x = quarts of 30% acid solution
y = quarts of 65% solution
x + y = 175
Each quart of the 30% solution contributes 0.30x quarts of acid; similarly, each quart of the 65% solution contributes 0.65y quarts of acid. So
0.3x + 0.65y = 0.55*175 = 96.25
That is, the chemist wants 175 quarts of a new mixture with a 55% concentration of acid, which comes out to 95.25 total quarts of acid. The total amounts of acid from either sides of this equation must balance out.
Now solve for x and y:
x + y = 175 ==> y = 175 - x
0.3x + 0.65y = 96.25 ==> 0.3x + 0.65(175 - x) = 96.25
==> 0.3x + 113.75 - 0.65x = 96.25
==> 17.5 = 0.35x
==> x = 50 ==> y = 125
