Question: The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 124. Find their present ages.

Answer: The present age of the father is 36 years and the present age of the son is 9 years.

Step-by-step solution

Step 1: Define variables and set up equations

Let $F$ be the present age of the father and $S$ be the present age of the son. The first statement gives us the sum of their ages. The second statement describes their ages five years ago and the product of those ages.

Step 2: Express one variable in terms of the other

From equation (1), we can express the father's age in terms of the son's age. This substitution will help us reduce the system to a single variable.

Step 3: Substitute and form a quadratic equation

Substitute the expression for $F$ into equation (2). Expand and simplify the equation to obtain a standard quadratic equation in terms of $S$ .

Step 4: Solve the quadratic equation for S

Use the quadratic formula $S = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve for $S$ . Calculate the discriminant and find the two possible values for $S$ .

Step 5: Determine valid ages and find father's age

We get two possible values for the son's age: 36 and 9. If the son's age is 36, the father's age would be 9, which is not logical. Therefore, the son's age must be 9 years. Substitute this value back into equation (1) to find the father's age.

Step 6: Verify the solution

Check if the calculated ages satisfy both original conditions. The sum of their ages is $36 + 9 = 45$ , and five years ago their ages were $31$ and $4$ , with a product of $31 \times 4 = 124$ . Both conditions are met.

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