The L.C.M. of two numbers is 48. The numbers are in the ratio 2 : 3. Find the sum of the two numbers.

Introduction / Context:
When two numbers are in a given ratio, they can be represented as multiples of a common factor k. Their LCM then depends on both the ratio and k. Given the LCM numerically, we can solve for k and then compute the actual numbers and their sum.

Given Data / Assumptions:

• LCM = 48.
• Numbers are in the ratio 2 : 3.
• Let the numbers be 2k and 3k, with k a positive integer.

Concept / Approach:
For 2k and 3k, the LCM is 6k (because 2 and 3 are coprime, the LCM of 2k and 3k is lcm(2,3)*k = 6k). Set 6k = 48 to find k. Then compute the numbers and their sum.

Step-by-Step Solution:

Verification / Alternative check:
LCM(16, 24) = 48 indeed (since 16 = 2^4, 24 = 2^3*3, LCM has 2^4*3 = 48). The ratio 16:24 simplifies to 2:3. Everything is consistent.

Why Other Options Are Wrong:

• 28, 32, 64, 36: Not equal to the sum of any pair in the ratio 2:3 whose LCM is 48.

Common Pitfalls:

• Computing LCM of 2k and 3k incorrectly as 3k or 2k.
• Picking numbers with the correct ratio but wrong LCM.

Final Answer:
40