Two numbers are in the ratio 3 : 4 and their least common multiple (LCM) is 180. What is the first number?

Introduction / Context:
The problem connects ratio with least common multiple (LCM). Represent numbers by a common multiplier and use known LCM behavior to find that multiplier, then recover the numbers.

Given Data / Assumptions:

• Numbers are 3k and 4k.
• LCM = 180.
• Find the first number (3k).

Concept / Approach:
For relatively prime 3 and 4, LCM(3k, 4k) = 12k. Equate 12k with 180 to find k, then compute 3k.

Step-by-Step Solution:
LCM(3, 4) = 12. LCM(3k, 4k) = 12k. 12k = 180 ⇒ k = 15. First number = 3k = 45.

Verification / Alternative check:
Second number = 4k = 60. LCM(45, 60) is 180, confirming the calculation.

Why Other Options Are Wrong:
15 or 20 are too small and do not give LCM 180 with their partners. 60 is the second number, not the first.

Common Pitfalls:
Confusing LCM with HCF, or forgetting that 3 and 4 are coprime which keeps LCM = product times k. Carefully apply LCM logic.

Final Answer:
45