Question

If you have 5 distinct positive integers and the median is 17 and the mean is 12, what are the 5 numbers? How do you know for sure? Can you find 5 distinct positive integers with a median of 17 and a mean of 10? What whole-number means will work for 5 distinct positive integers with a median of 17? What medians will work for 5 distinct positive integers with a mean of 12?

Answer 1

1) Part 1: 5 distinctive positive integers with median 17 and mean 12

If the median is  17, that means that one number is 17, two numbers are greater than 17 and two numbers are less than 17.

Remember that the five numbers are distint positive integers.

If the mean is 12, that means that the sum of the five numbers is 12 * 5 = 60

The minimum values could be 1 and 2, with which the partil sum of the three first terms would be 1 + 2 + 17 = 20.

So, the partial sum of the two greater terms is 60 - 20 = 40.

That leads to these possibilities:

1, 2, 17, 18, 22
1, 2, 17, 19, 21

If the two smaller terms are 1 and 3, the possibilities are:

1, 3, 17, 18, 21
1, 3, 17, 19, 20.

If the two smaller terms are 2 and 3, the only possibility is:

2, 3, 17, 18, 20

If the smaller terms are 1 and 4, the only possibility is

1, 4, 17, 18, 20

Other options are:

1, 5, 17, 18, 19

2, 4, 17, 18, 19

You can realize that you cannot use other set of numbers.

2) Part 2: five distinctive numbers with median 17 and mean 10.

sum of the five numbers = 5 * 10 = 50

Given that 17 + 18 + 19 = 54, it is not possible to find those five distinctive numbers.

3) Part 3: What whole-number means will work for 5 distinctive integers with a median of 17.

=>the minimum whole-number means that will work is the whole number equal or greater than [1+2+17+18+19] / 5 = 11.4 => 12.

So, the whole-number means that will work are 12 or more.

4) Part 4. What medians will work for 5 distinctive positive integers with a mean of 12:

total sum = 5*12 = 60.

The smallest median would be 3, because the smaller values would be 1 and 2. The highest median would be.

Also, 1 + 2 = 3 => the median + the two greatest values = 60 - 3 = 57.

=> 18, 19, 20

=> 1, 2,18, 19, 20

So the median has to be between 3 and 18, inclusive.