A person's present age is two-fifth of the age of his mother. After 8 years, he will be one-half, of the age of his mother. How old is the mother at present ?

Options

A. (a) 40 years
B. (b) 48 years
C. (c) 32 years
D. (d) 36 years

Explanation

To solve the problem step by step, let's denote the present age of the person as \( P \) and the present age of the mother as \( M \).

### Step 1: Set up the equations based on the information given.
From the problem, we know:
1. The person's present age is two-fifth of the age of his mother:
\[ P = \frac{2}{5}M \]

2. After 8 years, the person's age will be one-half of his mother's age:
\[ P + 8 = \frac{1}{2}(M + 8) \]

### Step 2: Substitute the first equation into the second equation.
Substituting \( P \) from the first equation into the second equation:
\[ \frac{2}{5}M + 8 = \frac{1}{2}(M + 8) \]

### Step 3: Clear the fraction by multiplying through by 10 (the least common multiple of 5 and 2).
Multiplying the entire equation by 10 gives:
\[ 10 \left(\frac{2}{5}M\right) + 10 \cdot 8 = 10 \left(\frac{1}{2}(M + 8)\right) \]
This simplifies to:
\[ 4M + 80 = 5(M + 8) \]

### Step 4: Expand and simplify the equation.
Expanding the right side:
\[ 4M + 80 = 5M + 40 \]

### Step 5: Rearrange the equation to isolate \( M \).
Subtract \( 4M \) from both sides:
\[ 80 = 5M - 4M + 40 \]
This simplifies to:
\[ 80 = M + 40 \]

### Step 6: Solve for \( M \).
Subtract 40 from both sides:
\[ M = 80 - 40 \]
Thus,
\[ M = 40 \]

### Conclusion:
The present age of the mother is \( 40 \) years.