@@Micboss1000 No, because you get to understand the logic on a deeper level, when you already understand the answer. That's how math gets framed, is by knowing the answer on something simple, and then working logic to construct a ratio for all other numbers to conform to. It's beautiful. But generally, you look at the simpler shapes, and construct a ratio through it, that will inherently get the answer on more difficult problems. You have to understand it fully in its simpler iterations, before you can use it for more complex ones.
@@sunandasengupta8113 You don't understand? You sometimes need easy calculated things to understand why and how the logic works. He did that deliberately. Like look over the whole thing, and see the nuances of the calculations. How everything, even in complicated form, fits together perfectly.
@ 8:20 Once you found that y=3, you can get the other side of the green rectangle by dividing y=3 into the area (27) and get 27/3 = 9... WITHOUT plugging y=3 into that ugly expression! Fun problem.
0:44 12 The width of the blue = 4 (20/5) Since the area of the pink =8, let's ASSUME THAT ITS WIDTH = 2 (or half the width of the blue, just by observation it is reasonable to make this assumption since 8 is so small compared to the other areas), then 'slice off' 2* 5 of the blue, or area of 10, leaving 10. This would leave a rectangle with dimensions x by 5 and a total area of 60 [ 10 (the remaining blue), 27+ 8 + 15). Hence, x = 12 (60/5). Let's check to see if my assumption was correct: If the width of the pink = 2, then the width of the green =3 (given that the length of the yellow =5, hence, the width of the yellow =3) Therefore, the length of the green is 9 (27/3), and the width of the yellow is 3 (x-9 or 12-9). So far, it has worked out for the green and yellow. Also if the length of the green = 9, and the length of the blue =5 (given), then the length of the pink= 4 ( and 4*2 =8, recall the assumption was that the width of the pink=2 Hence, x = 12,
Extend the pink rectangle's into the blue rectangle to split it into two parts. Call the bottom area 5a. Total area of the bigger rectangle formed by this is: 27+15+8+5a = 50+5a. Since the height is 5, the length x is (10+a) That length can also be expressed as: 5 + (8/a) + 3 = 8 + (8/a) So, 10 + a = 8 + 8/a Solve to get a = 2 So, x = 10+a = 12
Blue top side = 5 Yellow top side = 3 since area 15 and side of 5 Pink top = 4 since top is longer than side and area is 8, so it has to be 4 x 2 rectangle So yeah x = 12
Noticed the light blue rectangle is redundant. That shows there's another way to do it . But the solution on the clip is good if it didn't even have to use anything linked to the blue triangle
I go too Great Lengths to achieve something and sometimes do extreme things to achieve it! ...so, @ 7:50 I'm tweekin cuz sometimes I am coherent and other times noncoherent scatter brained and know that neutrons have positive or negative lengths. ...well, I mean scattered lengths that is. 🙂
Or 1/2 the blue square, plus the other rectangles = 60. Divide by 5, and guess what? x = 12 Yeah, I know. I know. They want to see if you can set up the quadratic. But, this is so much faster.
I can also find x from this 8/(x-8)+27/(x-3)=5 Wich 5 is right side of figure, x-8 is over side of S=8 figure, x-3 is over side of S=27. x=6 and 12. 6 is not suitable. Answer is x=12.
In case it's needed, call the vertical y, as in graphs. The x direction of yellow is 3, so x>8 The y direction of blue is 4 The x direction of grey is a x = 8+a Visually, grey looks like 4 by 2, so work with that initially. Test x=12 Green is 9 by 3 Yellow is 3 by 5 Grey is 4 by 2 Blue is 5 by 4 Don't knock the method: science has made assumptions for generations and then proved them later :)
Length of green rectangle is 27/(x-3). Length of pink rectangle. Is 8/(x-8). So 5=8/(x-8)+27/x-3 That gives a quadratic equation where x=8 and x=12, so x=12 since it can’t be = 8.
Just looking at the diagram it looks like pink is 2x4. Yellow is 5x3 so X is 3+4+5=12. And since the numbers all work out with no contradictions our original assumption for pink is correct and our answer is correct.
