The general rule of thumb in base 10 is to put the repeating part (In this case, it is "27") over all 9s. So, in this case, you would have 27/99. This would eventually reduce to 3/11.
This was a very neat trick. I solved it by brute force myself. I figured that the result was just a little more than 1/4, so I starting adding 1 to the numerator and dividing the bigger numerator with (4 * numerator) - 1). So 1/4=0.25. 2/7=0.2857. 3/11=0.272727 and Bingo
For some reason, we, (1970's) were NEVER taught this in High School. I had to learn this on my own. I did it algebraically. If it is 1 decimal repeating, use 9, 2 decimal use 99, 3 decimal 999 .... etc. And yes, I figured this out the way you show it in this fine video.
It would be clearer if you always put a horizontal bar over the last "27" in all cases that it applies. Then you wouldn't have to keep explaining that it repeats.
This is great! So if I understand, take the repeating portion of the decimal, move the decimal point to the right - that’s the numerator - now in the denominator, put a 9 under each digit above (16:46). Shouldn’t this have a name, like “John’s Rule of Nines?”
The method shown does work but there's a much easier method if there are no non repeating digits before the repeating digits, as here. For the numerator, write down the first string of repeating digits. Here that's 27. For the denominator, write as many nines as the number of digits in the numerator. There are two digits in the numerator so 99 is the denominator. 27/99 is 3/11. Again, if there ARE non repeating digits before the repeating digits, the solution is more complex because you would need to create a second fraction using the non repeating digits (using different rules) and then add the two fractions. Example: .037037. The numerator is 37 and the denominator is 999 since there are three digits in 037. 37/999 is 1/27. Using this method, .272727 can be converted to a fraction mentally.
@@nixxonnor Yes. Just break up 2.72727272727... into 2 + 0.727272727.... Then the desired fraction is 2 + 72/99 = 198/99 + 72/99 = 270/99. I wish I could take credit but I just expanded a little on the method presented by nickcellino1503. Thanks, Nick!
@@nixxonnor the 2 is a whole number so we focus on the right of the decimal to start. This gives us 72/99. which is 8/11. Now for the 2. We can have it as a mixed number fraction, which is 2 and 8/11. or You can have a top heavy fraction which would be 30/11. (think 5.25 well that is just 5 and 1/4. You could also say it is 21 quarters.)
3/11, worked it out before opening the video, so I'll be fascinated to see what your methods was. I did iit like this, 27/100, so just a little over 1/4. You need a repeating pattern and using integers, I went next up to 3/11 was the first try. I am sure that is not scientific though! Best wishes from George. PS: Not taught this, and was born in 1961 in England.
3/11. slightly bigger than1/4, less then 1/3. Did a few trial and errors to see the pattern between changing the numerator and denominator by a digit to see where the pattern goes. presto, 3/11. Now I'll see how it was meant to be done.
That is an interesting and clever trick. But I solved it in a different way. x = 0.27272727.. It was said that you could use a calculator, so I used it to get: 1/x = 3.66666666667 and I can imediately recognise the .6666666667 part as related to 2/3 So then 3 x (1/x) = 11 or 3/x = 11 -> x = 3/11 and my calculator checked that as correct A different denominator may not have been recognised so your method is better, thanks.
