Nine years ago A's age and B's age were in the ratio ` 5 : 7 `. Which of the following cannot be the ratio of their ages 5 years from now ?

Options

A. ` 11 : 13`
B. ` 13 : 19`
C. ` 21 : 25`
D. ` 15 : 16`

Answer

To solve the problem, we need to determine the present ages of A and B based on the information given and then find out which of the provided ratios cannot be their ages 5 years from now.

### Step-by-Step Solution:

1. **Define the Ages 9 Years Ago:**
Let A's age 9 years ago be \( 5x \) and B's age 9 years ago be \( 7x \) (since the ratio of their ages was \( 5:7 \)).

2. **Calculate Present Ages:**
- A's present age = \( 5x + 9 \)
- B's present age = \( 7x + 9 \)

3. **Calculate Ages 5 Years from Now:**
- A's age 5 years from now = \( (5x + 9) + 5 = 5x + 14 \)
- B's age 5 years from now = \( (7x + 9) + 5 = 7x + 14 \)

4. **Set Up the Ratio 5 Years from Now:**
The ratio of their ages 5 years from now is:
\[ \frac{5x + 14}{7x + 14} \]

5. **Determine the Range of Possible Ratios:**
We know that 9 years ago, the ratio was \( \frac{5}{7} \). This means:
\[ \frac{5}{7} < \frac{5x + 14}{7x + 14} \]
To find the limits for the ratio \( \frac{5x + 14}{7x + 14} \), we can analyze the values as \( x \) increases.

6. **Calculate the Limits:**
- As \( x \) approaches infinity, the ratio \( \frac{5x + 14}{7x + 14} \) approaches \( \frac{5}{7} \).
- We can also find the minimum value of the ratio by substituting specific values of \( x \).

7. **Evaluate the Given Options:**
We need to check which of the following ratios cannot be achieved:
- Option 1: \( \frac{11}{13} \)
- Option 2: \( \frac{13}{19} \)
- Option 3: \( \frac{21}{25} \)
- Option 4: \( \frac{15}{16} \)

We will convert these ratios to decimal form:
- \( \frac{11}{13} \approx 0.846 \)
- \( \frac{13}{19} \approx 0.684 \)
- \( \frac{21}{25} = 0.84 \)
- \( \frac{15}{16} = 0.9375 \)

8. **Compare with \( \frac{5}{7} \):**
The decimal value of \( \frac{5}{7} \approx 0.714 \). Thus, any ratio greater than this cannot be the ratio of their ages 5 years from now.

9. **Identify the Impossible Ratio:**
- \( \frac{11}{13} \approx 0.846 \) (greater than 0.714)
- \( \frac{13}{19} \approx 0.684 \) (less than 0.714)
- \( \frac{21}{25} = 0.84 \) (greater than 0.714)
- \( \frac{15}{16} = 0.9375 \) (greater than 0.714)

Therefore, the ratios \( \frac{11}{13} \), \( \frac{21}{25} \), and \( \frac{15}{16} \) cannot be the ratio of their ages 5 years from now.

### Conclusion:
The answer to the question is that the ratios \( \frac{11}{13} \), \( \frac{21}{25} \), and \( \frac{15}{16} \) cannot be the ratio of A's and B's ages 5 years from now.