The ratio of two numbers is 3 : 7 and their LCM is 630. Then, sum of the LCM…

Concept: For two numbers whose ratio is a : b with a and b co-prime (no common factor other than 1), the numbers can be written as ax and bx. Because a and b share no common factor, the HCF of the two numbers is x, and their LCM is a times b times x. Also, for any two numbers, the product of the LCM and the HCF always equals the product of the two numbers itself, which gives a useful independent check.

Application:

1. The ratio of the two numbers is 3 : 7, and 3 and 7 are co-prime, so the numbers can be written as 3x and 7x.

2. Since 3 and 7 share no common factor, the LCM of 3x and 7x is 3 times 7 times x, which is 21x.

3. The LCM is given as 630, so 21x = 630, which gives x = 30.

4. The HCF of 3x and 7x is x itself, so the HCF is 30.

5. The two numbers are 3x = 90 and 7x = 210.

6. Sum of the LCM and the HCF = 630 + 30 = 660.

Cross-check: The product of the two numbers is 90 times 210 = 18900. The product of their LCM and HCF is 630 times 30 = 18900. Since both match, the values of the LCM and HCF used above are confirmed correct.

So, the sum of the LCM and HCF of these two numbers is 660.