GMAT Data Sufficiency questions can take simple concepts like averages and have test-takers pausing or falling into traps because of the way they are worded, and the fact that you have to keep in mind what your goal is with Data Sufficiency—to find out whether or not you have sufficient information to answer the question!
Problem:
Each of the 8 numbers s, t, u, v, w, x, y and z is positive. Is the average (arithmetic mean) of s, t, u, v, w, x, y and z greater than 46?
(1) The average (arithmetic mean) of s, t, u, v and w is greater than 74.
(2) The average of x, y and z is greater than 120.
Solution:
Before evaluating the statements, you should reword the question. We are asked if the average of a list of numbers is greater than 46. Since average is equal to the sum of the terms divided by the number of terms, we can write this question as: is sum/8 > 46? This can be simplified to: is sum > 46(8) or is sum > 368?
Statement 1 tells us that the average of the first five numbers in our set is greater than 74. At first, this may seem insufficient, as it tells us nothing about x, y and z. However, if five of our numbers have an average greater than 74, it means that those numbers must sum to a result greater than 74 x 5, which equals 370. If the sum of five numbers is greater than 370 and the other three numbers must all be positive, the overall sum must still be greater than 370. If the sum is greater than 370, then it is also greater than 368. Therefore, based on statement 1, we can answer the question as ‘always yes,’ which is sufficient.
Statement 2 we must approach in a similar manner. Now we know that the final three numbers in our set must have an average greater than 120. This means they must have a sum greater than 360. The other five numbers in the set can be equal to 1 at the smallest, therefore the total sum must be greater than 365. As 365 is smaller than 368, the average may or may not be greater than 46. ‘Sometimes yes, sometimes no’ is insufficient. So our final Data Sufficiency answer choice is (A) or choice (1); the first statement is sufficient to answer the question, the 2nd is not.
~Bret Ruber
