Eight years ago ,the ratio of the ages of A and B was 4 : 5 and the of their ages ,12 years of their hence ,will be 13 : 15 .The present age (in years)of B is :

Options

A. 52
B. 46
C. 48
D. 56

Answer

To solve the problem step by step, let's define the present ages of A and B.

1. **Define Variables:**
Let the present age of A be \( a \) and the present age of B be \( b \).

2. **Set Up the First Equation (from 8 years ago):**
According to the problem, 8 years ago, the ratio of the ages of A and B was 4:5. Therefore, we can write:
\[ \frac{a - 8}{b - 8} = \frac{4}{5} \]
Cross-multiplying gives us:
\[ 5(a - 8) = 4(b - 8) \]
Expanding this, we have:
\[ 5a - 40 = 4b - 32 \]
Rearranging gives us:
\[ 5a - 4b = 8 \quad \text{(Equation 1)} \]

3. **Set Up the Second Equation (from 12 years hence):**
The problem states that 12 years from now, the ratio of their ages will be 13:15. Thus, we can write:
\[ \frac{a + 12}{b + 12} = \frac{13}{15} \]
Cross-multiplying gives us:
\[ 15(a + 12) = 13(b + 12) \]
Expanding this, we have:
\[ 15a + 180 = 13b + 156 \]
Rearranging gives us:
\[ 15a - 13b = -24 \quad \text{(Equation 2)} \]

4. **Solve the System of Equations:**
Now we have a system of two equations:
- Equation 1: \( 5a - 4b = 8 \)
- Equation 2: \( 15a - 13b = -24 \)

We can solve these equations using substitution or elimination. Let's use elimination.

Multiply Equation 1 by 3:
\[ 15a - 12b = 24 \quad \text{(Equation 3)} \]

Now subtract Equation 2 from Equation 3:
\[ (15a - 12b) - (15a - 13b) = 24 - (-24) \]
This simplifies to:
\[ b = 48 \]

5. **Find the Present Age of A:**
Substitute \( b = 48 \) back into Equation 1 to find \( a \):
\[ 5a - 4(48) = 8 \]
\[ 5a - 192 = 8 \]
\[ 5a = 200 \]
\[ a = 40 \]

6. **Conclusion:**
The present age of B is \( b = 48 \) years.