Four years ago, the ratio of the age of A and B was `2 : 3` and after four years it will become `5 : 7`. Find their present age.

Options

A. 36 years and 40 years
B. 20 years and 28 years
C. 40 years and 56 years
D. 36 years and 52 years

Answer

To solve the problem step-by-step, we will denote the present ages of A and B as \( a \) and \( b \) respectively.

### Step 1: Establish the equations based on the given ratios
Four years ago, the ratio of the ages of A and B was \( 2:3 \). This can be expressed as:
\[ \frac{a - 4}{b - 4} = \frac{2}{3} \]
Cross-multiplying gives us:
\[ 3(a - 4) = 2(b - 4) \]
Expanding this, we have:
\[ 3a - 12 = 2b - 8 \]
Rearranging gives us our first equation:
\[ 3a - 2b = 4 \quad \text{(Equation 1)} \]

### Step 2: Establish the second equation based on the future ratios
After four years, the ratio of the ages of A and B will be \( 5:7 \). This can be expressed as:
\[ \frac{a + 4}{b + 4} = \frac{5}{7} \]
Cross-multiplying gives us:
\[ 7(a + 4) = 5(b + 4) \]
Expanding this, we have:
\[ 7a + 28 = 5b + 20 \]
Rearranging gives us our second equation:
\[ 7a - 5b = -8 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of linear equations:
1. \( 3a - 2b = 4 \)
2. \( 7a - 5b = -8 \)

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

To eliminate \( b \), we can multiply Equation 1 by 5 and Equation 2 by 2:
\[ 15a - 10b = 20 \quad \text{(Equation 3)} \]
\[ 14a - 10b = -16 \quad \text{(Equation 4)} \]

Now, we subtract Equation 4 from Equation 3:
\[ (15a - 10b) - (14a - 10b) = 20 - (-16) \]
This simplifies to:
\[ a = 36 \]

### Step 4: Substitute back to find \( b \)
Now that we have \( a = 36 \), we can substitute this value back into Equation 1 to find \( b \):
\[ 3(36) - 2b = 4 \]
This simplifies to:
\[ 108 - 2b = 4 \]
Rearranging gives:
\[ -2b = 4 - 108 \]
\[ -2b = -104 \]
\[ b = 52 \]

### Conclusion
The present ages of A and B are:
- Age of A: \( 36 \) years
- Age of B: \( 52 \) years