Question

A person's present age is two-fifth of the age of his mother. After 8 years, he will be onehalf of the age of his mother. What is the present age of the mother?

• A. 40
• B. 30
• C. 60
• D. 50

Hint:

For age problems, set up equations for the present time and the future/past time. If the present ages are M and S, then after 'n' years, their ages will be M+n and S+n. Formulate the equations based on the relationships given and solve them simultaneously.

Answer 1

The Correct Option is A

Let the present age of the mother be M and the present age of the person be S.
From the first statement: "A person's present age is two-fifth of the age of his mother."
\[ S = \frac{2}{5}M \cdots(1) \] From the second statement: "After 8 years, he will be onehalf of the age of his mother."
After 8 years, the son's age will be \( S+8 \) and the mother's age will be \( M+8 \). \[ S+8 = \frac{1}{2}(M+8) \cdots(2) \]
Now we solve the system of two equations. Substitute the expression for S from equation (1) into equation (2):
\[ \frac{2}{5}M + 8 = \frac{1}{2}(M+8) \] Multiply the entire equation by 10 to clear the denominators:
\[ 10\left(\frac{2}{5}M\right) + 10(8) = 10\left(\frac{1}{2}(M+8)\right) \] \[ 4M + 80 = 5(M+8) \] \[ 4M + 80 = 5M + 40 \] \[ 80 - 40 = 5M - 4M \] \[ 40 = M \] The present age of the mother is 40 years.