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Black Squares

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by surajgarg » Thu Jul 29, 2010 9:58 am
The combined area of the two black squares is equal to 1000 square units. A side of the larger black square is 8 units longer than a side of the smaller black square. What is the combined area of the two white rectangles in square units?


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A. 928
B. 936
C. 948
D. 968
E. 972


OA B
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by aloneontheedge » Thu Jul 29, 2010 11:33 am
surajgarg wrote:The combined area of the two black squares is equal to 1000 square units. A side of the larger black square is 8 units longer than a side of the smaller black square. What is the combined area of the two white rectangles in square units?


Image

A. 928
B. 936
C. 948
D. 968
E. 972


OA B
I did using traditional method.
let the side be x of smaller square and x+8 be the side of bigger square
x^2 + (x+8)^2 = 1000
on solving x = 18
so,
the area of 2 rectangular wil be 936
Hence B.
After i worked out completely, I realised calculation wasnt required.
Smaller square = x,larger square = x + 8
x^2 + (x+8)^2 = 1000
2x^2 + 16x = 936 (stop here and go back to the diagram)----1
area of 2 rectangulars = 2[x(x+8)] -->2x^2+16x----2
equating 1 n 2 we get 936
Hope i'am clear
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by kvcpk » Thu Jul 29, 2010 11:59 am
After taking some cases.. I came up with the following generic formula
Area of white portion = Sum of areas of black portion -(difference between length of a side of squares)^2
= 1000-8^2 =936

Hope this helps!!
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by Anaira Mitch » Fri Dec 30, 2016 4:17 am
Assume the larger red square has a side of length x + 4 units and the smaller red square has a side of length x - 4 units. This satisfies the condition that the side length of the larger square is 8 more than that of the smaller square.

Therefore, the area of the larger square is (x + 4)2 or x^2 + 8x + 16. Likewise, the area of the smaller square is (x - 4)^2 or x^2 - 8x + 16. Set up the following equation to represent the combined area:

(x^2 + 8x +16) + (x^2 - 8x +16) = 1000
2x^2 + 32 = 1000
2x^2 = 968

It is possible, but not necessary, to solve for the variable x here.

The two white rectangles, which are congruent to each other, are each x + 4 units long and x - 4 units high. Therefore, the area of either rectangle is (x + 4)(x - 4), or x^2 - 16. Their combined area is 2(x^2 - 16), or 2x^2 - 32.

Since we know that 2x^2 = 968, the combined area of the two white rectangles is 968 - 32, or 936 square units. The correct answer is B.
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by [email protected] » Fri Dec 30, 2016 10:53 am
Hi All,

If you don't see the 'elegant' approach to this question, you can still solve it with some 'brute force' math (and a bunch of multiplication) - it's not 'pretty', but if you're comfortable doing math by hand then you can still get to the correct answer in 2 minutes:

We're told that the sum of two squares is 1000 and that the two numbers (before they are squared) differ by 8. Since the answer choices to the question are integers, it's likely that the two side lengths of the two squares are also integers. Let's list out some perfect squares and get a sense of how big the numbers can get...

20^2 = 400
30^2 = 900

Since the larger square is clearly bigger, it's area will be more than half of the 1000, so it's side length will fall into the range of 20-30. The side length of the smaller triangle will be 8 LESS... Now let's try to narrow things down a bit...

15^2 = 225
25^2 = 625
17^2 = does NOT end in a 5, so there's no way the sum would be 1000, but this example probably isn't too far away from the correct answer...

26^2 = 676
18^2 = 324

This IS the sum that we're looking for, so these MUST be the dimensions of the two squares. By extension, the dimensions of the two white rectangles are 26x18 and the area of each is 468. Since there are two of those rectangles, their total area = 468(2) = 936.

Final Answer: B

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