# How do you find a set of numbers that will satisfy the following conditions?

## The median of a set of 20 numbers is 24. -the range is 42. -to the nearest whole number the mean is 24. -no more than three numbers are the same.

Dec 2, 2017

3, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 45.

#### Explanation:

There are 20 numbers in the set. Since this is an even number of items, the median will be found as the result of averaging the two values in the center of the set. To ensure the median comes out to be 24, we can either have both numbers be 24, or we can choose two numbers that average to 24 (such as 23 and 25).

The range is the difference between the highest and lowest value. To be 42 requires that $\max - \min$ = 42.

The mean of the 20 numbers should come out to roughly 24. This means that the sum of the 20 numbers should be roughly 20*24 = 480.

If we use 24 and 24 as the innermost numbers, that leaves about 480 - 24 - 24 = 432 left for 18 numbers. We can "space out" the min and max around a center of 24 by using the values 3 and 45 (which places each half of the range above and below). That leaves 432 - 3 - 45 = 384 for 16 numbers.

As it turns out, 384 is perfectly divisible by 16, resulting in 24. It would be nice if we could say all 16 numbers are 24, but we're told that no more than 3 numbers are the same. Thus, we can simply space out the numbers in pairs around 24:
23 and 25, 22 and 26, 21 and 27, 20 and 28, 19 and 29, 18 and 30, 17 and 31, and 16 and 32.

This gives us:

3, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 45.