Why 100x... that is because of the repetition of 27. If the decimal was 0.666666666... you could use 10x = 6.66666666 and 9x = 6 so x = 6/9 = 2/3 If the decimal was 0.217217217... you would use 1000x = 217.217217217217... and 999x = 217 so x = 217 / 999 If the decimal was 0.134913491349... you would use 10000x = 1349.134913491349... and 9999x = 1349 so x = 1349/9999
This was a wonderful method and I am certain I was never shown this in school (1960's). I am also certain that I would never have been able to figure out this method on my own. But I did get the answer a different way. First, I assumed that since only the first two decimal places repeat, then the numerator and denominator are going to be small numbers. (I'm not sure if that assumption is always correct.) Then I took the fact that .27 is a little bit larger than 1/4. So the numerator and denominator are going to be only one or two digits, and I'll try fractions which are a little bit more than 1/4. Let's increase the N and D by 1. That's 2/8 (which = 1/4). Now, a little bit larger than 2/8 is 2/7. That didn't work. So let's add one more to the D and N to get 3/12. Now, a little bit more than 3/12 is 3/11. That turned out to be right. That worked with .272727..., but I'm not sure if it would work so well with any other two-digit repeating decimal. Edit: I've been playing around with this and discovered that ALL fractions with 11 as a denominator are two-digit repeating decimals! And there is no other denominator which does that. Also, each denominator has its own pattern of repeating decimals. And if you go to twice the denominator (say from 7 to 14) then that decimal will have the same number of repeating decimal places, only starting with the second decimal place, not the first (unless the fraction reduces). For example, 3/22 is .136363636.... 5/7 is .714285714285... but 5/14 is .3571428571428.... All 7ths have a six digit repeating string, and amazingly, it's the same six digits in the same order, only beginning with a different digit depending on the numerator! 1/7 is .142857142857... 2/7 is .285714285714... 3/7 is .428571428571... WOW! (To which, the math gods would say, "Well, duh, it does that because...".)
Now, a more interesting problem will be to figure out 1/12. This is because 1/12 = 0.083333333 ... This will involve multiple techniques, including what you demonstrate in this video.
As I explain above, the decimal 0.428571 can be converted to the fraction 428571/999999 which can indeed be expressed as a simplified fraction, which is 3/7.
My scientific calculator on WIndows 10 tells me that 7/31 = 0.22580645161290322580645161290323 Repeating unit is 225806451612903. How would you approach that?
You put that repeating part over all 9s. The only difference is that it will take a lot longer to reduce it. You basically have to go over all the prime numbers that are less than or equal to the square root of the part that is left over after all the reductions that you have done up to that point and more than the prime numbers that you already went through to make sure you finally got the reduced fraction for this repeating decimal. 2 no; 3 yes --> 3 times to get 8,363,201,911,589/37,037,037,037,037; 5 no; 7 yes but no since only the numerator is divisible by 7; 11, 13, 17, 19, 23, 29, 31 all no; 37 yes to get 226,032,484,097/1,001,001,001,001; 41 yes to get 5,512,987,417/24,414,658,561; all succeeding primes under 271 no; 271 yes to get 20,343,127/90,090,991; all succeeding primes under 2906161 no; 2906161 yes to get 7/31. Of course, you don't want to try to do this one by hand since you would have to go through many primes to reduce this. You might want to use a computer or the internet to find the prime factorization of the number 1,001,001,001,001, which is what I just did, to help you out with this one. LOL.
Actually, what I should have done is: 2 no; 3 yes --> 3 times to get 8,363,201,911,589/37,037,037,037,037; 5 no; 7 yes but no since only the numerator is divisible by 7 to get 7 times 1,194,743,130,227/37,037,037,037,037; 11, 13, 17, 19, 23, 29 all no; 31 no but yes since only the denominator is divisible by 31 to get 7 times 1,194,743,130,227/1,194,743,130,227 divided by 31; At this point, you should notice that there is a 1,194,743,130,227 in both the numerator and denominator. This should then easily reduce to 7/31. You really don't have to worry about primes when you have the same thing in both the numerator and denominator. As you can see, doing it this way was better than doing it the way I showed in the previous comment. It is better to show the isolation when you have a prime that is in one place but not the other as was the case with the 7 and the 31. Of course, I still needed to use the calculator anyway to try out all the primes that I did try.
Why not teach students logical short-circuit and very easy logical steps. If we have after decimal points only the period of the periodic fraction the student already have the numerator. The denominator has as much nine's as the number of digits in the numerator. That's it. 0.(27) = 27/99 0.(4) = 4/9 0.(15) = 15 /99 0.(123) = 123/999 and so on.
