The sum of the squares of three numbers is 532, and the ratio of the first to…

Concept: When two ratios share a common term, scale each ratio so that the shared term matches, then combine them into one three-part ratio. Express each quantity as that ratio's part times an unknown k, and use the extra numeric condition given (here, a sum of squares) to solve for k.

Applying this to the given numbers:

1. Let the first, second and third numbers be in the ratio x : y : z. It is given that x : y = 3 : 2 and y : z = 3 : 2.

2. Scale both ratios so the common term y matches in both: x : y = 3 : 2 = 9 : 6, and y : z = 3 : 2 = 6 : 4. So the combined ratio is x : y : z = 9 : 6 : 4.

3. Let the three numbers be 9k, 6k and 4k for some positive k.

4. Sum of squares: (9k)2 + (6k)2 + (4k)2 = 532, i.e. 81k2 + 36k2 + 16k2 = 532.

5. Combine like terms: 133k2 = 532, so k2 = 4, giving k = 2 — taking the positive root, since a ratio between quantities is conventionally expressed using positive values.

6. Second number = 6k = 6 × 2 = 12.

Cross-check: with k = 2 the three numbers are 18, 12 and 8; their squares sum to 324 + 144 + 64 = 532, matching the given total, and both 18 : 12 and 12 : 8 reduce to 3 : 2, as required.