Eight years ago, Ajay's age was 4/3 times that of Vijay. Eight years hence, Ajay's age will be 6/5 times that of Vijay. What is the present age of Ajay ?

Options

A. a. 41 yrs
B. b. 40 yrs
C. c. 37 yrs
D. d. 33 yrs

Answer

To solve the problem step by step, we need to set up equations based on the information provided about Ajay's and Vijay's ages.

### Step 1: Define Variables
Let:
- \( A \) = Present age of Ajay
- \( B \) = Present age of Vijay

### Step 2: Set Up the First Equation
According to the problem, eight years ago, Ajay's age was \( \frac{4}{3} \) times that of Vijay's age. Therefore, we can write the first equation as:
\[ A - 8 = \frac{4}{3}(B - 8) \]

### Step 3: Simplify the First Equation
To eliminate the fraction, we can multiply both sides of the equation by 3:
\[ 3(A - 8) = 4(B - 8) \]
Expanding both sides gives:
\[ 3A - 24 = 4B - 32 \]
Rearranging this, we get:
\[ 3A - 4B = -8 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
The problem also states that eight years hence, Ajay's age will be \( \frac{6}{5} \) times that of Vijay's age. Therefore, we can write the second equation as:
\[ A + 8 = \frac{6}{5}(B + 8) \]

### Step 5: Simplify the Second Equation
Again, we eliminate the fraction by multiplying both sides by 5:
\[ 5(A + 8) = 6(B + 8) \]
Expanding both sides gives:
\[ 5A + 40 = 6B + 48 \]
Rearranging this, we get:
\[ 5A - 6B = 8 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
Now we have a system of two equations:
1. \( 3A - 4B = -8 \)
2. \( 5A - 6B = 8 \)

We can solve these equations using substitution or elimination. Here, we will use elimination.

First, let's multiply Equation 1 by 5 and Equation 2 by 3 to align the coefficients of \( A \):
\[ 15A - 20B = -40 \quad \text{(Equation 3)} \]
\[ 15A - 18B = 24 \quad \text{(Equation 4)} \]

### Step 7: Subtract the Equations
Now, subtract Equation 4 from Equation 3:
\[ (15A - 20B) - (15A - 18B) = -40 - 24 \]
This simplifies to:
\[ -2B = -64 \]
Thus, we find:
\[ B = 32 \]

### Step 8: Substitute Back to Find Ajay's Age
Now that we have \( B \), we can substitute it back into either Equation 1 or Equation 2 to find \( A \). We'll use Equation 1:
\[ 3A - 4(32) = -8 \]
This simplifies to:
\[ 3A - 128 = -8 \]
Adding 128 to both sides gives:
\[ 3A = 120 \]
Dividing by 3 gives:
\[ A = 40 \]

### Final Answer
The present age of Ajay is \( \boxed{40} \).