The least common multiple (LCM) of two numbers is 48. The two numbers are in the ratio 2 : 3. Determine the sum of the numbers with full working.

Introduction / Context:
Reiterating the standard method: when numbers are in a simple ratio and the LCM is known, we scale the ratio terms to meet the LCM and then compute the actual numbers and their sum.

Given Data / Assumptions:

• LCM = 48
• Ratio = 2 : 3

Concept / Approach:
Let the numbers be 2k and 3k. For coprime 2 and 3, LCM(2k, 3k) = 6k. Match 6k to 48 to find k, then sum the two values.

Step-by-Step Solution:
6k = 48 ⇒ k = 8.Numbers: 16 and 24.Sum = 16 + 24 = 40.

Verification / Alternative check:
LCM(16, 24) = 48; ratio 16:24 simplifies to 2:3.

Why Other Options Are Wrong:
28, 32, 64, and 36 do not correspond to the correct scaled values that achieve LCM 48.

Common Pitfalls:
Forgetting that 2 and 3 are coprime so the LCM with scaling is 6k; arithmetic slips when solving for k.

Final Answer:
40