Problem Solving

[GMAT math practice question]

In the diagram, each of a, b, c, ... ,g represents one of the numbers 1, 2, 3,... and 7, with no repetition. These values correspond to the number of elements in each part of the sets. If sets A, B and C have the same number of elements and the sum of the numbers of elements of A, B and C is as small as possible, find the value of b+d+f.

A. 5
B. 7
C. 9
D. 11
E. 13

=>

The number of elements in set A is a + b + c + d. The number in set B is b + c + e + f, and the number of elements in set C is c + d + f + g. The sum of the sizes of these three sets is then a + 2b + 3c + 2d + e + 2f + g.

To minimise this total, c must have the smallest value as its coefficient is the largest. Thus, c = 1. The next smallest values must be given to b, d, and f as they have coefficients of 2. Thus, b,d and f are chosen from 2, 3 and 4.
It follows that b + d + f = 2 + 3 + 4 = 9.

Therefore, the answer is C.
Answer: C