Did anyone else just work out 3 (for the bottom right rectangle). Then look at the bottom length rectangle and know that it must be 3*9. So it must have been 12. Much easier and quicker!
I always estimate a ballpark for the answer before I start - just to make sure my answer at the end makes sense. In this case I said x looks like it should be 11, 12, or 13ish. In my head 6 plugged the 12 in for x and bingo, saw it worked without picking up the pencil. Of course, that doesn't always work out that way :-)
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Thanks Sir
That’s very useful and enjoyable
We are very interested from the exercises
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I was able to eyeball 12 as the correct answer fairly quickly, but it was nice to see how it was proven.
It's kind of a shame that you can eyeball the answer here, makes the math feel redundant.
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@@Micboss1000 No, because you get to understand the logic on a deeper level, when you already understand the answer. That's how math gets framed, is by knowing the answer on something simple, and then working logic to construct a ratio for all other numbers to conform to. It's beautiful. But generally, you look at the simpler shapes, and construct a ratio through it, that will inherently get the answer on more difficult problems. You have to understand it fully in its simpler iterations, before you can use it for more complex ones.
Ya. It was easy to calculate 12 . No need for such elaborate process.
@@sunandasengupta8113 You don't understand? You sometimes need easy calculated things to understand why and how the logic works. He did that deliberately. Like look over the whole thing, and see the nuances of the calculations. How everything, even in complicated form, fits together perfectly.
@ 8:20 Once you found that y=3, you can get the other side of the green rectangle by dividing y=3 into the area (27) and get 27/3 = 9... WITHOUT plugging y=3 into that ugly expression! Fun problem.
Thanks for the feedback ❤️
I thought the same way.
Easy and useful
0:44 12
The width of the blue = 4 (20/5)
Since the area of the pink =8, let's ASSUME THAT ITS WIDTH = 2 (or half the width of the blue, just by observation it is reasonable to make this assumption since 8 is so small compared to the other areas),
then 'slice off' 2* 5 of the blue, or area of 10, leaving 10.
This would leave a rectangle with dimensions x by 5 and a total area of 60 [ 10 (the remaining blue), 27+ 8 + 15).
Hence, x = 12 (60/5).
Let's check to see if my assumption was correct:
If the width of the pink = 2, then the width of the green =3 (given that the length of the yellow =5, hence, the width
of the yellow =3)
Therefore, the length of the green is 9 (27/3), and the width of the yellow is 3 (x-9 or 12-9). So far, it has worked out for the green and yellow.
Also if the length of the green = 9, and the length of the blue =5 (given), then the length of the pink= 4 ( and 4*2 =8, recall
the assumption was that the width of the pink=2
Hence, x = 12,
20/5=4 15/5=3
(x-3)y=27 (x-8)(5-y)=8 x-3=27/y
(27/y-5)(5-y)=8 135/y-27-25+5y=8 135/y-60+5y=0
5y²-60y+135=0 y²-12y+27=0 (y-3)(y-9)=0 y=9 is rejected , thus y=3
(x-3)*3=27 x-3=9 x=12
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8=b*(8/b)→ 27=(5+b)[5-(8/b)]→ b=4→ X=(5+4)+(15/5)=9+3=12 ud.
Buen rompecabezas. Gracias y un saludo cordial.
I got this one!
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0:10 blue ,pink and yellow pre math is in my pillow 😊
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X= 12
20=5x4=20
8=4x2=8
15=3x5=15
27=27/5-2=9x3=27
X=9+3=12
15/5 = 3
4 x 2 = 8
3 + 4 + 5 = 12
just by looking at it i had it 12. but its good to know the equation process.
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Just another solution:
W - width; H - height;
W_yellow = 15/5=3;
W_green = x - W_yellow = x - 3;
H_green = 27/W_green = 27/(x-3);
W_red = W_green - W_blue = (x-3) - 5 = x-8;
H_red = 8/W_red = 8/(x-8);
H_yellow = H_green + H_red = 27/(x-3) + 8/(x-8) = 5;
2 roots: x1 = 12 (accepted); x2 = 6 (rejected as x>8 at least)))
x= 12.
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Extend the pink rectangle's into the blue rectangle to split it into two parts. Call the bottom area 5a.
