GMAT Official Guide Quantitative Review 2019_ Book

4.5 Answer Explanations

PS00904 �
134.The figure shown above consists of three identical
circles that are tangent to each other. If the area of the
shaded region is 64岛-32冗，what is the radius of
each circle?
(A) 4
(8)^8
(Cl 16
(D)^24
(El 32

Geometry

Let r represent the radius of each circle. Then the

triangle shown dashed in the figure is equilateral

with sides 2r units long. The interior of the

triangle is comprised of the shaded region and

three circular sectors. The area of the shaded

region can be found as the area of the triangle

minus the sum of the areas of the three sectors.

Since the triangle is equilateral, 兀sside lengths

are in the proportions as shown in the diagram

below. The area of the interior of the triangle is

炉r)(r句=r飞．

r r

讨5

2r

Each of the three sectors has a central angle of

60 ° because the central angle is an angle of the

equilateral triangle. Therefore, the area of each

60 1

sector is—= -of the area of the circle. The

360 6

sum of the areas of the three sectors is then

3(.!. 冗r^2 )=.!.冗r^2. Thus, the area of the shaded

6 2

region is r^2 喜；矿=r^2 (.Jj卡）But, this

area is given as 64.Ji -32冗= 64 (.Ji卡）．

Thus r2 = 64, and r = 8.

Th e correct answer 1s B.

PS02053

135. In a numerical table with 10 rows and 10 columns,
     each entry is either a 9 or a 10. If the number of 9s in
     the nth row is n - l for each n from 1 to 10, what is
     the average (arithmetic mean) of all the numbers in the
     table?

(A) 9. 45

(B) 9. 50

(C) 9. 55

(D) 9. 65

(El 9.70

Arithmetic

There are (10)(10) = 100 entries in the table. In

rows 1, 2, 3, ... , 10, the number of 9s is 0, 1, 2, ... ,

9, respectively, giving a total of O + 1 + 2 + ... + 9 =

45 entries with a 9. This leaves a total of

100 - 45 = 55 entries with a 10. Therefore, the

sum of the 100 entries is 45(9) + 55(10) =

405 + 550 = 955, and the average of the

100

.. 955
entnes 1s—= 9.55
100
Th e correct answer 1s C.
PS08485
136. A pos巾ve integer n is a perfect number provided that
the sum of all the positive factors of n, including 1 and
n, is equal to 2n. What is the sum of the reciprocals of
all the positive factors of the perfect number 28?

(Al

1-4

(Bl

56

27

(Cl 2

(Dl^3

(El^4