I don't think 0.1666 is a repeating decimal. The 6's repeat, but the 1 is just hanging loose and not repeated at all. Even though 1 ./. 6 = .166666 . . . . this technique does not seem to work.
@@terry_willis You should know that ALL fractions have repeating segments when in decimal form -- it's that which separates them from "irrational" numbers.
@@saltydog584 Close enough? Is 28/99 better or perhaps 26/99 ??? Or what about 26½/99 or 27½/99 ??? The answer 27/99 is spot on, just needs a little simplification but 100% correct.
I understand the method and it's very clever, but I don't understand why it works. I'll try to meditate on that. I could feel my IQ go up a little bit while viewing the video
@@MrMousley True, but there is nothing wrong about a correct answer that only needs a little clean up. Because 8/16 = 1/2 so the equation is true. Type 8 / 16 in your calculator and you will probably get 0.5 Now repeat this with 1/2 ... also 0.5 ??? So both answers are true !
Oh dear John.This very same technique was used to prove 0.999.... =1 (which is wrong). Once you operate on a recurring decimal number ie x or / 10, 100, 1000 etc, you create a shift of digits in an infinite series which is no longer equal to your original series (even though both are still infinite). Subtle, but true I'm afraid......0.272727...is irrational.. As for using a calculator which doesn't have an infinite memory to prove your theory is unforgivable Try dividing 3 by 11 you do get 0.272727, but multiply this by 2 , and you'll find an error ie 0.5454......44
Love the way you work it all out but the explanation is far too long. I'm unsubscribing only because I waist too much time. Could be learning this a lot faster.....
The general rule of thumb in base 10 is to put the repeating part (In this case, it is "27") over all 9s. So, in this case, you would have 27/99. This would eventually reduce to 3/11.
For sure in the 60’s, we definitely did Not learn this!
Thank you for an interesting video!
Linda
I don't remember this either at school ( left school in 1969) or even in higher education.
Take any decimal and multiply it by 1/8 1/16 /32 /64 and so on till you get the closest to a whole number.
.2727272727x 11 = 2.99999997 or 3/11
"Encourage people not to give up on math." Bless you, sir!
This was a very neat trick. I solved it by brute force myself. I figured that the result was just a little more than 1/4, so I starting adding 1 to the numerator and dividing the bigger numerator with (4 * numerator) - 1). So 1/4=0.25. 2/7=0.2857. 3/11=0.272727 and Bingo
For some reason, we, (1970's) were NEVER taught this in High School.
I had to learn this on my own. I did it algebraically. If it is 1 decimal repeating, use 9, 2 decimal use 99, 3 decimal 999 .... etc.
And yes, I figured this out the way you show it in this fine video.
Nice review! I haven’t seen that since high school and didn’t remember it.
It would be clearer if you always put a horizontal bar over the last "27" in all cases that it applies. Then you wouldn't have to keep explaining that it repeats.
I actually figured out what you were doing before you finished explaining it, but I thought it was really v clever.
This is great! So if I understand, take the repeating portion of the decimal, move the decimal point to the right - that’s the numerator - now in the denominator, put a 9 under each digit above (16:46). Shouldn’t this have a name, like “John’s Rule of Nines?”
The method shown does work but there's a much easier method if there are no non repeating digits before the repeating digits, as here. For the numerator, write down the first string of repeating digits. Here that's 27. For the denominator, write as many nines as the number of digits in the numerator. There are two digits in the numerator so 99 is the denominator. 27/99 is 3/11.
Again, if there ARE non repeating digits before the repeating digits, the solution is more complex because you would need to create a second fraction using the non repeating digits (using different rules) and then add the two fractions.
Example: .037037. The numerator is 37 and the denominator is 999 since there are three digits in 037. 37/999 is 1/27.
