brackets, and so that has to be resolved first. This gives 10, which divided into 60 gives you 6, which is the correct answer.
WRONG. 5(2) is not a bracketed priority and is exactly the same as 5×2
Brackets only group and give priority to operations INSIDE the symbol not outside the symbol...
The correct answer is 24
@@RS-fg5mf So , it should have been written as 5x2, NOT as it was written to create a confusion. I thought mathematicians were meant to be precise :)
@@paulbiddlecombe3279 don't blame the expression for your failure to understand it correctly....
60 boxes are delivered equally to 5 locations. Each box contains 7 winter coats and 5 of those coats are childrens coats.
How many adult coats did each location receive?? 60÷5(7-5)= 24 adult coats.
60 adult coats are delivered equally to 5 locations. Each location had 7 people waiting for coats. 5 of these people are children... How many coats could each adult receive?? 60÷(5(7-5))= 6 coats per adult.
60÷5(7-5) EQUALS 60÷5×2
@@paulbiddlecombe3279 the only people confused are the people that do not understand basic arithmetic. The symbols used to represent each operation changes nothing. I really do not understand why so many people don’t understand the answer is 24
Paul, You are correct. They are wrong. I am a chemist and in chemistry we would get 6. Here's why: The expression written as 5(7-5) is to be treated as a single expression because it shows a relationship between the 5 and the 7 and the 5 and the 5. This expression IS TO BE TREATED AS A SINGLE UNIT because the parenthesis is used to create the relationship. So 5(7-5) = (5x7)-(5x5) = (35)-(25)=10. The rewrite of the expression would be 60/5(7-5) = 60/10 = 6 The 5(7-5) is to be treated as a single expression. It is NOT (60/5) x 2. And it is NOT 60÷35-25 either. It's 60/5(7-5)
I love the emphasis that 24 is the answer according to the modern interpretation of rules of operation.
It really drives home the point that order of operation is a convention rather than a fundamental law of mathematics.
It's important to keep to the convention to avoid difficulties communicating with others.
But in theory, all mathematics should be possible if addition/subtraction came before multiplication/division. We would just write things differently (and I suspect would be forced to use much more parentheses.)
This is why the Order of Operations and the various properties and axioms of math were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses when dealing with inline infix notation....
As for Addition having priority over Multiplication, if that were the rule it would work but it wouldn't make much sense since Multiplication is shorthand notation for repeated Addition...
As it stands the Order of Operations and the various properties (LAWS) and axioms of math work logically and consistently across the board...
@@RS-fg5mf I was thinking more along the lines of we have 2 sets of eggs.
Example: One is 6x2, the other is 12x8
Writing it out as 6x2+12x8 without multiplication as a higher priority than addition, we would need (6x2)+(12x8)
In order for addition to make sense, we would need a setup more like:
Example 2: We have an L shaped array of eggs container that is 3 rows by 2 columns. We then add 2 more rows and 5 columns. How many eggs can the container hold?
Writing it out as described would be 3+2x2+5.
In this case though, you need to solve addition first. So parentheses are required for our current order of operations (3+2)x(2+5), but would not be required if addition was higher priority.
In the real world though, example 1 is far more common than example 2, which is why we developed the multiplication priority.
That's my best guess anyway. Your guess about multiplication being shorthand for repeated multiplication may also be correct.
@@plentyofpaper I understand what you're saying. What I'm saying is it wouldn't make logical sense to do it that way since Multiplication is shorthand for repeated addition...
(5×3)+(4×2) would still equal 3+3+3+3+3+2+2+2+2
@@RS-fg5mf Yup. More than one way to reach the same conclusion.
The mistake is that once the operations INSIDE the parenthesis are done, they should be replaced by the correct operand. 2(3) is the lazy way to say 2x3.
6 when I graduated but wth, nothing makes sense anymore.
Yep. I thought this was easy. 6. Then I read comments. 😵😵
WHAT??? You GRADUATED and you got that wrong? Get a refund for all you spent to get that graduation. U woz robd...
