Two numbers are in the ratio of 3 : 5, their LCM is 90. Find the numbers.

Concept: When two numbers are in the ratio a : b and a and b are co-prime (they share no common factor other than 1), the two numbers can always be written as ak and bk for a single common multiplier k. Because a and b share no factor, the HCF of the pair is k and the LCM of the pair is a × b × k. So for such a ratio the LCM is always the product of the ratio terms multiplied by the common multiplier.

Application

1. Let the two numbers be 3k and 5k, where k is the common multiplier.

2. 3 and 5 are co-prime, so LCM(3k, 5k) = 3 × 5 × k = 15k.

3. The LCM is given as 90, so 15k = 90.

4. k = 90 ÷ 15 = 6.

5. The two numbers are 3 × 6 = 18 and 5 × 6 = 30.

Cross-check

By prime factorisation, 18 = 2 × 32 and 30 = 2 × 3 × 5. Taking the highest power of each prime gives LCM = 2 × 32 × 5 = 90, and the common prime factors give HCF = 2 × 3 = 6, which is exactly the multiplier k. Dividing 18 : 30 by 6 gives 3 : 5, so both conditions stated in the question are satisfied.

Contrast

Every pair listed above is in the 3 : 5 ratio, so the ratio by itself cannot separate them; only the common multiplier that makes 15k equal to 90 does.