Three numbers in ratio with a sum condition: Three numbers are in the ratio 3 : 4 : 5. The sum of the largest and the smallest equals the sum of the remaining number and 52. Find the smallest number.

Introduction / Context:
This is a direct application of ratio representation with an extra linear relationship. Identifying which term is largest and smallest from the ratio lets you set up a simple equation in one variable.

Given Data / Assumptions:

• Numbers ∝ 3 : 4 : 5 ⇒ let them be 3k, 4k, 5k.
• Largest + smallest = middle + 52.

Concept / Approach:
From 3k, 4k, 5k, the largest is 5k and the smallest is 3k; the “remaining number” is 4k. Use 5k + 3k = 4k + 52 to solve for k and then compute the smallest, 3k.

Step-by-Step Solution:
5k + 3k = 4k + 52 ⇒ 8k = 4k + 52.4k = 52 ⇒ k = 13.Smallest = 3k = 39.

Verification / Alternative check:
Numbers are 39, 52, 65. Check: 65 + 39 = 104 and 52 + 52 = 104, condition satisfied.

Why Other Options Are Wrong:

• 20 and 27 are smaller than the correct smallest derived from k = 13.
• 52 is the middle number, not the smallest.

Common Pitfalls:

• Misidentifying the “remaining number” (it is the middle one, 4k).

Final Answer:
39