Ratio of two numbers is 3 : 2. If the LCM of the numbers is 60, then the…

Concept:

When two numbers are in the ratio a : b with a and b coprime (no common factor besides 1), they can be written as a×x and b×x. Because a and b share no common factor, the LCM of the two numbers is simply a × b × x — both ratio terms multiply directly with x to form the LCM.

Application:

1. Let the two numbers be 3x and 2x, since their ratio is 3 : 2 and 3, 2 are coprime.

2. The LCM of 3x and 2x is 3 × 2 × x = 6x.

3. Given LCM = 60, so 6x = 60, which gives x = 10.

4. Smaller number = 2x = 2 × 10 = 20.

Cross-check:

The two numbers are 3 × 10 = 30 and 2 × 10 = 20. Since 30 = 2 × 3 × 5 and 20 = 22 × 5, their LCM is 22 × 3 × 5 = 60, confirming the working.