The ratio of two numbers is 3 : 4 and their HCF is 4. What will be their LCM?

Concept: If two numbers are in the ratio a : b (in lowest terms) and their HCF is H, the numbers themselves are aH and bH. Because a and b share no common factor, the relationship LCM × HCF = Product of the two numbers always holds, so LCM = (aH × bH) / H = H × a × b.

Applying this to the problem:

1. Let the two numbers be 3M and 4M, where M is their HCF, since 3 and 4 have no common factor.

2. The HCF is given as 4, so M = 4. The numbers are 3 × 4 = 12 and 4 × 4 = 16.

3. Using LCM = (Product of numbers) / HCF: LCM = (12 × 16) / 4.

4. 12 × 16 = 192, and 192 / 4 = 48, so the LCM is 48.

Cross-check: Since 3 and 4 are co-prime, the LCM can also be found directly as HCF × 3 × 4 = 4 × 12 = 48, the same result. Also, 48 is divisible by both 12 (48 ÷ 12 = 4) and 16 (48 ÷ 16 = 3), confirming it is indeed their least common multiple.