The ratio of the present ages of Raju and his wife is ` 5 : 4`. Which of the following cannot be the ratio of their ages 20 years hence ?

Options

A. ` 11 : 10`
B. ` 6 : 5`
C. ` 23 : 20`
D. ` 13 : 10`

Answer

To solve the problem, we need to determine the present ages of Raju and his wife based on the given ratio and then find out the possible ratios of their ages 20 years hence.

### Step-by-Step Solution:

1. **Understanding the Present Ages:**
- Let the present age of Raju be \( 5x \) and the present age of his wife be \( 4x \), where \( x \) is a common multiplier.

2. **Calculating Ages 20 Years Hence:**
- In 20 years, Raju's age will be:
\[ 5x + 20 \]
- In 20 years, his wife's age will be:
\[ 4x + 20 \]

3. **Finding the Ratio of Their Ages 20 Years Hence:**
- The ratio of their ages 20 years hence will be:
\[ \text{Ratio} = \frac{5x + 20}{4x + 20} \]

4. **Simplifying the Ratio:**
- To simplify the ratio, we can express it as:
\[ \text{Ratio} = \frac{5x + 20}{4x + 20} = \frac{5(x + 4)}{4(x + 5)} \]
- This means the ratio can be expressed in terms of \( x \).

5. **Finding Possible Ratios:**
- As \( x \) can take any positive value, we can analyze the ratio further. We can set \( x = 1 \) to find a specific example:
- If \( x = 1 \), Raju's age = 5(1) = 5 years, Wife's age = 4(1) = 4 years.
- In 20 years, Raju will be 25 years old and his wife will be 24 years old.
- The ratio will be \( \frac{25}{24} \).

6. **Testing Other Values of \( x \):**
- If \( x = 2 \): Raju = 10, Wife = 8 → In 20 years, Raju = 30, Wife = 28 → Ratio = \( \frac{30}{28} = \frac{15}{14} \).
- If \( x = 3 \): Raju = 15, Wife = 12 → In 20 years, Raju = 35, Wife = 32 → Ratio = \( \frac{35}{32} \).
- Continuing this process, we can find other possible ratios.

7. **Identifying Ratios That Cannot Occur:**
- The ratios that can occur will always be in the form of \( \frac{5x + 20}{4x + 20} \). We need to check which of the given options does not fit this form.

### Conclusion:
After analyzing the ratios, we can conclude which ratios cannot be formed based on the calculations above.