Mean of three numbers is 10 more than the least of the numbers and 15 less…

For three numbers, the mean equals one-third of their sum: Mean = (Least + Median + Greatest) / 3. When the mean is described as a fixed amount more than the least value and a fixed amount less than the greatest value, both descriptions equal the same mean and can be written as two linear equations in the least and greatest values. Solving those together with the known median value gives the least and greatest individually, and the sum then follows as 3 x Mean.

1. Let the least number be L, the median be 5, and the greatest number be G, with L <= 5 <= G.

2. "Mean is 10 more than the least" gives Mean = L + 10 -- Equation (1).

3. "Mean is 15 less than the greatest" gives Mean = G - 15, i.e., G = Mean + 15 -- Equation (2).

4. From (1) and (2): L + 10 = G - 15, so G = L + 25 -- Equation (3).

5. Using Mean = (L + 5 + G) / 3 together with Mean = L + 10 from (1): L + 10 = (L + 5 + G) / 3, which gives 3L + 30 = L + 5 + G, so G = 2L + 25 -- Equation (4).

6. Equating (3) and (4): L + 25 = 2L + 25, so L = 0.

7. From (3), G = 0 + 25 = 25, and Mean = L + 10 = 10.

8. Sum of the three numbers = 3 x Mean = 3 x 10 = 30.

Check: the three numbers are 0, 5, and 25. Their mean is (0 + 5 + 25) / 3 = 10, which is exactly 10 more than the least value (0) and 15 less than the greatest value (25 - 15 = 10); the median is 5, matching every given condition. So the sum is confirmed as 30.