Explanation:

To solve this problem, let's use algebra:

- Let the two numbers be \(5x\) and \(11x\).

- Initially, the ratio is 5:11, so these numbers align with \(5x:11x\).

- Increase each by 10, making them \(5x + 10\) and \(11x + 10\).

- The new ratio is 7:13.

Set up the equation:

$$

\frac{5x + 10}{11x + 10} = \frac{7}{13}

$$

Cross-multiply to solve:

$$

13(5x + 10) = 7(11x + 10)

$$

Simplify and solve for \(x\):

$$

65x + 130 = 77x + 70 \\

12x = 60 \\

x = 5

$$

Substitute back:

- Original numbers: 5x = 25, 11x = 55.

- Sum = 25 + 55 = 80.

- Option 1: 80

- Option 2: 32

- Option 3: 48

- Option 4: 160