Question

The L.C.M of two numbers is 48. The numbers are in the radio 2:3. find the sum of the numbers

Answer 1

I'll try to establish some truths/facts from the given information. Then proceed accordingly.

1. The numbers are in the ratio 2:32:3. Let they be of the form 2x2x and 3x3x(formally {2x,3x∣x∈N}{2x,3x∣x∈N}) . So their LCM must be of the form 6x6x.

2. Solving for 6x=486x=48, we arrive at the result x=8x=8.

3. The sum of the numbers must be 2x+3x=5x2x+3x=5x. Hence the sum is surely 5×8=405×8=40.

Thus the sum of the two numbers is 40 in this case. A fun thing to notice is that during the whole process, you don't really need to find the two numbers. Taking an arbitrary constant to convert the ratio into operable variables works in that favour.

Or Simply,

Let the two numbers be 2n and 3n

Their LCM = 48 and their HCF = x

LCM*HCF = 2x*3x

48x = 6x^2, or

48 = 6x, or

x = 8

So the numbers are 16 and 24.

The sum of the two numbers is 40.

Hope This Helps :)

Answer 2

sum of the numbers is 40