If the average test score of four students is 85, which of the following scores could a fifth student receive such that the average of all five scores is greater than 84 and less than 86?

Indicate all such scores.

A. 88

B. 86

C. 85

D. 83

E. 80

The sum of the first four students scores is 85*4 = 340.

If the average of all 5 students is 84, then the sum would be 84*5 = 420.

If the average of all 5 is 86, then the sum would be 86*5 = 430.

So for the average to be between 84 and 86, the sum has to be between 420 and 430. Since we already have 340 accounted for, the sum has to be between (420 - 340) and (430 - 340), or between 80 and 90.

Choice A, B, C, and D are all between 80 and 90, but E is not.

Since the average of four scores is 85. We want to add a score that makes the average of five scores fall between 84 and 86. Whenever we add a value equal to the average the average stays the same.
So adding a score of 85 will not affect the average of five scores. To make the average more than 85 we need a score greater than 85. to make the average less than 85 we need a score less than 85.
A score of 84 is 1 less than the old average 85. We can take this difference and divide it by 5 and subtract that from the old average 85. we see that 1/5 is 0.20. When we subtract 0.20 from 85 we get 84.8 which is greater than 84 we get one answer value.

When we add 88 we see that difference between old average 85 and 80 is 3. Now divide the difference 3 by 5, giving us 0.60 which we can add to old average 85. Now 85+0.6 =85.6 this works as well as fifth score. We can test other numbers this way and see that only 80 does not work. So A,B,C,D is the answer.