how many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt

50 ounces of 15% saline solution contains 0.15*50 = 7.5 ounces of salt.

Let x be the amount of pure water you add to this solution. Then you end up with (x + 50) ounces of this new solution. The amount of salt stays the same.

You want the new solution to have a concentration of 10%. This means you need

7.5 / (x + 50) = 0.1

==> 7.5 = 0.1 (x + 50)

==> 7.5 = 0.1x + 5

==> 2.5 = 0.1x

==> 2.5/0.1 = 0.1x/0.1

==> x = 25