The mean of three numbers is 6 more than the least of the numbers and 8 less…

Concept: For three numbers arranged in ascending order, the median is the middle value, and the mean equals one third of the sum of all three numbers. When the mean is described in terms of how much more or less it is than the least and greatest values, those descriptions become two linear equations in the two unknown extreme values, which can be solved simultaneously alongside the known median.

1. Let the three numbers be x, 7, and y, where x is the least, 7 is the given median, and y is the greatest, so x ≤ 7 ≤ y.

2. Mean = (x + 7 + y)/3. Since the mean is 6 more than the least: (x + 7 + y)/3 = x + 6, which simplifies to 2x − y = −11 —(i)

3. Since the mean is 8 less than the greatest: (x + 7 + y)/3 = y − 8, which simplifies to x − 2y = −31 —(ii)

4. From (i), y = 2x + 11. Substituting into (ii): x − 2(2x + 11) = −31 ⇒ x − 4x − 22 = −31 ⇒ −3x = −9 ⇒ x = 3.

5. Substituting x = 3 back into y = 2x + 11 gives y = 2(3) + 11 = 17.

6. Sum of the three numbers = x + 7 + y = 3 + 7 + 17 = 27.

Cross-check: Mean = 27/3 = 9. Least + 6 = 3 + 6 = 9 ✓, and Greatest − 8 = 17 − 8 = 9 ✓ — both given conditions hold, confirming the sum is 27.