Three numbers are in the ratio 1 : 2 : 3 and their H.C.F. is 12. What are the three numbers?

Introduction / Context:
This is a straightforward ratio and H.C.F. question. It tests whether you can correctly scale a given ratio by an appropriate factor derived from the highest common factor to reconstruct the actual numbers. Such questions are common in basic arithmetic sections of aptitude exams.

Given Data / Assumptions:

• The ratio of three numbers is 1 : 2 : 3.
• The H.C.F. of the three numbers is 12.
• We must determine the actual three numbers.

Concept / Approach:
If three numbers are in the ratio 1 : 2 : 3, we can represent them as k, 2k, and 3k for some positive integer k. The H.C.F. of k, 2k, and 3k is k times the H.C.F. of 1, 2, and 3. Since H.C.F.(1, 2, 3) is 1, the overall H.C.F. is k. We are told that the H.C.F. is 12, so we can directly identify k as 12 and then compute the three numbers.

Step-by-Step Solution:
Step 1: Let the three numbers be k, 2k, and 3k.Step 2: The H.C.F. of k, 2k, and 3k is H.C.F.(1, 2, 3) * k = 1 * k = k.Step 3: Given that the H.C.F. is 12, we have k = 12.Step 4: Substitute k = 12 into the expressions for the numbers.Step 5: First number = k = 12.Step 6: Second number = 2k = 24.Step 7: Third number = 3k = 36.

Verification / Alternative check:
Check the ratio: 12 : 24 : 36 simplifies to 1 : 2 : 3 when each term is divided by 12.Check the H.C.F.: H.C.F.(12, 24, 36) is 12, since 12 divides all three and no larger number does. Both conditions are satisfied, so the numbers 12, 24, 36 are correct.

Why Other Options Are Wrong:
Option a (4, 8, 12) has H.C.F. 4, not 12. Option b (5, 10, 15) has H.C.F. 5. Option c (10, 20, 30) has H.C.F. 10. None of these match the required H.C.F. of 12, even though they may share the same ratio pattern. Only option d satisfies both the ratio and H.C.F. conditions.

Common Pitfalls:
Students sometimes pick numbers that preserve the ratio but do not check the H.C.F. explicitly. It is essential to verify both given conditions. Another typical error is scaling by the wrong factor or miscomputing the H.C.F. Always verify the ratio and the H.C.F. at the end.

Final Answer:
The three numbers are 12, 24, 36.