The sum of the squares of three numbers is 532 and the ratio of the first and the second as also of the second and the third is $3 : 2$. The third number is

### Concept & Continuous Proportion When $A:B = B:C$, the terms are in continuous proportion. Combine them to form a single ratio string $A:B:C$. ### Step-by-Step Solution 1. **Given:** Ratio of first to second = $3:2$, second to third = $3:2$. 2. Let the numbers be $A, B, C$. $A:B = 3:2$ and $B:C = 3:2$. 3. Make the middle term $B$ common. Multiply $A:B$ by 3 and $B:C$ by 2. 4. $A:B = 9:6$ and $B:C = 6:4$. Thus, $A:B:C = 9:6:4$. 5. Let the numbers be $9x, 6x, 4x$. 6. Sum of squares: $(9x)^2 + (6x)^2 + (4x)^2 = 81x^2 + 36x^2 + 16x^2 = 133x^2$. 7. Equate to 532: $133x^2 = 532 \implies x^2 = 4 \implies x = 2$. 8. The third number is $4x = 4(2) = 8$. ### Exam Strategy & Shortcut Since the combined ratio is $9:6:4$, the third number must be a multiple of 4. Look at the options: 8, 12, 20 are all multiples, so substitution or the full method is required. 133 is roughly a quarter of 532 ($133 \times 4 = 532$), so $x = 2$ can be spotted quickly. ### Common Pitfall Forgetting to square the $x$ variable alongside the coefficients, leading to linear equations instead of quadratic ones. ### Final Answer Therefore, the correct answer is **8**.