15 gallons of 35% peroxide solution contains 0.35 × 15 = 5.25 gallons of hydrogen peroxide. The rest of the volume is made up of 15 - 5.25 = 9.75 gallons of water.
If we add [tex]x[/tex] gallons of water to the solution, the amount of hyrdrogen peroxide in the solution stays the same, but the volume of water increases to [tex]9.75+x[/tex] gallons, and the total volume increases to [tex]15+x[/tex] gallons.
To get a concentration of 8% in this diluted solution, we add [tex]x[/tex] gallons of water such that
[tex]\dfrac{5.25}{15+x} = 0.08[/tex]
Solve for [tex]x[/tex].
[tex]5.25 = 0.08 (15 + x)[/tex]
[tex]5.25 = 1.2 + 0.08x[/tex]
[tex]0.08x = 4.05[/tex]
[tex]x = 50.625[/tex]
To get the desired concentration, one must add 50.625 gallons of water, and one will end up with a total volume of 65.625 gallons of solution.
