Present ages of A and B are in the ratio 7:4, which of the following can be the difference between their ages after 8 years?

Options

A. 5 years
B. 10 years
C. 15 years
D. All of these

Answer

To solve the problem step by step, we will determine the present ages of A and B based on the given ratio and then find the possible age differences after 8 years.

### Step-by-Step Solution:

1. **Understand the Ratio**:
The present ages of A and B are in the ratio of 7:4. This means if we let the present age of A be \(7k\) and the present age of B be \(4k\), where \(k\) is a positive integer.

2. **Calculate the Present Age Difference**:
The difference in their present ages can be calculated as:
\[ \text{Difference} = \text{Age of A} - \text{Age of B} = 7k - 4k = 3k \]

3. **Determine the Age Difference After 8 Years**:
After 8 years, the ages of A and B will be:
- Age of A after 8 years: \(7k + 8\)
- Age of B after 8 years: \(4k + 8\)

The difference in their ages after 8 years will still be:
\[ \text{Difference after 8 years} = (7k + 8) - (4k + 8) = 7k + 8 - 4k - 8 = 3k \]
Thus, the difference in their ages remains \(3k\) even after 8 years.

4. **Finding Possible Values for the Difference**:
Now, we need to check which of the given options can be expressed as \(3k\) where \(k\) is an integer. This means that the difference must be a multiple of 3.

5. **Evaluate the Options**:
Let's evaluate the options:
- **Option 1**: If the difference is 5, then \(3k = 5\) implies \(k = \frac{5}{3}\) (not an integer).
- **Option 2**: If the difference is 10, then \(3k = 10\) implies \(k = \frac{10}{3}\) (not an integer).
- **Option 3**: If the difference is 15, then \(3k = 15\) implies \(k = 5\) (an integer).
- **Option 4**: If the difference is 20, then \(3k = 20\) implies \(k = \frac{20}{3}\) (not an integer).

6. **Conclusion**:
The only option that gives an integer value for \(k\) is Option 3, which corresponds to a difference of 15 years. Therefore, the difference between their ages after 8 years can be 15 years.

### Final Answer:
The difference between their ages after 8 years can be **15 years**.