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Difficulty:
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40%
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A square shown above is divided into 5 rectangles, each having the same area. If the upper left rectangle has a width of 5 units, then what is the area of the square?
A. 100
B. 169
C. 225
D. 400
E. 625
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Attachment:
Untitled.png
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Posts: 731
Updated on: 17 Aug 2021, 07:06
Originally posted by HarshaBujji on 16 Aug 2021, 07:29.
Last edited by HarshaBujji on 17 Aug 2021, 07:06, edited 1 time in total.
Last edited by HarshaBujji on 17 Aug 2021, 07:06, edited 1 time in total.
Bunuel
Please ref. then attachment for better understanding.
Let us assume the side of the square is s,
So total area of the square is \(s^2\)
As we know all the rectangles inscribed in the square are of the same area A,
=> \(s^2\) = 5A,
So area of each rectangle is \(s^2\)/5;
So Ar[ABDE] = Ar[BCGF] = \(s^2\)/5;
We know AD=5; So AB= Ar[ABDE]/AD = \(s^2\)/25;
Similarly, We know CG=s; So BC= Ar[BCGF]/CG = s/5;
AC = AB + BC = \(s^2\)/25 + s/5;
We also know AC=s [Since side of the square]
Hence s= \(s^2\)/25 + s/5;
So s=20 or 0;
As s=0 is not possible; s=20;
Hence area of square \(s^2\) = 400;
IMO D
Attachments
PB10.PNG [ 10.06 KiB | Viewed 9364 times ]
Official Solution:
A square shown above is divided into 5 rectangles, each having the same area. If the upper left rectangle has a width of 5 units, then what is the area of the square?
A. \(100\)
B. \(169\)
C. \(225\)
D. \(400\)
E. \(625\)
Say the length of the purple rectangle is \(x\). Then the area of all 5 rectangles will be \(5x\).
Say the length of the red rectangle is \(y\). Then the combined area of red, yellow, and green rectangles will be \(xy\), as well as \(5x+5x+5x=15x\).
So, \(xy=15x\). This gives: \(y=15\).
The side of the square is \(5+y=20\) and the area is \(20^2=400\).
Answer: D
_________________
A square shown above is divided into 5 rectangles, each having the same area. If the upper left rectangle has a width of 5 units, then what is the area of the square?
A. \(100\)
B. \(169\)
C. \(225\)
D. \(400\)
E. \(625\)
Say the length of the purple rectangle is \(x\). Then the area of all 5 rectangles will be \(5x\).
Say the length of the red rectangle is \(y\). Then the combined area of red, yellow, and green rectangles will be \(xy\), as well as \(5x+5x+5x=15x\).
So, \(xy=15x\). This gives: \(y=15\).
The side of the square is \(5+y=20\) and the area is \(20^2=400\).
Answer: D
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General Discussion
avijitsrivastava
Joined: 04 Jul 2021
Last visit: 18 Nov 2021
Posts: 3
GMAT 1: 720 Q50 V38
GMAT 2: 750 Q51 V40
WE:Management Consulting (Consulting)
GMAT 2: 750 Q51 V40
Posts: 3
Assume the square to be of side L and the light blue rectangle to be of horizontal length X. Then, as per the directions of the question,
LX = (L^2)/5 (Area of blue rectangle is one-fifth of the total area of the square)
Hence,
X = L/5
Now comparing the purple rectangle with the blue one
5*(L-X) = LX
=> 4L = (L^2)/5
=> L = 20
Therefore, area of the square will be:
L^2 = 400
Answer: (D) 400
LX = (L^2)/5 (Area of blue rectangle is one-fifth of the total area of the square)
Hence,
X = L/5
Now comparing the purple rectangle with the blue one
5*(L-X) = LX
=> 4L = (L^2)/5
=> L = 20
Therefore, area of the square will be:
L^2 = 400
Answer: (D) 400
Kinshook
Major Poster
Joined: 03 Jun 2019
Last visit: 10 Sep 2026
Posts: 6,126
Location: India
Schools: HBS '28 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
Given: A square shown above is divided into 5 rectangles, each having the same area.
