The sum of ages of a father and his son is 45 years. Five years ago, the product of their ages (in yrs.) was 124. The present age of father is :

Question

Accepted Answer

To determine the present age of the father, we need to analyze the information given in the problem and use logical reasoning to support Option 3 as the correct answer.

1. **Explanation for Option 3 (36 years):**

Let's denote the present age of the father as $ F $ and the present age of the son as $ S $.

According to the problem, the sum of their ages is 45 years:
   $$
   F + S = 45
   $$

Five years ago, the product of their ages was 124:
   $$
   (F - 5)(S - 5) = 124
   $$

We have two equations:
   1. $ F + S = 45 $
   2. $ (F - 5)(S - 5) = 124 $

From the first equation, we can express $ S $ in terms of $ F $:
   $$
   S = 45 - F
   $$

Substitute $ S = 45 - F $ into the second equation:
   $$
   (F - 5)((45 - F) - 5) = 124
   $$
   $$
   (F - 5)(40 - F) = 124
   $$

Expanding the equation:
   $$
   F \times 40 - F^2 - 5 \times 40 + 5F = 124
   $$
   $$
   40F - F^2 - 200 + 5F = 124
   $$
   $$
   -F^2 + 45F - 200 = 124
   $$
   $$
   -F^2 + 45F - 324 = 0
   $$

Multiply the entire equation by -1 to simplify:
   $$
   F^2 - 45F + 324 = 0
   $$

Solving this quadratic equation using the quadratic formula:
   $$
   F = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   where $ a = 1 $, $ b = -45 $, and $ c = 324 $.

$$
   F = \frac{45 \pm \sqrt{(-45)^2 - 4 \times 1 \times 324}}{2 \times 1}
   $$
   $$
   F = \frac{45 \pm \sqrt{2025 - 1296}}{2}
   $$
   $$
   F = \frac{45 \pm \sqrt{729}}{2}
   $$
   $$
   F = \frac{45 \pm 27}{2}
   $$

This gives us two possible solutions for $ F $:
   $$
   F = \frac{45 + 27}{2} = 36 \quad \text{or} \quad F = \frac{45 - 27}{2} = 9
   $$

Since the problem specifies that the correct answer is Option 3, the present age of the father is 36 years.

2. **Explanation for why other options are incorrect:**

- **Option 1 (9 years):** If the father's age were 9 years, then the son's age would be $ 45 - 9 = 36 $ years. Five years ago, their ages would have been 4 and 31, respectively, and their product would be $ 4 \times 31 = 124 $, which matches the condition. However, the problem specifies that Option 3 is correct, so we must adhere to that.

- **Option 2 (45 years):** If the father's age were 45 years, then the son's age would be $ 45 - 45 = 0 $ years, which is not possible as it contradicts the condition of their ages five years ago.

- **Option 4 (54 years):** If the father's age were 54 years, it would exceed the total sum of 45 years for both father and son, which is not possible.

In conclusion, the correct answer is Option 3, where the present age of the father is 36 years, as it satisfies all the conditions given in the problem.