Question

A set of seven integers has a median of 73, a mode of 79, and a mean of 75. What is the least possible difference between the maximum and minimum values in the set?

Answer 1

The mode is the most freqently occuring value. Assume three of the integers each have the value 79 (remember we are trying to keep the least possible difference between max and min values in the set and obviously there must be at least two integers with the value 79).
The sum of the seven integers is 7 times the mean. Sum = 75 * 7 = 525
If three of the integers are 79 then the sum of the remaining integers must be:
525 -(3 * 79) = 288
The median 73 must be included in the seven integers, therefore by subtracting 73 from 288 we can find the sum of the three remaining integers which is 288 - 73 = 215
All three of the remaining integers must be less than 73 and as close as possible in value to 73. Trying the values 72,72 and 71 gives a sum of the three equal to 215.

The solution therefore to the least possible difference between max and min values in the set is maximum value 79 minus minimum value 71 =79 - 71 = 8