Question

At a certain company, a test was given to a group of men and women seeking promotions. If the average (arithmetic mean) score for the group was 80, was the average score for the women greater than 85?
(1) The average score for the men was less than 75.
(2) The group consisted of more men than women.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

Answer: (C)

Both statements together are sufficient, but neither statement alone is sufficient.

• The average score for a group of men and women was $80$. Letting $x$ be the average score for the women, determine whether the value of $x$ is greater than $85$.

1. The average score for the men was less than $75$. Let the average score for the men be $74$. Then we would get a variable value of $x$ where it may or may not be greater than $85$. NOT sufficient.
2. For statement 2, we will get the same case as statement 1. Taking (1) and (2) together, let $M$ be the sum of the $m$ men’s scores and let $W$ be the sum of the $w$ women’s scores it follows that $M+W=80(m+w)=80m+80w.$
3. From (1) it follows that $75m>M$ and from (2) it follows that $5m>5w.$ Hence, the average score for the women is greater than $85$