Using this method, .272727 can be converted to a fraction mentally.
Does your method work for 2.7272727... as well? (30/11). What do you do with the 2 and the repeating decimals .727272...
@@nixxonnor Yes. Just break up 2.72727272727... into 2 + 0.727272727.... Then the desired fraction is 2 + 72/99 = 198/99 + 72/99 = 270/99. I wish I could take credit but I just expanded a little on the method presented by nickcellino1503. Thanks, Nick!
@@nixxonnor the 2 is a whole number so we focus on the right of the decimal to start. This gives us 72/99. which is 8/11. Now for the 2. We can have it as a mixed number fraction, which is 2 and 8/11. or You can have a top heavy fraction which would be 30/11.
(think 5.25 well that is just 5 and 1/4. You could also say it is 21 quarters.)
Interesting! I don't remember learning this in school,
3/11, worked it out before opening the video, so I'll be fascinated to see what your methods was. I did iit like this, 27/100, so just a little over 1/4. You need a repeating pattern and using integers, I went next up to 3/11 was the first try. I am sure that is not scientific though!
Best wishes from George.
PS: Not taught this, and was born in 1961 in England.
I did much the same. .25=1/4. .33 = 4/12
Therefore try 3/11
3/11. slightly bigger than1/4, less then 1/3. Did a few trial and errors to see the pattern between changing the numerator and denominator by a digit to see where the pattern goes. presto, 3/11. Now I'll see how it was meant to be done.
NOT ABOUT THIS. Other sites say that PEMDAS does not work for all . . . I understand your voice and presentation. . . . thanks
Agree.
That is an interesting and clever trick.
But I solved it in a different way.
x = 0.27272727.. It was said that you could use a calculator, so I used it to get:
1/x = 3.66666666667 and I can imediately recognise the .6666666667 part as related to 2/3
So then 3 x (1/x) = 11 or 3/x = 11 -> x = 3/11 and my calculator checked that as correct
A different denominator may not have been recognised so your method is better, thanks.
I defaulted to an approximation (same as done with pi). ~27/100. There's my fraction.
This is 0.27 recurring. 2 recurring decimals for every decimal ÷9. Do the answer is 27/99 =3/11. Simple. Didn't take me ages
Thank you
Yes I can...
x = 0.2727272727.... then 100x = 27.272727272727...
So 100x - x = 99x = 27 and resulting in a nice fraction x = 27/99 = 3/11
Why 100x... that is because of the repetition of 27.
If the decimal was 0.666666666... you could use 10x = 6.66666666 and 9x = 6 so x = 6/9 = 2/3
If the decimal was 0.217217217... you would use 1000x = 217.217217217217... and 999x = 217
so x = 217 / 999
If the decimal was 0.134913491349... you would use 10000x = 1349.134913491349... and 9999x = 1349
so x = 1349/9999
Just when I thougth I knew what to do with a one step linear equation you come up with this. :(
Great! :(
Very interesting, I was never aware of this trick.
This was a wonderful method and I am certain I was never shown this in school (1960's). I am also certain that I would never have been able to figure out this method on my own. But I did get the answer a different way.
First, I assumed that since only the first two decimal places repeat, then the numerator and denominator are going to be small numbers. (I'm not sure if that assumption is always correct.)
Then I took the fact that .27 is a little bit larger than 1/4.
So the numerator and denominator are going to be only one or two digits, and I'll try fractions which are a little bit more than 1/4. Let's increase the N and D by 1. That's 2/8 (which = 1/4). Now, a little bit larger than 2/8 is 2/7. That didn't work.
So let's add one more to the D and N to get 3/12. Now, a little bit more than 3/12 is 3/11. That turned out to be right. That worked with .272727..., but I'm not sure if it would work so well with any other two-digit repeating decimal.
Edit: I've been playing around with this and discovered that ALL fractions with 11 as a denominator are two-digit repeating decimals! And there is no other denominator which does that.