The thing that I don't understand that they're doing in the "modern" interpretation of this problem is that they are just dropping the parentheses after calculating (7-5), so 5(2) becomes just a regular multiplication, not a parentheses calculation now?
Or everyone else got it wrong. Wouldnt be the first time 😘
@@towmlvb3423 5(7-5) is like a single sentence therefore the answer is 6. If it was written like 60/5*(7-5) then that will be different. Like seriously is it that hard to understand that the way u write it will determine the answer?
Solve Parenthesis means to remove the parenthesis from the equation first. Before left to right the parenthesis marks need to have been solved (removed from the equation). You can't move on to the next step in either PEMDAS or BODMAS until the parenthesis have been solved and thus removed. 60/5(2) still contains parenthesis (brackets) and it would be outside of BOTH methods to continue while the presence of a parenthesis/brackets still exist. Regardless of the fact that it is a multiplication it is still a bracketed expression. 60/5(7-2) and 60/5*(7-2) are not the same expression. This is also why it is not ambiguous.
There's one guy at the bottom of the comment section that says it's 3 lol
Walter Ostrowski damn you really gonna call him out like that
Ded
I want to like this, but you're at 123 lol
Hahaha
Hahahhahahahaha I don’t know why I lost it to your comment, made my day!
I was right, what a great use of my time to find out. Also, I once saw a man with binoculars with binoculars.
The most popular guy at the nudist colony didn't have binoculars at all.
He was the one carrying the 10 glazed donuts and two cups of coffee!
Why did his binoculars have binoculars?
I was taught calculations are done in a specific order. Given Inside parentheses is always done first.
1. Multiplication
2. Division
3. Addition
4. Subtraction.
So my answer is 6, & that's what my teacher, who was never wrong, would accept.
Your teacher was wrong if they agree with the wrong answer.
@Charles Mosley the equation has a division, a multiplication and subtraction operation. Which of these operations do you find ambiguous?
B. S. G.: Maybe your teacher was never wrong, but if you answered 6 you are wrong.
NO... his teacher was right. And the way he described the order of operations is correct. The YT content creator also stated the order correctly - but then... for totally INEXPLICABLE reasons, what he entered in Google was a DIFFERENT EQUATION. Sheesh!
If you view 5(7-5) as 5 being factored out of (35-25) you end up with the second binary tree. This is what we were still taught in the 60's and 70s, the parenthetical surrounding 5(7-5) is inferred, and this is NOT the same as 5*(7-5) where the parenthetical is not inferred.
I feel like I was lied to my whole life. In school, the way I was taught, it would be 6. The parentheses isn't solved till the exponent is factored in, then a multiplication that is before it. That was the way they taught it in the 80's at my school.
I'm 90's gen, This is what I was taught too...
If I remember correctly when I was in junior high at the end of the 60s the part of the equation in parentheses was solved first then the rest of the equation.
I was taught this in the 60's ; it was 24.
Correct order of operations:
ua-cam.com/users/shortsMaPZGyudFzo
60/5(7-5); expression
A=60/5=12; first operation and step.
B=(7-5)=2; second step, group, do together.
A(B)=12(2)=24; final step.
Or use Distributive multiplication
A(7)-A(5)=84-60=24; parenthesis content not done first or together.
Video : ua-cam.com/video/y9h1oqv21Vs/v-deo.html, great message at end.
___
BIDMAS
Brackets refers to any part of the equation that is in brackets. These should always be complete first.
Indices simply means to the power of. For example, 3² or 5³.
Division and Multiplication: Starting from the left, work these out in the order that they appear in the equation. If multiplication appears first you should complete this before division.
Addition and Subtraction: Also start from the left and work these out in the order that they appear in the equation. If subtraction appears before addition, you should complete this first.
The way they taught us in school in the 80's was Bedmas; Brackets, Exponents, Division and Multiplication, Addition and Subtraction. And left to right. It seems there were a lot of schools teaching different ways of interpreting the order.
I sincerely hope that at your school they weren't calling that '5' an exponent. And therein lies the rub.