A person's present age is `2/5`th 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. 32 years
B. 36 years
C. 40 years
D. 48 years

Answer

Let's solve the problem step by step.

### Step 1: Define Variables
Let the present age of the mother be \( x \) years.

### Step 2: Express the Person's Age
According to the problem, the person's present age is \( \frac{2}{5} \) of the mother's age. Therefore, we can express the person's age as:
\[ \text{Person's age} = \frac{2}{5}x \]

### Step 3: Set Up the Age After 8 Years
After 8 years, the person's age will be:
\[ \text{Person's age after 8 years} = \frac{2}{5}x + 8 \]
And the mother's age after 8 years will be:
\[ \text{Mother's age after 8 years} = x + 8 \]

### Step 4: Set Up the Equation
According to the problem, after 8 years, the person will be half the age of the mother. Therefore, we can set up the equation:
\[ \frac{2}{5}x + 8 = \frac{1}{2}(x + 8) \]

### Step 5: Solve the Equation
Now, let's solve the equation step by step.

1. Multiply both sides by 10 to eliminate the fractions:
\[ 10\left(\frac{2}{5}x + 8\right) = 10\left(\frac{1}{2}(x + 8)\right) \]
This simplifies to:
\[ 4x + 80 = 5(x + 8) \]

2. Distribute on the right side:
\[ 4x + 80 = 5x + 40 \]

3. Rearranging the equation:
\[ 80 - 40 = 5x - 4x \]
This simplifies to:
\[ 40 = x \]

### Step 6: Conclusion
The present age of the mother is \( x = 40 \) years.

### Final Answer
The mother is currently **40 years old**.

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