Total area of the bigger rectangle formed by this is: 27+15+8+5a = 50+5a. Since the height is 5, the length x is (10+a)
That length can also be expressed as: 5 + (8/a) + 3 = 8 + (8/a)
So, 10 + a = 8 + 8/a
Solve to get a = 2
So, x = 10+a = 12
Blue top side = 5
Yellow top side = 3 since area 15 and side of 5
Pink top = 4 since top is longer than side and area is 8, so it has to be 4 x 2 rectangle
So yeah x = 12
I am always surprised to see how you solve 2nd degree equations.
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Noticed the light blue rectangle is redundant. That shows there's another way to do it . But the solution on the clip is good if it didn't even have to use anything linked to the blue triangle
I go too Great Lengths to achieve something and sometimes do extreme things to achieve it! ...so, @ 7:50 I'm tweekin cuz sometimes I am coherent and other times noncoherent scatter brained and know that neutrons have positive or negative lengths. ...well, I mean scattered lengths that is. 🙂
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Thanks for the feedback ❤️
Or 1/2 the blue square, plus the other rectangles = 60. Divide by 5, and guess what? x = 12 Yeah, I know. I know. They want to see if you can set up the quadratic. But, this is so much faster.
I can also find x from this 8/(x-8)+27/(x-3)=5
Wich 5 is right side of figure, x-8 is over side of S=8 figure, x-3 is over side of S=27. x=6 and 12. 6 is not suitable. Answer is x=12.
In case it's needed, call the vertical y, as in graphs.
The x direction of yellow is 3, so x>8
The y direction of blue is 4
The x direction of grey is a
x = 8+a
Visually, grey looks like 4 by 2, so work with that initially.
Test x=12
Green is 9 by 3
Yellow is 3 by 5
Grey is 4 by 2
Blue is 5 by 4
Don't knock the method: science has made assumptions for generations and then proved them later :)
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Let's find x:
.
..
...
....
.....
Let's use the labels x and y for the horizontal and vertical side lengths of the rectangles, respectively. Then we obtain:
x(b) = 5 ; y(b) = A(b)/x(b) = 20/5 = 4
y(y) = 5 ; x(y) = A(y)/y(y) = 15/5 = 3
x(b) + x(p) = x(g)
y(g) + y(p) = y(y)
x(p)*y(p) = A(p)
x(g)*y(g) = A(g)
5 + x(p) = x(g)
y(g) + y(p) = 5
x(p)*y(p) = 8
x(g)*y(g) = 27
x(p)*y(p) = 8
(5 + x(p))*(5 − y(p)) = 27
x(p)*y(p) = 8
25 − 5*y(p) + 5*x(p) − x(p)*y(p) = 27
25 − 5*y(p) + 5*x(p) − 8 = 27
5*x(p) − 5*y(p) = 10
x(p) − y(p) = 2
x(p)*y(p) = 8
x(p) − y(p) = 2
(2 + y(p))*y(p) = 8
2*y(p) + y²(p) = 8
2*y(p) + y²(p) − 8 = 0
y(p) = −1 ± √[(−1)² + 8] = −1 ± √(1 + 8) = −1 ± √9 = −1 ± 3
Since y(p)>0, the only useful solution is:
y(p) = −1 + 3 = 2
x(p) = A(p)/y(p) = 8/4 = 2
y(g) = 5 − y(p) = 5 − 2 = 3
x(g) = A(g)/y(g) = 27/3 = 9
Now we are to calculate the value of x:
x = x(g) + x(y) = 9 + 3 = 12
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ab = 8 => a = 8/b
x = 8 + a => x = 8 + 8/b
27 = (x - 3)(5 - b)
27 = 5x - bx - 15 + 3b
42 = 5x + 3b - bx
x = 8 + 8/b => bx = 8b + 8
42 = 5x + 3b - (8b + 8)
50 = 5x - 5b => b = x - 10
x = 8 + 8/b = 8 + 8/(x - 10)
(x - 8)(x - 10) = 8
x² - 18x + 72 = 0
(x - 6)(x - 12) = 0
b = x - 10 => x > 10 => *x = 12*
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another way [ but something similar ]
5x = 27 + 15 + 8 + 5h
(where bh = 8)
5x = 50 + 5h => h = x - 10
(x - 3)(5 - h) = 27
5x - xh - 15 + 3h = 27
xh - 3h - 5x + 42 = 0
x(x - 10) - 3(x - 10) - 5x + 42 = 0
x² - 10x - 3x + 30 - 5x + 42 = 0
x² - 18x + 72 = 0
(x - 6)(x - 12) = 0
h = x - 10 > 0 => x > 10 => *x = 12*
first?, why does it even mattter
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Some people will never use math. Others will. Engineers need to be able to figure out stuff like this all the time.
