If you cut the 21-area with the right side of the unknown rectangle, then you have an area 14+28+right part of 21 = (105+35)/35 * 14 = 56, so right part of 21 is 14, and left part of 21 is 7 Then, unknown area is (14+28)/14 * 7 = 21
The two presented methods seem like two variations on the same method. An alternative method would be to work with relative side length of all the component rectangles. And while that's the method I used, I prefer the method presented.
Ok.. before looking... '14' is 1 x 14, 2 x 7 '28' is 1 x 28, 2 x 14, 4 x 7 '21' is 1 x 21, 3 x 7 '35' is 1 x 35, 5 x 7 '105' is 1 x 105, 3 x 35, 5 x 21, 7 x 15 '14' and '28' have the same width, so it has to be either 7 or 14, but '14' has the same height as '35' which means '14' and '35' have to have a height of 1. So '14' is 1 x 14 and '28' is 2 x 14. Since '21' is wider than '14' it has to be 1 x 21. Which means the blue rectangle is 3 x 7 (height of '14' = 1 + height of '28' = 2 and width of '21' (21) - width of '28' (14) = 7) So area is 21.
105 is 3•35, which implies green rectangle is 3× height of purple rectangle, as widths are the same. 28 is 2•14 which implies yellow rectangle is 2× height and red rectangle is 1× height of gray rectangle, as those two rectangles combined are same height as green and gray is same height as purple. 21 ia 1.5•14, which implies red rectangle is 3/2 width and blue rectangle is 1/2 width of gray and yellow rectangles, as gray and red rectangles are same height and blue, yellow and gray rectangles combined are same width as red. As blue rectangle is same height as gray and yellow and 1/2 width, blue is 1/2 gray + yellow = (14+28)/2 = 42/2 = 21
Extend the line segment for the top side of the red rectangle to the right to divide the green rectangle into two rectangles, let's call them top green and bottom green. The gray rectangle and the yellow rectangle have the same base but different heights. The yellow rectangle has twice the area of the gray rectangle, so the height of the yellow rectangle is twice that of the gray rectangle. The purple rectangle and the top green rectangle have the same base but the purple rectangle has the same height as the gray rectangle and the top green rectangle has the same height as the yellow rectangle. So, the top green rectangle, with twice the height but the same base, has twice the area of the purple rectangle, or 70. This leaves 35 for the bottom green rectangle. Because the bottom green rectangle and the purple rectangle have the same area and base, they must have equal heights. So, the gray rectangle and the red rectangle have the same height. The blue, gray and yellow rectangles combine into a single rectangle with the same base as the red rectangle, but 3 times the height, so three times the area = 63. Deduct the areas of the gray rectangle (14) and yellow rectangle (28), so the blue rectangle has area 21, as PreMath also found.
I cut the red rectangle in two to match the rectangles above it and used the four given rectangles to work out the right red piece was 14, leaving 7 on the left. Then I cut the blue one to match gray and yellow, and comparing the columns, got 7 & 14 for the two blue pieces, giving a total of 21.
We find the Golden Ratio when we divide a line into two parts s that : * the whole length divided by the long part IS ALSO EQUAL TO *the long part divided by the short part ...approximately = 1.618 🙂
Dzeltenais 28 ir 2 reizes lielāks neka pelēkais 14, ja uz zaļā atliksim divreiz lielāku nekā violetais 2*35=70 apakšā pretim 21 arī paliks 35 , tātad arī augšā pret violeto 35 pelēkais 14+x=21 , x=7 gaiši zilais 3x=21.
green area 15x 7 , violet area 5x7 , yellow area 4x7 , grey area 2x7 , red area 3x7 , so longer side is value 22 , smaller side is long 12 , area of bigger rectangle is 264 , so area of blue rectangle is 264-21-28-14-35-105 is 61
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If you cut the 21-area with the right side of the unknown rectangle, then you have an area 14+28+right part of 21 = (105+35)/35 * 14 = 56, so right part of 21 is 14, and left part of 21 is 7
Then, unknown area is (14+28)/14 * 7 = 21
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Nice, though I'd start by dividing everything by 7, to make calculations simpler.
That's a good idea!
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Nice exercise
Keep up your great work
Thank you, I will ❤️
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Great explanation 👍
Thanks for sharing 😊
My pleasure 😊
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35 : 105 = 14 : (28 + 21x)
35(28 + 21x) = 105 * 14
980 + 735x = 1470
735x = 490
x = 2/3
=> right part of red rectangle = 2/3 * 21 = 14 square units
=> left part of red rectangle = 1/3 * 21 = 7 square units
A : 7 = (14 + 28) : 14
14A = 7 * 42
A = 294 / 14 = 21 square units
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تمرين جميل جيد . رسم واضح مرتب . شرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا . تحياتنا لكم من غزة فلسطين .
