4 years ago, the ratio of ages of A and B was 9 : 11 and 6 years hence, the ratio of ages of A and B will be 7 : 8. Find the present age of B.

Options

A. 22 years
B. 24 years
C. 26 years
D. 37 years

Explanation

To solve the problem, we need to find the present age of B based on the given ratios of ages of A and B at different times. Here’s a step-by-step breakdown of the solution:

### Step 1: Define Variables
Let the present age of A be \( A \) and the present age of B be \( B \).

### Step 2: Set Up the Equations
According to the problem:
- Four years ago, the ratio of ages of A and B was 9:11.
- Six years hence, the ratio of ages of A and B will be 7:8.

From the first condition (4 years ago):
\[ \frac{A - 4}{B - 4} = \frac{9}{11} \]
Cross-multiplying gives:
\[ 11(A - 4) = 9(B - 4) \]
Expanding this, we get:
\[ 11A - 44 = 9B - 36 \]
Rearranging gives us:
\[ 11A - 9B = 8 \quad \text{(Equation 1)} \]

From the second condition (6 years hence):
\[ \frac{A + 6}{B + 6} = \frac{7}{8} \]
Cross-multiplying gives:
\[ 8(A + 6) = 7(B + 6) \]
Expanding this, we get:
\[ 8A + 48 = 7B + 42 \]
Rearranging gives us:
\[ 8A - 7B = -6 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have a system of two equations:
1. \( 11A - 9B = 8 \)
2. \( 8A - 7B = -6 \)

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

Multiply Equation 2 by 9 and Equation 1 by 7 to align the coefficients of \( B \):
\[ 9(8A - 7B) = 9(-6) \implies 72A - 63B = -54 \quad \text{(Equation 3)} \]
\[ 7(11A - 9B) = 7(8) \implies 77A - 63B = 56 \quad \text{(Equation 4)} \]

Now, subtract Equation 3 from Equation 4:
\[ (77A - 63B) - (72A - 63B) = 56 + 54 \]
This simplifies to:
\[ 5A = 110 \implies A = 22 \]

### Step 4: Substitute Back to Find B
Now substitute \( A = 22 \) back into Equation 1:
\[ 11(22) - 9B = 8 \]
\[ 242 - 9B = 8 \]
\[ -9B = 8 - 242 \]
\[ -9B = -234 \implies B = 26 \]

### Conclusion
The present age of B is \( \boxed{26} \).