If 48 is the mean proportion between x and 192, what is the value of x?

Concept — The mean proportional (geometric mean) between two quantities is the middle term that turns the three quantities into a continued proportion: for a, m and b it is the value that makes a : m = m : b true. Cross-multiplying that equality gives m2 = a × b, so m = √(a × b), and rearranging for the first term gives a = m2 ÷ b.

Application — Here the middle term is 48 and the third term is 192, so the continued proportion to solve is x : 48 = 48 : 192.

1. Write the defining relation of a mean proportional: m2 = a × b, with m = 48, a = x and b = 192.

2. Substitute the given values: 482 = x × 192.

3. Evaluate the square: 2304 = 192x.

4. Divide both sides by 192: x = 2304 ÷ 192 = 12.

Cross-check — Substituting 12 back into the definition gives √(12 × 192) = √2304 = 48, which is the mean proportional the question states. The two ratios agree as well: 12 : 48 = 1 : 4 and 48 : 192 = 1 : 4.

So the first term is 12.