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If the average of five positive integers is 16, and the

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by Max@Math Revolution » Mon Jun 18, 2018 12:48 am

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[GMAT math practice question]

If the average of five positive integers is 16, and the largest of the integers is 40, then the median of the five integers could be which of the following?

I. 10
II. 15
III. 20

A. I only
B. II only
C. III only
D. I & II only
E. I, II & III only
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by Max@Math Revolution » Wed Jun 20, 2018 12:35 am
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Since the average of the five positive integers is 16, their sum is 5*16 = 80. So, the four smallest numbers must sum to 80 - 40 = 40.

As the integers are positive, the smallest possible median occurs if the integers are 1,1,1,37 and 40. The largest possible median occurs if the third and fourth integers are as large as possible. For this to occur, the two smallest integers must be as small as possible, that is, 1 and 1. In this case, the two remaining integers add to 40 - (1 + 1) = 38. The largest possible median occurs if the two remaining integers are equal to 38/2 = 19. Therefore, the median lies between 1 and 19, inclusive. So, 20 is not the median.

If the numbers are 1,1,10,28 and 40, the median is 10, and if the numbers are 1,1,15, 23 and 40, the median is 15.

Therefore, the answer is D.

Answer: D
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by Scott@TargetTestPrep » Wed Jun 20, 2018 4:05 pm
Max@Math Revolution wrote:[GMAT math practice question]

If the average of five positive integers is 16, and the largest of the integers is 40, then the median of the five integers could be which of the following?

I. 10
II. 15
III. 20

A. I only
B. II only
C. III only
D. I & II only
E. I, II & III only
The sum of the integers is 16 x 5 = 80, and the largest integer is 40, so we have a sum of 40 for the remaining 4 integers.

If all the 4 remaining integers were 10, then 10 could be the median. Let's test 15. If the median were 15, we'd have 1 integer, from the remaining 4, equal to or larger than 15 and the rest equal to or smaller than 15.

a + b + 15 + c = 40

Since c could be 15 and a + b could be 10, 15 also works.

Let's test 20.

a + b + 20 + c = 40

We see that c must be at least 20, which means a + b = 0. However, the integers are positive, so a + b can't be 0. Therefore, 20 cannot be the median.

Answer: D

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