The ratio between the ages of A and B at present is 2 : 3. Five years hence the ratio of their ages will be 3 : 4. What is the present age of A ?

Options

A. 10 years
B. 15 years
C. 25 years
D. Data inadequate

Answer

To solve the problem step by step, we will follow the given information and set up equations based on the ratios of ages.

### Step 1: Define the present ages of A and B
Let the present age of A be \( 2x \) and the present age of B be \( 3x \), based on the ratio of their ages which is \( 2:3 \).

### Step 2: Set up the equation for their ages in 5 years
In 5 years, the age of A will be \( 2x + 5 \) and the age of B will be \( 3x + 5 \). According to the problem, the ratio of their ages at that time will be \( 3:4 \). Therefore, we can set up the equation:
\[ \frac{2x + 5}{3x + 5} = \frac{3}{4} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 4(2x + 5) = 3(3x + 5) \]

### Step 4: Expand both sides of the equation
Expanding both sides, we get:
\[ 8x + 20 = 9x + 15 \]

### Step 5: Rearrange the equation to isolate \( x \)
Now, we will rearrange the equation to isolate \( x \):
\[ 8x + 20 - 15 = 9x \]
\[ 5 = 9x - 8x \]
\[ 5 = x \]

### Step 6: Find the present age of A
Now that we have \( x = 5 \), we can find the present age of A:
\[ \text{Present age of A} = 2x = 2 \times 5 = 10 \]

### Conclusion
The present age of A is \( 10 \) years.