M37-101

Question

https://gmatclub.com/gmat-practice-tests/resources/import/img/M37-101.png 
 
Five identical circles shown above have their centers equally spaced on a straight horizontal line. Another line connects the bottom of the first circle and the top of the fifth circle. If the area of the grey region, under the line enclosed by the circles, is equal to 40 and the overlapping area between any two adjacent circles is equal to 5. What is the area of one circle?

A.  12 
B.  20 
C.  25 
D.  30 
E.  60

Bunuel · Answer

Official Solution:

https://gmatclub.com/gmat-practice-tests/resources/import/img/M37-101.png 
 
Five identical circles shown above have their centers equally spaced on a straight horizontal line. Another line connects the bottom of the first circle and the top of the fifth circle. If the area of the grey region, under the line enclosed by the circles, is equal to 40 and the overlapping area between any two adjacent circles is equal to 5. What is the area of one circle?

A.  12 
B.  20 
C.  25 
D.  30 
E.  60

The figure given is symmetrical around the line crossing it, so if the area of the grey region is 40, then the area of the whole figure is 80.
 
The total area of five circles IF they were not overlapping would be the area of the figure PLUS four times the area of one overlap (each of the four overlaps belong to two circles). So, the area of five circles is  80 + 4*5=100 . The area of one circle is therefore, 20.
 
Note that while this question uses basic knowledge of lines and figures, it is actually not a Geometry question. You do not need to calculate the area of the circle, even if you may have wanted to. There are 8 questions within GMAT Prep Focus Edition that use similar principles.  Here is one example .

Answer: B

MayaLove · Answer

I solved it as :

5 (πr^2) - 20 = 2 (40) 
5 (πr^2) = 100
(πr^2) = 20

adityabadaya · Answer

This is a great question that’s helpful for learning.

RKhatri · Answer

Hi 
the question mentions area under the line in the grey region. That means that greay region includes shaded parts also. and over lapping parts also. Why have you not included the overlapping region in the value 40?

Bunuel · Answer

The 40 is the actual area under the line, and it already includes the overlaps correctly. We only adjust for overlaps when calculating the total area of 5 full circles, because overlaps would be double-counted there. You can check an alternative solution here: https://gmatclub.com/forum/12-days-of-c ... 42332.html Hope it helps.

Sunny3897 · Answer

I like the solution - it’s helpful.

ShlokGupta22 · Answer

How do we know that given figure is symmetrical?

Bunuel · Answer

The symmetry comes from the fact that the circles are identical and their centers are equally spaced.

GulfTube · Answer

Since the diagonal line is symmetrical, the shaded region (with overlaps) is 2 1/2 circles. 
if 2 1/2 circles have area of 40, then 1 circle has area of 16.

Since 16 is area of one circle with overlap, we add 5 to it. 16+5 = 21 ~ 20 . Answer B

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