Divide the sum of \( \frac{65}{13} \) and \( \frac{5}{7} \) by their difference.

To solve this problem, we first need to find the sum and difference of the two fractions \( \frac{65}{13} \) and \( \frac{5}{7} \). Starting with the sum: \[ \frac{65}{13} = 5 \] Now, we find the sum: \[ 5 + \frac{5}{7} \] To add these, we convert 5 into a fraction with a denominator of 7: \[ 5 = \frac{35}{7} \] Therefore, \[ \frac{35}{7} + \frac{5}{7} = \frac{35 + 5}{7} = \frac{40}{7} \] Next, we calculate the difference: \[ 5 - \frac{5}{7} \] Again, we convert 5 into a fraction: \[ 5 = \frac{35}{7} \] Thus, the difference is: \[ \frac{35}{7} - \frac{5}{7} = \frac{35 - 5}{7} = \frac{30}{7} \] Now, we need to divide the sum by the difference: \[ \frac{\frac{40}{7}}{\frac{30}{7}} \] Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{40}{7} \times \frac{7}{30} = \frac{40 \cdot 7}{7 \cdot 30} = \frac{40}{30} \] Now, we can simplify \( \frac{40}{30} \): \[ \frac{40}{30} = \frac{4}{3} \] Thus, the final answer is: \[ \boxed{\frac{4}{3}} \]