Two numbers are such that the ratio between them is 3:5 but if each is…

Concept: When two quantities are in the ratio p:q, they can be written as pk and qk for a common multiplier k. If the same fixed amount is added to (or subtracted from) both, the new ratio gives a single equation in k that can be solved directly.

Application: Let the two numbers be 3a and 5a, since their ratio is 3:5.

1. Increasing each number by 10 gives 3a + 10 and 5a + 10, and the new ratio is 5:7, so (3a + 10)/(5a + 10) = 5/7.

2. Cross-multiplying: 7(3a + 10) = 5(5a + 10).

3. Expanding both sides: 21a + 70 = 25a + 50.

4. Solving for a: 70 - 50 = 25a - 21a, so 20 = 4a, giving a = 5.

5. Substituting a = 5 back: the numbers are 3(5) = 15 and 5(5) = 25.

Cross-check: 15:25 simplifies to 3:5, the original ratio. Adding 10 to each gives 25 and 35, and 25:35 simplifies to 5:7, the new ratio — both conditions hold.

Result: The two numbers are 15 and 25.