The ratio of the present ages of P and Q is 3 : 4.\nFive years ago, the ratio of their ages was 5 : 7.\nWhat are the present ages of P and Q?

Introduction / Context:
This question involves present and past age ratios. We know the current ratio of ages of P and Q and their ratio five years ago. Using these two ratio conditions, we can set up a simple algebraic equation and find their exact ages.

Given Data / Assumptions:

• Present ratio P : Q = 3 : 4.
• Five years ago, the ratio of their ages was 5 : 7.
• We need to determine the present ages of P and Q.

Concept / Approach:
Let the present ages be multiples of the ratio numbers. If P : Q = 3 : 4, we can write P = 3k and Q = 4k for some positive constant k. Five years ago, the ages would have been P - 5 and Q - 5, and their ratio at that time is given as 5 : 7. This gives one equation in the unknown k, which we solve.

Step-by-Step Solution:
Let P = 3k and Q = 4k. Five years ago, ages were P - 5 = 3k - 5 and Q - 5 = 4k - 5. Given that (3k - 5) / (4k - 5) = 5 / 7. Cross multiply: 7(3k - 5) = 5(4k - 5). 21k - 35 = 20k - 25. 21k - 20k = -25 + 35. k = 10. Therefore P = 3k = 30 and Q = 4k = 40. So P is 30 years old and Q is 40 years old.

Verification / Alternative check:
Check the given ratios. Present ratio: 30 : 40 simplifies to 3 : 4, which matches. Five years ago the ages were 25 and 35. The ratio 25 : 35 simplifies to 5 : 7, which matches the condition. Therefore the solution is fully consistent.

Why Other Options Are Wrong:
Options with 40 and 30 reversed, or 30 and 50, or 50 and 30 do not match both ratio conditions simultaneously. For example, if P were 40 and Q were 30, the present ratio would be 4 : 3, not 3 : 4, so that cannot be correct.

Common Pitfalls:
A common mistake is to forget to subtract 5 from both ages for the past ratio, or to treat 3 and 4 as the actual ages instead of ratio units. Another issue is failing to simplify and check both ratios after finding k. Always verify your solution against every condition in the problem.

Final Answer:
The present ages are 30 years for P and 40 years for Q.