If the mean proportion of 24 and A is 72, then what is the value of A?

Concept: The mean proportional (geometric mean) of two positive numbers a and c is the number b satisfying a : b :: b : c, i.e. b2 = a × c.

1. Let the two given numbers be 24 and A, with mean proportional 72.

2. By the mean proportional relation, 722 = 24 × A.

3. Compute 722 = 5184.

4. So 24 × A = 5184, giving A = 5184 ÷ 24 = 216.

Cross-check: 24 × 216 = 5184, and √5184 = 72, which matches the given mean proportional. Hence A = 216.