## At a certain company…

## Solution:

We are given that a group of men and woman at a company received an average score of 80 on a test. We must determine whether the average score for the women was greater than 85.

We must recognize that this is a weighted average problem. The weighted average of two different sets of data points will be closer to the average of the set of data points with the greater number of observations. Thus, if there are more women than men, the overall weighted average of the test score will be closer to the average score of the women than that of the men. The same idea applies if there are more men than women.

**Statement One Alone: **

The average score for the men was less than 75.

Statement one gives us a range of the average test score for the men (less than 75). However, since we do not know anything about the number of men or women, we still do not have enough information to determine whether the average score for the women was greater than 85. Statement one is not sufficient to answer the question. We can eliminate answer choices A and D.

**Statement Two Alone: **

The group consisted of more men than women.

From statement two we know that there are more men than women. However, without any information about the average score of the men, we cannot determine whether the average score for the women was greater than 85. Statement two is not sufficient to answer the question. We can eliminate answer choice B.

**Statements One and Two Together: **

From statements one and two we know that the average score for the men was less than 75 and that the group consisted of more men than women. We also know that the total weighted average of the entire group was 80.

Since there are more men than women we know that the overall weighted average is closer to the average of the men than to the average of the women. However, we are given that the average of the men was less than 75. Thus, the average score for the women must be greater than 85.

This conclusion may be difficult to see, so let’s explain it further. Let’s assume that the average score for the men was exactly 75. We know that the weighted average for the entire group is 80. If there were an equal number of men and women, it would follow that the average score for the women was 85. However, we know that there were actually more men than women and that the average score for the men was actually less than 75. Thus, the only way that the weighted average could be 80 would be if the average score for the women were greater than 85.

We can support our conclusion with some actual numbers:

Let’s say the average score for the men was exactly 75. Since 80 – 75 = 5, the men’s average is 5 units from the weighted average of 80. Since there are more men than women, the men’s average is closer to the weighted average than is the women’s average. It follows that the women’s average must be more than 5 units from the weighted average of 80. That is, the women’s average must be greater than 80 + 5 = 85.

However, the average score for the men was actually less than 75, so let’s say it was 74. We see that the men’s average is 6 units below the weighted average, which means the women’s average must be more than 6 units above the weighted average. That is, the women’s average must be greater than 80 + 6 = 86. Therefore, regardless of how much the men’s average was less than 75, we see that the women’s average will always be greater than 85.

**Answer: C**