Also, each denominator has its own pattern of repeating decimals. And if you go to twice the denominator (say from 7 to 14) then that decimal will have the same number of repeating decimal places, only starting with the second decimal place, not the first (unless the fraction reduces).
For example, 3/22 is .136363636....
5/7 is .714285714285... but 5/14 is .3571428571428....
All 7ths have a six digit repeating string, and amazingly, it's the same six digits in the same order, only beginning with a different digit depending on the numerator!
1/7 is .142857142857...
2/7 is .285714285714...
3/7 is .428571428571...
WOW!
(To which, the math gods would say, "Well, duh, it does that because...".)
That's pretty amazing. Great job.
got it. the fraction conversion is 27/99 or 3/11 thanks for the fun
Very nice trick. I think I learn something else a few years ago... I just can't remember. :)
My reasoning: this decimal is almost 3/10, but more than 3/12, so it had to be 3/11 and my calculator said I was right. :)
Now, a more interesting problem will be to figure out 1/12.
This is because 1/12 = 0.083333333 ...
This will involve multiple techniques, including what you demonstrate in this video.
It's not that much more complicated...
10x = 0.83333...
x = 0.08333...
subtracting...
9x = 0.75
9x / 9 = (3/4) / 9
x = 1/12
I used a slide rule.
this was a tough one.
pretty easy just realize this is only for numbers that can be expressed as a fraction.
I learned that in the 8th grade. I can't believe i got this answer in seconds. Wow!
I would not want to I What would be the reason?
Instantly recognised. How about 0.428571 recurring?
Charlie/Golf?
@@Oblitus1 Delta/Golf
As I explain above, the decimal 0.428571 can be converted to the fraction 428571/999999 which can indeed be expressed as a simplified fraction, which is 3/7.
Funny and insane, but good. It looks familiar to me. Very clever.
3/11, not very difficult to fathom out!!! :o
I never learned this. I'm sure of that. I don't think anybody ever tried to teach it to me, either. Even in college algebra.
27／99=3／11
Hey i learnt something.
Mr UA-cam Mathsman should put a bar over himself when he keeps repeating himself
I've seen that before. Something over 11 I think. Yep.
3/11.
My scientific calculator on WIndows 10 tells me that 7/31 = 0.22580645161290322580645161290323
Repeating unit is 225806451612903. How would you approach that?
You put that repeating part over all 9s. The only difference is that it will take a lot longer to reduce it. You basically have to go over all the prime numbers that are less than or equal to the square root of the part that is left over after all the reductions that you have done up to that point and more than the prime numbers that you already went through to make sure you finally got the reduced fraction for this repeating decimal.
2 no; 3 yes --> 3 times to get 8,363,201,911,589/37,037,037,037,037; 5 no; 7 yes but no since only the numerator is divisible by 7; 11, 13, 17, 19, 23, 29, 31 all no; 37 yes to get 226,032,484,097/1,001,001,001,001; 41 yes to get 5,512,987,417/24,414,658,561; all succeeding primes under 271 no; 271 yes to get 20,343,127/90,090,991; all succeeding primes under 2906161 no; 2906161 yes to get 7/31.
Of course, you don't want to try to do this one by hand since you would have to go through many primes to reduce this. You might want to use a computer or the internet to find the prime factorization of the number 1,001,001,001,001, which is what I just did, to help you out with this one. LOL.
Actually, what I should have done is:
2 no; 3 yes --> 3 times to get 8,363,201,911,589/37,037,037,037,037; 5 no; 7 yes but no since only the numerator is divisible by 7 to get 7 times 1,194,743,130,227/37,037,037,037,037; 11, 13, 17, 19, 23, 29 all no; 31 no but yes since only the denominator is divisible by 31 to get 7 times 1,194,743,130,227/1,194,743,130,227 divided by 31; At this point, you should notice that there is a 1,194,743,130,227 in both the numerator and denominator. This should then easily reduce to 7/31. You really don't have to worry about primes when you have the same thing in both the numerator and denominator. As you can see, doing it this way was better than doing it the way I showed in the previous comment. It is better to show the isolation when you have a prime that is in one place but not the other as was the case with the 7 and the 31. Of course, I still needed to use the calculator anyway to try out all the primes that I did try.
