Numbers with Fixed Ratio and a Composite Condition Two numbers are in the ratio 3 : 2. If 10 and the sum of the two numbers are added to their product, the result is 16 squared (256). What is the smaller number?

Introduction / Context:
Here, ratio information lets us parametrize the numbers in terms of a single variable. The combined condition involving product, sum, and a constant then determines that variable precisely.

Given Data / Assumptions:

• Let the numbers be 3k and 2k.
• Product + sum + 10 = 256.
• We need the smaller number (2k).

Concept / Approach:
Compute product and sum in terms of k, form a quadratic equation, and solve for k. Select the positive integer solution that makes sense in the context and then report 2k.

Step-by-Step Solution:
Product = (3k)(2k) = 6k^2.Sum = 3k + 2k = 5k.Condition: 6k^2 + 5k + 10 = 256 → 6k^2 + 5k − 246 = 0.Discriminant = 5929 = 77^2; k = (−5 ± 77)/12 → k = 6 (positive), or negative root (discard).Smaller number = 2k = 12.

Verification / Alternative check:
Numbers are 18 and 12. Product 216; sum 30; product + sum + 10 = 256 = 16^2, satisfied.

Why Other Options Are Wrong:
14, 16, and 18 are not equal to 2k with k = 6; 10 breaks the ratio with the corresponding partner.

Common Pitfalls:
Mistaking which is smaller; arithmetic slips forming the quadratic; forgetting to add the 10 constant.

Final Answer:
12