Suppose ten distinct, positive integers have a median of 10. ("Distinct integers" means that no two integers are the same.)

What is the smallest the average of those ten integers could be?

Could you please explain your answer so i can understand how to go about this problem.

SmartMathMan
Dec 13, 2017

#1**+2 **

If the median is 10, the fifth and sixth values must sum to 20, because in an even set of values, the median is the sum of the middle two values divided by 2

We want the sum of the first four digits and the last four digits to be as small as possible

1,2,3,4 would work for the first four

If we let the fifth number be 5 the sixth would have to be 15 and the smallest that the remaining values could be is 16,17,18, 19.....but the average of this set = 10

The minimum average is achieved when the two numbers that sum to 20 are odds that differ by 2, i.e., 9 and 11

So...the data set that produces the smallest average is

1,2,3,4,9,11,12,13,14,15 = 8.4

CPhill
Dec 13, 2017