# Arithmetic mean of 4 numbers word problem

Encountered this in a timed online test, only allowing 20 seconds per question:

The average (arithmetic mean) of 4 numbers is greater than 8 but less than 14. Which of the following could not be the sum of these four numbers?

Answer choices: 44, 53, 38, 31, 48

8 x 4 is 32 and 14 x 4 is 56, so the sum should be somewhere between 32 and 56, NOT inclusive, is that correct (thus the answer being 31)

I've seen a variation of this question where one of the answers was 32 itself. If the average is greater than 8, wouldn't that mean it's >32 - not inclusive?

What is this question looking for?

• Your answer is correct, and yes, if the average is greater than $8$, the sum must be strictly greater than $32$. The point is apparently to see whether you understand the relationship between the mean and the sum, and whether you can do some very elementary arithmetic quickly. Oct 5 '20 at 6:34

You're correct. Since the average must be greater than $$8$$, it cannot include $$8$$. And since $$8\cdot 4 = 32$$, when adding all the terms, you can't have $$32$$. This also means you cannot have $$56$$; all numbers must be between $$32$$ and $$56$$ non-inclusive, so $$33, 34, \dots, 54, 55.$$