3/11 in my head.
Most will never need to be able to figure this out...
Why not teach students logical short-circuit and very easy logical steps.
If we have after decimal points only the period of the periodic fraction the student already have the numerator. The denominator has as much nine's as the number of digits in the numerator. That's it.
0.(27) = 27/99
0.(4) = 4/9
0.(15) = 15 /99
0.(123) = 123/999
and so on.
My answer is 3/11
3/11
What about 0.166666... = 1/6. How to you get that type?
I don't think 0.1666 is a repeating decimal. The 6's repeat, but the 1 is just hanging loose and not repeated at all. Even though 1 ./. 6 = .166666 . . . . this technique does not seem to work.
@@terry_willis I did look it up, and there are indeed slight modifications of the method for leading numbers before the repeat. Quite interesting...
@@terry_willis You should know that ALL fractions have repeating segments when in decimal form -- it's that which separates them from "irrational" numbers.
This is the same technique that proves that 0.999... is equal to 1
To elborate
let x= 0.999...
10x = 9.999...
-x / 0.999... from both sides
9x = 9
x = 1
27/99 = 3/11
3/11. Did it by trying to and fail
27/99 = 0.272727....
explanation went on too long - usually your slow and careful discussion was helpful - but this time it was just dragging it out.
I use speed 2X and he's a little fast but quite understandable.
1/x button.
Factor the 27. Its got to be 9 or 3
neh
1/000000004
2/27
No way...
27/99
Close
@@saltydog584 Close enough? Is 28/99 better or perhaps 26/99 ??? Or what about 26½/99 or 27½/99 ???
The answer 27/99 is spot on, just needs a little simplification but 100% correct.
@@panlomito yes you are right
@@panlomito although perhaps 27/99 is really 3/11
I understand the method and it's very clever, but I don't understand why it works. I'll try to meditate on that.
I could feel my IQ go up a little bit while viewing the video
27272727/100000000
Maybe 27/99...
27/99 ... which is 3/11
@@MrMousley True but still 27/99 is a 100% correct answer!
@@panlomito Yes .. but saying 27/99 instead of 3/11 is the same as saying 8/16 instead of just saying 1/2
@@MrMousley True, but there is nothing wrong about a correct answer that only needs a little clean up. Because 8/16 = 1/2 so the equation is true. Type 8 / 16 in your calculator and you will probably get 0.5
Now repeat this with 1/2 ... also 0.5 ??? So both answers are true !
Oh dear John.This very same technique was used to prove 0.999.... =1 (which is wrong). Once you operate on a recurring decimal number ie x or / 10, 100, 1000 etc, you create a shift of digits in an infinite series which is no longer equal to your original series (even though both are still infinite). Subtle, but true I'm afraid......0.272727...is irrational.. As for using a calculator which doesn't have an infinite memory to prove your theory is unforgivable Try dividing 3 by 11 you do get 0.272727, but multiply this by 2 , and you'll find an error ie 0.5454......44
You are very confused ...
Most boring explanation I have ever listened to 14:52
Love the way you work it all out but the explanation is far too long. I'm unsubscribing only because I waist too much time. Could be learning this a lot faster.....
The correct answer is 2.72727272727..../10 It's that simple.
No, it is not the correct answer because that is the decimal you started anyway. The task is to find the FRACTION and that answer is 27/99 or 3/11.
@@panlomito The fractions 27/99 or 3/11 are in no way related to the correct answer to this question.
1/3.666666703
Thank you
3/11
27/100
27/100
3/11