PSAT Math Multiple-Choice Question 46: Answer and Explanation

Question: 46

A student took five tests. He scored an average (arithmetic mean) of 80 on the first three tests and an average of 90 on the other two. Which of the following must be true?

I. The student scored more than 85 on at least one test.

II. The average (arithmetic mean) score for all five tests is less than 85.

III. The student scored less than 80 on at least two tests.

• A. I only
• B. II only
• C. I and II
• D. II and III

Correct Answer: C

Explanation:

C

The question asks what must be true among three statements about a set of averages. For averages, use the formula T = AN, in which T is the total, A is the average, and TV is the number of things. For the first three tests, 80 is the average and 3 is the number of things. The formula becomes T = (3)(80) = 240. Use the formula again for the last two tests, with 90 as the average and 2 as the number of things to get T = (2)(90) = 180. Now, use the formula one more time to find the average for all five tests. The total for all 5 tests is 240 + 180 = 420, so the formula becomes 420 = A(5). Divide both sides by 5 to get 84 = A. Now evaluate each statement and use Process of Elimination. The final two tests prove that the student must have scored more than 85 on at least one test. If the student scored the same on each of the second two tests, his score would be 90 on each. He could have also scored 0 on one test and 180 on the other test, or any other combination of two numbers whose sum is 180. No matter what, there's at least one test on which the student scored more than 85, so (I) must be true. Eliminate (B) and (D). No remaining answer choices include (III), so only consider (II). The student's average for all five tests was 84, which is less than 85. Statement (II) is true. The correct answer is (C).