Question

Sum of the present ages of a father and his son is 48 years. If the product of their ages 5 years back is 165, what is the present age of the father?

• A. 38 years
• B. 36 years
• C. 31 years
• D. 28 years

Hint:

Let the father's age be \(f\) and the son's age be \(48-f\), then form a quadratic from the product of their ages 5 years back.

Answer 1

The Correct Option is A

Step 1: Work with the ages 5 years back as a pair of numbers.
Let the father's age 5 years back be $p$ and the son's age 5 years back be $q$. Their present ages are $p+5$ and $q+5$.

Step 2: Turn both given facts into equations in $p$ and $q$.
The present ages add up to 48:
\[ (p+5) + (q+5) = 48 \]
\[ p+q = 38 \]
The product of the ages 5 years back is 165:
\[ pq = 165 \]

Step 3: Find $p$ and $q$ as roots of a quadratic.
Any two numbers with a known sum and known product are the roots of $t^2 - (\text{sum})t + (\text{product}) = 0$:
\[ t^2 - 38t + 165 = 0 \]

Step 4: Solve this quadratic.
\[ t = \frac{38 \pm \sqrt{38^2 - 4(165)}}{2} = \frac{38 \pm \sqrt{1444-660}}{2} = \frac{38 \pm \sqrt{784}}{2} = \frac{38 \pm 28}{2} \]
So $t = 33$ or $t = 5$. These are the two ages 5 years back, one for the father and one for the son.

Step 5: Convert back to present ages and pick the father's.
Adding 5 to each root gives present ages of $33+5=38$ and $5+5=10$. Since the father is the older of the two, his present age is 38. This matches the sum check ($38+10=48$) and the product check ($33 \times 5 = 165$).
\[ \boxed{38 \text{ years}} \]