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Excellent explanation 👍
Glad it was helpful!
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The two presented methods seem like two variations on the same method. An alternative method would be to work with relative side length of all the component rectangles. And while that's the method I used, I prefer the method presented.
Good point!
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Ok.. before looking...
'14' is 1 x 14, 2 x 7
'28' is 1 x 28, 2 x 14, 4 x 7
'21' is 1 x 21, 3 x 7
'35' is 1 x 35, 5 x 7
'105' is 1 x 105, 3 x 35, 5 x 21, 7 x 15
'14' and '28' have the same width, so it has to be either 7 or 14, but '14' has the same height as '35' which means '14' and '35' have to have a height of 1.
So '14' is 1 x 14 and '28' is 2 x 14. Since '21' is wider than '14' it has to be 1 x 21.
Which means the blue rectangle is 3 x 7 (height of '14' = 1 + height of '28' = 2 and width of '21' (21) - width of '28' (14) = 7)
So area is 21.
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105 is 3•35, which implies green rectangle is 3× height of purple rectangle, as widths are the same.
28 is 2•14 which implies yellow rectangle is 2× height and red rectangle is 1× height of gray rectangle, as those two rectangles combined are same height as green and gray is same height as purple.
21 ia 1.5•14, which implies red rectangle is 3/2 width and blue rectangle is 1/2 width of gray and yellow rectangles, as gray and red rectangles are same height and blue, yellow and gray rectangles combined are same width as red.
As blue rectangle is same height as gray and yellow and 1/2 width, blue is 1/2 gray + yellow =
(14+28)/2 = 42/2 = 21
Thanks for video.Good luck sir!!!!!!!!!
So nice of you, dear❤️
35：105=1:3， so 14x3-28=42-28=14, thus 21-14=7, then the answer is 7x3=21.😊
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That’s very nice,
The second method is more simple and easy .
Thanks Sir for like these exercises.
Always welcome❤️
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Extend the line segment for the top side of the red rectangle to the right to divide the green rectangle into two rectangles, let's call them top green and bottom green. The gray rectangle and the yellow rectangle have the same base but different heights. The yellow rectangle has twice the area of the gray rectangle, so the height of the yellow rectangle is twice that of the gray rectangle. The purple rectangle and the top green rectangle have the same base but the purple rectangle has the same height as the gray rectangle and the top green rectangle has the same height as the yellow rectangle. So, the top green rectangle, with twice the height but the same base, has twice the area of the purple rectangle, or 70. This leaves 35 for the bottom green rectangle.
Because the bottom green rectangle and the purple rectangle have the same area and base, they must have equal heights. So, the gray rectangle and the red rectangle have the same height.
The blue, gray and yellow rectangles combine into a single rectangle with the same base as the red rectangle, but 3 times the height, so three times the area = 63. Deduct the areas of the gray rectangle (14) and yellow rectangle (28), so the blue rectangle has area 21, as PreMath also found.
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I cut the red rectangle in two to match the rectangles above it and used the four given rectangles to work out the right red piece was 14, leaving 7 on the left. Then I cut the blue one to match gray and yellow, and comparing the columns, got 7 & 14 for the two blue pieces, giving a total of 21.
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We find the Golden Ratio when we divide a line into two parts s that :
* the whole length divided by the long part
IS ALSO EQUAL TO
*the long part divided by the short part
...approximately = 1.618 🙂
28/14=2》105/35=3=(2a+28+21)/(a+14)》a=7》Azul =a+2a =3a =3×7=21
Gracias y un saludo cordial.
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1st method is easy...
And also 35/105 can be expressed as ⅓
Dzeltenais 28 ir 2 reizes lielāks neka pelēkais 14, ja uz zaļā atliksim divreiz lielāku nekā violetais 2*35=70 apakšā pretim 21 arī paliks 35 , tātad arī augšā pret violeto 35 pelēkais 14+x=21 , x=7 gaiši zilais 3x=21.
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Second method is relatively simple.
35 and 105 are not needed at all in order to understand what will be 21
105/35 =3 28/14=2 21/1=21
Simple logic
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❤🥂👍😀
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Je ne vois aucune différence entre "ces 2 methodes" ^^
21
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My answer is 21.
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green area 15x 7 , violet area 5x7 , yellow area 4x7 , grey area 2x7 , red area 3x7 , so longer side is value 22 , smaller side is long 12 , area of bigger rectangle is 264 , so area of blue rectangle is 264-21-28-14-35-105 is 61
21
21