Question

The combined age of a mother and daughter duo is 42 years. The product of their ages, 5 years back, was 60 years. What is the present age of the mother?

• A. 40 years
• B. 38 years
• C. 32 years
• D. None of these

Hint:

When solving age-related problems, set up equations for the present and past ages, and use the information to form a quadratic equation if needed.

Answer 1

The Correct Option is D

Let the present age of the mother be \( M \) and the present age of the daughter be \( D \). The following equations can be written from the problem statement:
1. \( M + D = 42 \) (combined age of mother and daughter).
2. \( (M - 5)(D - 5) = 60 \) (the product of their ages 5 years ago).
From the first equation, we can express \( M \) in terms of \( D \):
\[ M = 42 - D \] Substitute this into the second equation:
\[ (42 - D - 5)(D - 5) = 60 \] \[ (37 - D)(D - 5) = 60 \] Expanding this:
\[ 37D - 185 - D^2 + 5D = 60 \] \[ -D^2 + 42D - 185 = 60 \] \[ -D^2 + 42D - 245 = 0 \] Multiplying through by -1:
\[ D^2 - 42D + 245 = 0 \] Solving this quadratic equation using the quadratic formula:
\[ D = \frac{-(-42) \pm \sqrt{(-42)^2 - 4(1)(245)}}{2(1)} \] \[ D = \frac{42 \pm \sqrt{1764 - 980}}{2} \] \[ D = \frac{42 \pm \sqrt{784}}{2} \] \[ D = \frac{42 \pm 28}{2} \] Thus, \( D = 35 \) or \( D = 7 \).
If \( D = 7 \), then \( M = 42 - 7 = 35 \). However, this doesn't satisfy the condition for the product of their ages 5 years ago. Therefore, the present age of the mother is:
\[ M = 35 + 5 = 40 \, \text{years}. \] So, the present age of the mother is \( 40 \, \text{years} \).