Remember that water is a 0% solution of argyrol (since there's no argyrol in it). Let's write an equation where x = ounces of water.
The amount of argyrol in a given solution is (percent)(amount). So:
There are 0.0x = 0 ounces of argyrol in the water solution.
There are 0.15(20) = 3 ounces of argyrol in the 15% solution.
There are 0.1(20 + x) = 0 + 3 ounces of argyrol in the resulting 10% solution.
Use that last equation and solve for x:
0.1(20 + x) = 3
2 + 0.1x = 3
0.1x = 1
x = 10 ounces of water