Four years ago the ratio of the ages of Salman and Aamir was 5:4 respectively. Eight years hence the ratio of their ages will be 13:11 respectively. What is Aamir's present age ?

Options

A. 32 years
B. 40 years
C. 38 years
D. 36 years

Answer

To solve the problem, we need to set up equations based on the information given about the ages of Salman and Aamir.

### Step 1: Define the variables
Let:
- A = Aamir's present age
- S = Salman's present age

### Step 2: Set up the first equation based on the information given
According to the problem, four years ago, the ratio of their ages was 5:4. This can be expressed as:
\[ \frac{S - 4}{A - 4} = \frac{5}{4} \]
Cross-multiplying gives:
\[ 4(S - 4) = 5(A - 4) \]
Expanding this, we get:
\[ 4S - 16 = 5A - 20 \]
Rearranging gives us our first equation:
\[ 4S - 5A = -4 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation based on the future ages
Eight years hence, the ratio of their ages will be 13:11. This can be expressed as:
\[ \frac{S + 8}{A + 8} = \frac{13}{11} \]
Cross-multiplying gives:
\[ 11(S + 8) = 13(A + 8) \]
Expanding this, we get:
\[ 11S + 88 = 13A + 104 \]
Rearranging gives us our second equation:
\[ 11S - 13A = 16 \quad \text{(Equation 2)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( 4S - 5A = -4 \) (Equation 1)
2. \( 11S - 13A = 16 \) (Equation 2)

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

First, we can multiply Equation 1 by 11 and Equation 2 by 4 to align the coefficients of S:
\[ 44S - 55A = -44 \quad \text{(Equation 3)} \]
\[ 44S - 52A = 64 \quad \text{(Equation 4)} \]

Now, subtract Equation 4 from Equation 3:
\[ (44S - 55A) - (44S - 52A) = -44 - 64 \]
This simplifies to:
\[ -3A = -108 \]
Dividing both sides by -3 gives:
\[ A = 36 \]

### Step 5: Find Salman's present age
Now that we have Aamir's present age, we can substitute \( A = 36 \) back into Equation 1 to find Salman's age:
\[ 4S - 5(36) = -4 \]
\[ 4S - 180 = -4 \]
\[ 4S = 176 \]
\[ S = 44 \]

### Conclusion
Aamir's present age is **36 years**.