12 Days of Christmas GMAT Competition - Day 1: Five identical circles

Question

12 Days of Christmas GMAT Competition with Lots of Fun https://gmatclub.com/forum/download/file.php?id=59427 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 https://gmatclub.com/blog/wp-content/uploads/2020/12/12-days-of-christmas.jpg M37-101 1.png

Bunuel · Accepted Answer

Official Solution: https://gmatclub.com/forum/download/file.php?id=88245 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. Answer: B GMAT-Club-Forum-hhm37anx.png

Pooki · Answer

Answer B: area of grey = 40 , so area of haft of 5 circle = 40+ 5*2= 100 , hence area of 1 cirle = 100/5=20

jle21 · Answer

Option B is the correct answer.

The line connecting the bottom of the first circle to the top circle cuts the total area of our figure in half (if we rotate the figure 180 degrees, the green and grey areas would overlap each other) so if the grey area is 40, then the total area covered by our figure is double that, or 80.

Now to solve the problem of the area of one circle, we can find the area of all five circles combined IF they were not overlapping and divide by five. Since the overlapping area between any two adjacent circles is 5, then that means that for each overlapping area we are missing 5 to the maximum possible area covered by our five circles (when they don't overlap). Since there are four overlapping areas between the five circles, we are missing 20 from the total non-overlapping area. We add that 20 to the 80 from our figure and conclude that, if none of the circles were overlapping, then the combined area of all five circles would be 100.

Since we are interested in the area of one circle, divide the 100 units by 5 and we have 20.

Eswar42 · Answer

Since, the line symmetrically divides the enclosed area, so the total enclosed area is 2*40 = 80
Lets get the total area of 5 circles with no overlap = 80+4*5 = 100, (here 4 is for number of overlaps)
the area of one circle = 100/5 = 20
So answer is B

RishS94 · Answer

The line divides figure in equal parts.

Green Area  =  Grey Area  =40
Total Area of figure = 80

Now add overlapped area = 80 +(5*4) = 100
Area of each circle = 100/5 =20

Deepakjhamb · Answer

Image

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

We need to find area of 1 circle , all circles are identical given . The best approach would be thus to find area of 5 circles from the figure and divide by 5 to give area of 1 circle .

Now we are given some grey shaded portion which actually divides the area of( 5 circles - overlap ) into 2 parts

So area of(5 circles) - overlap = 40*2 = 80

so area of 5 circles = 80 + overlap
5 is the area of overlap of adjacent circles

so area of 5 circles = 80 + 4 *5 = 100

so area of 1 circle = 100/5 = 20

so answer is b) 20

GCMEMBER · Answer

Let the circles be A, B, C, D, and E from left to right

The grey region of A and E plus one overlap region makes one circle
The grey region of B and D plus one overlap region makes one circle
The grey region of C makes half circle.

Total circles = 2.5
Area = 2.5*πr^2 = grey area + two overlaps
2.5*πr^2 = 40 + 2*5
πr^2=20

Area of one circle = 20

Option B

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rocky620 · Answer

Since the line is connecting the Bottom and Top of first and fifth circle respectively, it divides the figure in two equal parts.
The area under the line is 40 units so the total area will be 80 units.

From the above image total area =>  2x + 4y + 3z = 80 ........eq1

Area of a circle =>  x+y  = 2y+z => x-y=z.........eq2
 y=5  (given)......eq3

Substituting values of z and y from eq2 & eq3 in eq1:

2x + 4*5 + 3*(x-5) = 80
2x + 20 + 3x - 15 = 80
5x + 5 = 80
5x = 75
so x= 15

Area of one circle = x + y =>15 + 5 =>  20

Option B

bb · Answer

Possible Solutions:

Total area = 40*2=80 but there are 4 overlaps so the total area is 100 and when dividied by 5, it is 20
The line intersects the circles in the middle and so we have 2.5 circles shaded which first made me think it was 40/2.5 = 16 but I forgot about the shaded areas. There are 2 shaded areas, so it is 50/2.5 = 20
You can make a triangle and find the side (not great/ideal).
You can back solve using the numbers and circle area formula - PR^2. Also probably longer than other approaches but when you can't find an elegant solution, brute force can help.

P.s. Tip to those who found the answer to be "about 12" - it is exact on the GMAT. Unless the answer is exactly 12, it is not right. So if it is 16 or 12.6667 or 12.3333, that's a hint for you that you did not find the right answer

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harshitasinghal · Answer

Doesn't the total area include the overlapping area of the circles as well? So technically, the overlapping areas are being double counted so (total area - overlapping area)/5 = 60/5 = 12 = area of 1 circle?

Bunuel · Answer

No. Here, 80 is the area of the union of the five circles, so the overlap parts are already counted only once there, not twice. That is exactly why you add the four overlaps back once to get the sum of the five individual circle areas.

12 Days of Christmas GMAT Competition - Day 1: Five identical circles

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