how much of x% concentration do you add to your y% concentration to get z% concentration?

Dimensional analysis is an important tool.

Here's an example. Suppose the problem is: "How much 85% alcohol do you have to add to 1 kilogram of 40% alcohol to get 55% alcohol?" Ethanol has a cute property where "Mixing equal volumes of ethanol and water results in only 1.92 volumes of mixture", which would be a distraction here, so let's specify that we're working with mass fractions, aka alcohol by weight here.

We're starting with 1 kg of 40% alcohol, which contains .4 kg alcohol in 1 kg of total fluid. We're adding N kg of 85% alcohol, which contains (.85 * N) kg alcohol in N kg of total fluid. So we'll end up with (.4 + .85 * N) kg of alcohol in (1 + N) kg of total fluid. We want 55% alcohol, so this gives us an equation:

(.4 + .85 * N) kg alcohol / (1 + N) kg total fluid = 55% alcohol

We specified that "55% alcohol" means dividing kilograms of alcohol by kilograms of total fluid, so the division is correct here. (Dimensional analysis means more than just making sure that you don't try to add kilograms to liters. Dividing kg total fluid by kg alcohol would give us a dimensionless number, but not one that could be referred to as "percent alcohol". Same for dividing kg alcohol by kg water.)

Now we can nuke the units and grind out the math:

(.4 + .85 * N) / (1 + N) = .55
(.4 + .85 * N) = .55 * (1 + N)
.4 + .85 * N = .55 + .55 * N
.4 + .85 * N - .55 * N = .55
.85 * N - .55 * N = .55 - .4
.3 * N = .15
N = .15 / .3
N = .5

Verify:

1 kg of 40% alcohol contains 1 * .4 = .4 kg alcohol
.5 kg of 85% alcohol contains .5 * .85 = .425 kg alcohol
.4 kg alcohol + .425 kg alcohol = .825 kg alcohol
1 kg total fluid + .5 kg total fluid = 1.5 kg total fluid
.825 kg alcohol / 1.5 kg total fluid = .55 YAY!