Problem Solving

Sets M and N contain exactly m and n elements, respectively. What is the value of n?

(1) 7m=8n
(2) The intersection of M and N contains exactly 0.4m elements.

Mo2men wrote:Sets M and N contain exactly m and n elements, respectively. What is the value of n?

(1) 7m=8n
(2) The intersection of M and N contains exactly 0.4m elements.

(1) NOT SUFFICIENT. We know m & n must be positive integers, but there are numerous pairs for which this equation holds. m = 8, n = 7; m = 16, n = 14; etc.

(2) NOT SUFFICIENT. This tells us that exactly 2/5th of the total number of elements in Set M are also common to Set N. However, it tells us nothing about the values of m and n.

Combined (1) and (2). NOT SUFFICIENT. From (1), we know that m must be a multiple of 8. From (2), we know, that m must be a multiple of 5 (in order to get an integer value for 2m/5). Thus m must be a multiple of 40 (LCM of 5 and 8). There are infinite number of multiples of 40. We cannot obtain a specific value of either m or n.

Answer choice: E

Mo2men wrote:Sets M and N contain exactly m and n elements, respectively. What is the value of n?

(1) 7m=8n
(2) The intersection of M and N contains exactly 0.4m elements.

Statement 1:
n = (7/8)m.
Since n must be an integer, m must be a MULTIPLE OF 8.
If m=8, then n = (7/8)(8) = 7.
If m=16, then n = (7/8)(16) = 14.
Since n can be different values, INSUFFICIENT.

Statement 2:
Since (0.4)m = (2/5)m, and the number of elements common to M and N must be an INTEGER, m must be a MULTIPLE OF 5.
If m=5, then the number of elements common to M and N = (2/5)(5) = 2.
Implication:
It's possible that M and N have exactly 2 elements in common, with the result that n could be any integer such that n≥2.
INSUFFICIENT.

Statements combined:
Since m must be both a multiple of 8 and a multiple of 5, m must be a MULTIPLE OF 40.
If m=40, then n = (7/8)(40) = 35.
If m=80, then n = (7/8)(80) = 70.
Since n can be different values, INSUFFICIENT.

The correct answer is E.

S1: n is some multiple of 7; NOT SUFFICIENT

S2: This only gives us the intersection; NOT SUFFICIENT

Together, we know that m is a multiple of 8 and divisible by 5. That doesn't tell us anything about the actual value of either number, so we're still stuck.