Asked: If the upper left rectangle has a width of 5 units, then what is the area of the square?
Attachment:
Screenshot 2021-08-16 at 8.57.45 PM.png [ 16.61 KiB | Viewed 9740 times ]
Since the area of all rectangles is same.
5(x + y) = ty = 2tx = (2t+5)z
y = 2x
15x = 2tx
t = 7.5
15x = 20z
3x = 4z
Since all rectangles combined form a square
2t+5 = x+ y+z
20 = x + 2x + 3x/4 = 15x/4
x = 80/15 = 16/3
y = 32/3
z = 4
Area of the square = (2t+5)^2 = 20^2 = 400
IMO D
Posts: 60
I would go backward from the answer choices:
Option A is 100 as area of square:
Area Square (S): 100
Side of Square : 10
Long Side of Red (R): 5
Area of Purple(P): 100/5=20
Long Side of P: 20/5=4
Area of Red + Yellow + Green= R * P = 5*4=20 ....(1)
But since Red + Yellow + Green must be equal area as 3 * Area of Purple (i.e. 3*20=60) ... (2)
From 1 and 2 A option is not correct.
On similar terms and since 100 led to a small combined area of the 3 rectangles next chose a bigger option like. Like D is 400 and more likely to the answer.
Area Square (S): 400
Side of Square : 20
Long Side of Red (R): 15
Area of Purple(P): 400/5=80
Long Side of P: 80/5=16
Area of Red + Yellow + Green= R * P = 15*16=240 ....(3)
But since Red + Yellow + Green must be equal area as 3 * Area of Purple (i.e. 3*80=240) ... (4)
From 3 and 4 we see these matches hence D is the right answer.
Option A is 100 as area of square:
Area Square (S): 100
Side of Square : 10
Long Side of Red (R): 5
Area of Purple(P): 100/5=20
Long Side of P: 20/5=4
Area of Red + Yellow + Green= R * P = 5*4=20 ....(1)
But since Red + Yellow + Green must be equal area as 3 * Area of Purple (i.e. 3*20=60) ... (2)
From 1 and 2 A option is not correct.
On similar terms and since 100 led to a small combined area of the 3 rectangles next chose a bigger option like. Like D is 400 and more likely to the answer.
Area Square (S): 400
Side of Square : 20
Long Side of Red (R): 15
Area of Purple(P): 400/5=80
Long Side of P: 80/5=16
Area of Red + Yellow + Green= R * P = 15*16=240 ....(3)
But since Red + Yellow + Green must be equal area as 3 * Area of Purple (i.e. 3*80=240) ... (4)
From 3 and 4 we see these matches hence D is the right answer.
The area of the square is given as l^2.
Let the area of the rectangle be considered as 5x such that say, x is the length of the rectangle.
Also, since the area of the rectangles is same. So, all the areas would be 5x.
Now let us consider only red, yellow and green rectangles. The sum of their areas would be 15x.
Also, the area can be written as x*y(where y is the vertical length of the three rectangles(r, y & g combined).)
So,
15x=x*y
thus the vertical length would be 15
Hence, the side of square would be (15+5) and
area of square would be 20^2=400
Let the area of the rectangle be considered as 5x such that say, x is the length of the rectangle.
Also, since the area of the rectangles is same. So, all the areas would be 5x.
Now let us consider only red, yellow and green rectangles. The sum of their areas would be 15x.
Also, the area can be written as x*y(where y is the vertical length of the three rectangles(r, y & g combined).)
So,
15x=x*y
thus the vertical length would be 15
Hence, the side of square would be (15+5) and
area of square would be 20^2=400
Bunuel
Let the area of the rectangles be to A to E as shown. => \(A=B=C=D=E\)
Given that \(B+C+D=3*A\)
A=5*b
B+C+D=x*b
=> x*b=3*5*b
x=15
So, each side of the square = 5+15
Area = 20*20=400
Attachments
squares cloured.png [ 7.83 KiB | Viewed 8324 times ]
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