The respective ratio of Avinash is present age and Shashi’s present age is 9 : 5. Avinash is 54 years old at present, how many years ago was the respective ratio of their ages7:3?

Options

A. 12
B. 6
C. 9
D. 15

Answer

To solve the problem step by step, we will follow these instructions:

1. **Understand the given ratio and ages**:
- The ratio of Avinash's present age to Shashi's present age is given as 9:5.
- Avinash's current age is 54 years.

2. **Calculate Shashi's present age**:
- Since the ratio of their ages is 9:5, we can set up the equation:
\[ \frac{Avinash's \, age}{Shashi's \, age} = \frac{9}{5} \]
- Let Shashi's age be \( S \). Then:
\[ \frac{54}{S} = \frac{9}{5} \]
- Cross-multiplying gives:
\[ 9S = 54 \times 5 \]
\[ 9S = 270 \]
\[ S = \frac{270}{9} = 30 \]
- Thus, Shashi's present age is 30 years.

3. **Set up the equation for ages 'x' years ago**:
- Let \( x \) be the number of years ago when the ratio of their ages was 7:3.
- Therefore, Avinash's age \( x \) years ago was \( 54 - x \) and Shashi's age \( x \) years ago was \( 30 - x \).
- The ratio of their ages at that time can be set up as:
\[ \frac{54 - x}{30 - x} = \frac{7}{3} \]

4. **Cross-multiply to solve for x**:
- Cross-multiplying gives:
\[ 3(54 - x) = 7(30 - x) \]
- Expanding both sides:
\[ 162 - 3x = 210 - 7x \]

5. **Rearranging the equation**:
- Bring all terms involving \( x \) to one side and constants to the other:
\[ 7x - 3x = 210 - 162 \]
\[ 4x = 48 \]
\[ x = \frac{48}{4} = 12 \]

6. **Conclusion**:
- Therefore, the number of years ago when the respective ratio of their ages was 7:3 is **12 years**.