PSAT Math Multiple-Choice Question 133: Answer and Explanation

Question: 133

A high school class is selling barrels of popcorn to raise money. The histogram below shows the number of students that sold each quantity of barrels.

Which of the following is true?

I. The mode number of barrels sold is equal to the median number of barrels sold.

II. The median number of barrels sold is equal to the mean number of barrels sold.

III. The mode number of barrels sold is equal to the range of the number of barrels sold.

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

Correct Answer: B

Explanation:

B

The question asks for a true statement based on the data in a histogram. Read the numbered statements carefully and use Process of Elimination. There are various statistical measures in the statements, so start with the easiest to calculate: mode and range. Statement (III) says that the mode is equal to the range. The mode is the most commonly occurring value, so in this case it will be the bar with the most students, which is 5. The range of a list of values is the greatest value minus the least value, so in this case, that is 5 - 0 = 5. Because the mode and range are equal, (III) is true. Eliminate (A) and (C), which do not contain statement (III). Compare the remaining answer choices. Choice (D) includes statement (II), which indicates that the median is equal to the mean. The median is the value in the middle of all the values, when all the values are arranged consecutively. There are 6 students who sold 0 barrels of popcorn, 2 who sold 1 barrel, 3 who sold 2, 3 who sold 3, 7 who sold 4, and 9 who sold 5. Therefore, there are 6 + 2 + 3 + 3 + 7 + 9 = 30 students total. Because there is an even number of students, the median will be average of the 15^th and 16^th students. Starting from the lower number of barrels of popcorn, there are 6 + 2 + 3 = 11 students who sold 0 to 2 barrels. The next 3 students sold 3 barrels, bringing the list up to 14 students who sold 0 to 3 barrels. Because 7 students sold 4 barrels of popcorn, the 15^th and 16^th student both sold 4 barrels of popcorn. Therefore, the median is 4. To find the mean or average, use the formula T = AN, in which T is the total, A is the average, and N is the number of things. To find the total number of barrels, multiply the number of students who sold a certain number of barrels by the number of barrels they sold and add up the results. This becomes (6 × 0) + (2 × 1) + (3 × 2) + (3 × 3) + (7 × 4) + (9 × 5), which is 0 + 2 + 6 + 9 + 28 + 45 = 90. Therefore, T = 90 and N = 30 for the number of students. The average formula becomes 90 = A(30). Divide both sides by 30 to get 3 = A. This does not match the median, so statement (II) is false. Eliminate (D) which contains statement (II). The correct answer is (B).