The ratio of the present ages of P and Q is 8 : 5. After 4 years, their ages will be in the ratio 4 : 3 respectively. What will be the ratio of the age of P after 7 years from now to the present age of Q?

Introduction / Context:
This question combines present and future age ratios of two people and then asks for a different ratio involving another time shift. It is a good test of your ability to manage multiple time based changes and ratios using algebra. Problems like this are common in competitive exams where ratio and proportion and age based reasoning are tested together.

Given Data / Assumptions:

• The present age ratio of P and Q is 8 : 5.
• After 4 years, their ages will be in the ratio 4 : 3.
• We are asked for the ratio of P age after 7 years from now to the present age of Q.
• All ages are in years and positive.

Concept / Approach:
First, we express the present ages in terms of a common multiple. Then we use the future ratio after 4 years to form an equation and solve for the common multiple. Once the actual present ages of P and Q are known, we can compute P age after 7 years and then form the requested ratio with Q present age. The problem demonstrates how time shifts can be handled systematically with simple algebra.

Step-by-Step Solution:
Step 1: Let the present age of P be 8x years and the present age of Q be 5x years. Step 2: After 4 years, P age will be 8x + 4 years and Q age will be 5x + 4 years. Step 3: At that time, the ratio of their ages is given as 4 : 3, so (8x + 4) / (5x + 4) = 4 / 3. Step 4: Cross multiply to obtain 3 * (8x + 4) = 4 * (5x + 4). Step 5: Expand both sides: 24x + 12 = 20x + 16. Step 6: Rearrange the equation: 24x - 20x = 16 - 12, so 4x = 4. Step 7: Divide both sides by 4 to get x = 1. Step 8: Therefore, present ages are P = 8 * 1 = 8 years and Q = 5 * 1 = 5 years. Step 9: P age after 7 years will be 8 + 7 = 15 years. Step 10: We need the ratio of P age after 7 years to Q present age, that is 15 : 5. Step 11: Simplify 15 : 5 by dividing both terms by 5 to get 3 : 1.

Verification / Alternative check:
As a quick check, verify the given after 4 years ratio. After 4 years, P will be 8 + 4 = 12 years and Q will be 5 + 4 = 9 years. The ratio 12 : 9 simplifies to 4 : 3, which matches the problem statement. Next, P age after 7 years is 15 years and Q present age is 5 years, so the ratio 15 : 5 simplifies correctly to 3 : 1. Therefore, the solution is consistent.

Why Other Options Are Wrong:
Option A (3 : 2) would imply that P after 7 years is only 1.5 times Q present age, which contradicts the calculated values of 15 and 5.
Option B (1 : 2) suggests that P would be younger or much smaller in comparison to Q, which is inconsistent with the determined ages.
Option C (2 : 1) does not match the exact ratio of 15 : 5, which simplifies to 3 : 1, not 2 : 1.

Common Pitfalls:
Students sometimes confuse which ages are used in the final ratio and may incorrectly use Q age after 7 years instead of Q present age. Another common mistake is to forget to add the correct number of years to both ages when forming the future ratio. Careful reading of the question and clear labeling of each age expression helps avoid such errors.

Final Answer:
The required ratio is 3 : 1.