The ratio of two numbers is 3:4 and their HCF is 4. Their LCM is:

Concept: When two numbers are in the ratio a : b (with a and b having no common factor other than 1) and their HCF is h, the numbers themselves are a×h and b×h, and their LCM is a×b×h — the product of the ratio terms and the HCF.

1. The ratio 3 : 4 has terms 3 and 4, which share no common factor other than 1, so the two numbers can be written as 3x and 4x, where x is their HCF.

2. Since HCF(3x, 4x) = x when 3 and 4 are coprime, and the HCF is given as 4, x = 4.

3. The two numbers are therefore 3 × 4 = 12 and 4 × 4 = 16.

4. LCM = 3 × 4 × 4 = 48, using LCM = (ratio term 1) × (ratio term 2) × HCF.

Cross-check: For any two numbers, HCF × LCM = product of the numbers. Here, HCF × LCM = 4 × 48 = 192, and 12 × 16 = 192 — the two match, confirming the LCM is correct.

Hence, the LCM of the two numbers is 48.