The ratio of present ages of X and Y is 4:5. Which of the following can't be the ratio of ages of X and Y, 20 years ago ?

Options

A. A) 2 : 5
B. B) 8 : 15
C. C) 9 : 10
D. D) 3 : 5

Answer

To solve the problem, we need to determine which of the given ratios cannot represent the ages of X and Y 20 years ago, given that the present ratio of their ages is 4:5.

### Step-by-Step Solution:

1. **Define Present Ages**:
Let the present age of X be \( 4k \) and the present age of Y be \( 5k \), where \( k \) is a positive constant.

2. **Calculate Ages 20 Years Ago**:
- The age of X 20 years ago: \( 4k - 20 \)
- The age of Y 20 years ago: \( 5k - 20 \)

3. **Formulate the Ratio 20 Years Ago**:
The ratio of their ages 20 years ago can be expressed as:
\[ \text{Ratio} = \frac{4k - 20}{5k - 20} \]

4. **Simplify the Ratio**:
To simplify the ratio, we can express it in terms of \( k \):
\[ \frac{4k - 20}{5k - 20} = \frac{4(k - 5)}{5(k - 4)} \]

5. **Analyze Possible Ratios**:
We need to check which of the given ratios cannot be formed by the expression above. The possible ratios given in the options are:
- A) \( \frac{9}{10} \)
- B) \( \frac{20}{5} \) (which simplifies to \( 4 \))
- C) \( \frac{8}{15} \)
- D) \( \frac{3}{5} \)

6. **Check Each Option**:
- For \( \frac{9}{10} \):
\[ 9(5k - 20) = 10(4k - 20) \]
Simplifying gives:
\[ 45k - 180 = 40k - 200 \implies 5k = -20 \implies k = -4 \text{ (not possible)} \]
- For \( \frac{20}{5} \):
\[ 20(5k - 20) = 5(4k - 20) \]
Simplifying gives:
\[ 100k - 400 = 20k - 100 \implies 80k = 300 \implies k = 3.75 \text{ (possible)} \]
- For \( \frac{8}{15} \):
\[ 8(5k - 20) = 15(4k - 20) \]
Simplifying gives:
\[ 40k - 160 = 60k - 300 \implies 20k = 140 \implies k = 7 \text{ (possible)} \]
- For \( \frac{3}{5} \):
\[ 3(5k - 20) = 5(4k - 20) \]
Simplifying gives:
\[ 15k - 60 = 20k - 100 \implies 5k = 40 \implies k = 8 \text{ (possible)} \]

7. **Conclusion**:
The only ratio that cannot be formed from the present ages of X and Y is \( \frac{9}{10} \).

### Final Answer:
The ratio that cannot be the ages of X and Y 20 years ago is \( \frac{9}{10} \).