## Average Problem

#### Problem

For a set of five whole numbers, the mean is 4, the mode is 1, and the median is 5. What are the five numbers?

#### Solution

As the mode is 1, there must be at least two 1's. But because the median is 5, the third number must be 5, and so we have the set of numbers {1, 1, 5, `x`, `y`}.

If the mean is 4, the sum of the numbers must be 45 = 20; that is, 1 + 1 + 5 + `x` + `y` = 20 `x` + `y` = 13.

Without loss of generality (WLOG), let `x` `y`, and if `x` = `y`, we get `x` + `x` = 13, 2`x` = 13, `x` = 6.5. Clearly `x` 5, and so 5 `x` 6.5

However, if `x` = 5, we would have two modal values: 1 and 5. Hence we deduce that `x` = 6, `y` = 7, and the set of five numbers must be {1, 1, 5, 6, 7}.

Problem ID: 232 (31 Jul 2005) Difficulty: 1 Star