Length of green rectangle is 27/(x-3). Length of pink rectangle. Is 8/(x-8). So 5=8/(x-8)+27/x-3 That gives a quadratic equation where x=8 and x=12, so x=12 since it can’t be = 8.
Just looking at the diagram it looks like pink is 2x4. Yellow is 5x3 so X is 3+4+5=12. And since the numbers all work out with no contradictions our original assumption for pink is correct and our answer is correct.
x=12 units
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Did anyone else just work out 3 (for the bottom right rectangle). Then look at the bottom length rectangle and know that it must be 3*9.
So it must have been 12. Much easier and quicker!
Could solve for X directly by using the rectangle of area 8 and obtaining x2 - 18x + 72 = 0.
H_yel = 6
W_blue = 4, W_yel = 3, W_gr = x -3, W_red = x - 8
H_red = 8/(x-8), x > 8
H_gr = H_yel - H_red = 5 - 8(x-8)
A_gr = W_gr * H_gr
27 = (x-3) [5 - 8(x-8)]
27 = 5x - 15 - [8x/(x-8)] + 24/(x-8)
42 = [5x(x-8) - 8x + 24 ] / (x-8)
42(x-8) = 5x^2 -40x -8x +24
42x - 336 = 5x^2 - 48x + 24
5x^2 - 90x + 360 = 0
x^2 - 18x + 72 = 0
(x -6)(x-12) = 0
x> 8 so x = 12
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12
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1 min for me
My way of solution ▶
20= 5*t
t= 4
15= 5*u
u= 3
a+b= 5
b+c= 4
⇒
a+b= 5
-b-c= -4
a-c= 1
a= c+1
we can write a= c+1
b*d= 8......................(I)
a(5+d)= 27..............(II)
a= c+1
⇒
(c+1)(5+d)= 27.......(III)
c+1+b= 5
b+c= 4
b= 4-c
if we put this in the equation (I) above:
b*d= 8
(4-c)*d= 8
d= 8/(4-c)
⇒
(c+1)*(5+d)= 27.......(III)
if we put d= 8/(4-c) we get:
(c+1)*[5+ 8/(4-c)] = 27
(c+1)*(20-5c+8)/(4-c)= 27
(c+1)(28-5c)= 27(4-c)
28c-5c²+28-5c= 108-27c
5c²-27c+108-28-23c=0
5c²-50c+80= 0
c²-10c+16=0
Δ= 100-4*1*16
Δ= 36
√Δ= 6
c₁= (10+6)/2
c₁= 8
c₂= (10-6)/2
c₂= 2
if c= 8
b= 4-c would be negative❗
⇒
c= 2
b= 4-c
b= 4-2
b= 2
d= 8/(4-c)
d= 8/(4-2)
d= 8/2
d= 4
x= 5+d+u
x= 5+4+3
x= 12 length units
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12 blir svaret
I always estimate a ballpark for the answer before I start - just to make sure my answer at the end makes sense. In this case I said x looks like it should be 11, 12, or 13ish. In my head 6 plugged the 12 in for x and bingo, saw it worked without picking up the pencil. Of course, that doesn't always work out that way :-)
STEP-BY-STEP RESOLUTION PROPOSAL :
Let's check these Solutions :
01) Sum of All Areas = 70
02) 20 / 5 = 4
03) 15 / 5 = 3
04) 27 = 3 * 9 (?)
05) 8 = 4 * 2 (?)
06) Left Side = 4 + 3 (?)
07) The only possible solution herein is x = 9 + 3 = 12
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12
12