The following figure contains three squares with areas of 100, 16 and 49 lying side by side as shown. By how much should the area of the middle square be reduced in order that the total length PQ of the resulting three squares is 19?

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Q: 146 (IAS/1996)
The following figure contains three squares with areas of 100, 16 and 49 lying side by side as shown. By how much should the area of the middle square be reduced in order that the total length PQ of the resulting three squares is 19?

question_subject: 

Maths

question_exam: 

IAS

stats: 

0,3,7,3,3,4,0

keywords: 

{'middle square': [0, 1, 0, 0], 'total length pq': [0, 1, 0, 0], 'squares': [0, 0, 1, 0], 'area': [0, 0, 0, 1], 'following figure': [0, 3, 2, 4], 'areas': [0, 0, 1, 0]}

The question asks how much you should reduce the area of the middle square so that the total length PQ of the resulting three squares is 19. To understand the answer, it`s helpful to remember that the side length of a square equals the square root of its area.

In the original figure, you have squares with areas of 100, 16, and 49. Their respective side lengths are 10 (sqrt(100)), 4 (sqrt(16)) and 7 (sqrt(49)). This total length is 21, which is 2 units too long. Since you want to reduce the size of the 4-unit length (middle) square, you need to reduce its side length by exactly 2 units.

Option 1 (12) is the correct answer because the area of a square with a side length of 2 (the reduced length) is 4, and if you subtract this from the original 16, you get 12.

The other options (2, 4) are incorrect because they do not reflect the correct reduction in area for the middle square to result in a total length of 19 units.

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