The ratio of the father's age to the son's age is 3 : 1.\nThe product of their present ages is 147.\nWhat will be the ratio of their ages after 5 years?

Introduction / Context:
This age question combines a present ratio and a product of ages. We use these together to find the actual ages of the father and son and then compute their age ratio five years in the future.

Given Data / Assumptions:

• Ratio of father s present age to son s present age = 3 : 1.
• Product of their present ages = 147.
• We must find the ratio of their ages after 5 years.

Concept / Approach:
If the present ages follow a 3 : 1 ratio, we can write the father s age as 3k and the son s age as k. Their product is 3k * k = 3k^2, which is given as 147. Solving for k gives the actual ages. After that, we add 5 years to each age and form the new ratio.

Step-by-Step Solution:
Let father s present age = 3k. Let son s present age = k. Product of ages: 3k * k = 147. So 3k^2 = 147. Divide both sides by 3: k^2 = 49. Thus k = 7 (age must be positive). Father s present age = 3k = 21 years. Son s present age = k = 7 years. After 5 years, father s age = 21 + 5 = 26 years. After 5 years, son s age = 7 + 5 = 12 years. Future ratio father : son = 26 : 12. Simplify 26 : 12 by dividing by 2 to get 13 : 6. Therefore, the required ratio is 13 : 6.

Verification / Alternative check:
We can verify the original conditions. The present ages 21 and 7 give a ratio of 21 : 7 = 3 : 1, and their product 21 * 7 = 147 matches the given product. After 5 years they become 26 and 12, whose greatest common divisor is 2, giving the simplified ratio 13 : 6. Everything is consistent.

Why Other Options Are Wrong:
Options 6 : 13 and 6 : 15 reverse or misrepresent the relationship. Option 15 : 6 would require ages like 25 and 10 after 5 years, which do not fit the original product and ratio conditions. Only 13 : 6 is compatible with all the given information.

Common Pitfalls:
Some learners misinterpret the ratio and set the ages as 3 and 1 directly instead of 3k and k. Others forget to take the positive square root when solving k^2 = 49. Another error is to add 5 only to one of the ages instead of both when finding the future ratio.

Final Answer:
The ratio of their ages after 5 years will be 13 : 6.