SAT Math Multiple Choice Practice Question 42: Answer and Explanation
20. In a set of five positive whole numbers, the mode is 90 and the average (arithmetic mean) is 80. Which of the following statements is false?
A. The number 90 must appear two, three, or four times in the set.
B. The number 240 cannot appear in the set.
C. The number 80 must appear exactly once in the set.
D. The five numbers must have a sum of 400.
E. The median cannot be greater than 90.
Correct Answer: C
C. Take each statement one at a time. Choice (A) is true. The mode appears most often, which means the set has to have two, three, or four 90s. Choice (B) requires you to remember this formula: Total = Number × Mean. In this case, the five numbers must add up to 5 × 80 = 400. Because you know the set has at least two 90s, which add up to 180, the other three numbers must add up to 220. But because the numbers are all positive, you wouldn't have enough room for the last two numbers if 240 were in the set, which makes Choice (B) true. Choice (C) is false because making a list whose average is 80 without including any 80s in the list is easy. Try it yourself and see.
You could, of course, stop there (and you probably should, to save time, on the real test). But I want you to give you your money's worth, so I continue through the rest of the answer choices. Choice (D) is definitely true; you used this fact already when you checked Choice (A). In Choice (E), the median of an odd number of numbers is the middle number. So if the median were greater than 90, then three of the five numbers would have to be greater than 90. But you already know from Choice (A) that the set must include at least two 90s. You can't have a list with two 90s and three numbers greater than 90 and still have the average be 80. Thus, the median can't be greater than 90.