The fourth proportional to 9, a and 6a is 54. What is the positive value of a?

Concept: The fourth proportional to three quantities p, q, and r is the value x that completes the proportion p : q :: r : x, so cross-multiplication gives x = (q times r) / p.

Application: Here p = 9, q = a, and r = 6a, and the fourth proportional x is given as 54.

1. Write the proportion: 9 : a :: 6a : 54.

2. Cross-multiply the proportion: 9 times 54 = a times 6a.

3. Simplify the right side: a times 6a = 6a2.

4. So 6a2 = 486, which gives a2 = 81.

5. Taking the positive square root, a = 9.

Cross-check: Substituting a = 9 gives the three numbers 9, 9, and 54; the proportion 9 : 9 :: 54 : 54 holds because both ratios equal 1, confirming the value.