The ratio of the number of $5-notes to the number of $10-notes is 5 : 6. When four $5-notes were exchanged for $10-notes, the ratio became 8 : 13. How much money was there?

Let's denote the number of $5-notes as 5x and the number of$10-notes as 6x according to the first ratio.

When four $5-notes were exchanged for$10-notes, the number of $5-notes became 5x - 4 and the number of$10-notes became 6x + 2 (since each $5-note exchanged gives us half of a$10-note).

According to the second ratio, we can write the equation (5x - 4) / (6x + 2) = 8 / 13.

Solving this equation gives us x = 18.

So, the number of $5-notes is 5x = 5 * 18 = 90 and the number of$10-notes is 6x = 6 * 18 = 108.

Therefore, the total amount of money is $5 * 90 +$10 * 108 = $450 +$1080 = $1530.

Step 1: Let's denote the number of $5-notes as 5x and the number of$10-notes as 6x, according to the first ratio given in the problem.

Step 2: According to the problem, four $5-notes were exchanged for$10-notes. This means the number of $5-notes decreased by 4 and the number of$10-notes increased by 2 (since each $10-note is worth two$5-notes). So, the new number of $5-notes is 5x - 4 and the new number of$10-notes is 6x + 2.

Step 3: The problem states that after this exchange, the ratio of $5-notes to$10-notes became 8 : 13. So, we can set up the following equation: (5x - 4) / (6x + 2) = 8 / 13.

Step 4: Solve this equation for x. Cross-multiplying gives 65x - 52 = 48x + 16. Simplifying this gives 17x = 68, so x = 4.

Step 5: Substitute x = 4 back into the expressions for the number of $5-notes and$10-notes. The original number of $5-notes was 5x = 20 and the original number of$10-notes was 6x = 24.

Step 6: To find the total amount of money, multiply the number of each type of note by its value and add the results. The total amount of money was 20 * $5 + 24 *$10 = $100 +$240 = $340.