Two numbers are in the ratio 3:5. If 12 is added to both the numbers, then the…

When a fixed quantity is added to both terms of a ratio, the new ratio is a different fraction of the same numbers. Represent the original numbers as multiples of a common variable using the given ratio, add the fixed quantity to each, set the resulting fraction equal to the new ratio, and solve the resulting linear equation for the variable.

1. Let the two numbers be 3x and 5x, since they are in the ratio 3:5.

2. After 12 is added to each number, the numbers become (3x + 12) and (5x + 12), and their ratio is given as 5:7.

3. Write this as an equation: (3x + 12) / (5x + 12) = 5/7.

4. Cross-multiply: 7(3x + 12) = 5(5x + 12).

5. Expand both sides: 21x + 84 = 25x + 60.

6. Collect the x-terms: 84 - 60 = 25x - 21x, which gives 24 = 4x, so x = 6.

7. The two original numbers are 3x = 18 and 5x = 30, so their sum is 18 + 30 = 48.

Cross-check: with x = 6, adding 12 to each number gives 30 and 42, and 30:42 reduces to 5:7, which matches the ratio stated in the question, confirming the sum is 48.