The ratio of two numbers is 3 : 4 and their H.C.F. is 4. What is the L.C.M. of these two numbers?

Introduction / Context:
This is a straightforward question that combines ratio and highest common factor (H.C.F.) to determine the actual numbers and then asks for their least common multiple (L.C.M.). It is a common pattern in aptitude tests because it connects basic ratio knowledge with H.C.F. and L.C.M. properties.

Given Data / Assumptions:

• The ratio of two numbers is 3 : 4.
• Their H.C.F. is 4.
• We must find the L.C.M. of the two numbers.

Concept / Approach:
If two numbers are in the ratio 3 : 4, we can write them as 3k and 4k for some positive integer k. The H.C.F. of 3k and 4k is k times the H.C.F. of 3 and 4. Since 3 and 4 are co-prime, their H.C.F. is 1, so the overall H.C.F. is k. We are given that this H.C.F. is 4, so k = 4. Once we know the numbers, we can compute their L.C.M. using either prime factorization or the formula involving product and H.C.F.

Step-by-Step Solution:
Step 1: Let the two numbers be 3k and 4k.Step 2: The H.C.F. of 3k and 4k is k * H.C.F.(3, 4) = k * 1 = k.Step 3: Given H.C.F. = 4, so k = 4.Step 4: Therefore, the actual numbers are 3 * 4 = 12 and 4 * 4 = 16.Step 5: Now find L.C.M.(12, 16).Step 6: Prime factors: 12 = 2^2 * 3, 16 = 2^4.Step 7: Take highest powers: 2^4 and 3^1.Step 8: L.C.M. = 2^4 * 3 = 16 * 3 = 48.

Verification / Alternative check:
We can also use the relation L.C.M. = (product) / H.C.F.Product of numbers = 12 * 16 = 192.Given H.C.F. = 4, so L.C.M. = 192 / 4 = 48. Both methods give the same result, confirming that the L.C.M. is 48.

Why Other Options Are Wrong:
Option a (12) and option b (16) are the smaller factors and cannot be the least common multiple since they do not include all prime powers needed to cover both 12 and 16. Option c (24) is a common multiple but not the least one that covers the largest factor 16 multiplied by 3. Only 48 satisfies the condition of being divisible by both 12 and 16 and being the least such positive integer.

Common Pitfalls:
Some students mistakenly treat the ratio numbers (3 and 4) themselves as the original numbers or misinterpret H.C.F. in that context. Others miscompute the L.C.M. of 12 and 16 by not taking the highest power of 2. Always reconstruct the original numbers carefully from the ratio before calculating H.C.F. and L.C.M.

Final Answer:
The L.C.M. of the two numbers is 48.