Ratios, Percents, and Proportions
One setup solves them all: part over whole, cross multiply
Three costumes, one machine
Ratios, percents, and proportions look like different topics, but they are the same move wearing different costumes. Each one is a statement that two fractions are equal. Set up the proportion, cross multiply, and solve. On the enhanced ACT this cluster shows up constantly in word problems about prices, maps, recipes, and mixtures.
A percent is just a ratio with 100 as the whole: part / whole = percent / 100. If you can write that one line correctly, every percent problem on the test is arithmetic.
Percent change: divide by the original
Percent increase or decrease is always the change divided by the original value, not the new one. A price going from $80 to $92 changed by $12, and 12 divided by 80 is 15 percent. Dividing by 92 instead gives about 13 percent, and that wrong number will be waiting in the choices.
Discounts run backwards with division. If a shirt sells for $42 after 30 percent off, the original price is 42 / 0.70, because $42 is 70 percent of the original. Adding 30 percent of 42 back on is the trap.
Ratios with real totals
When a ratio like 3:5 comes with a real count, find the value of one ratio unit. If boys to girls is 3:5 and there are 24 boys, one unit is 24 divided by 3, which is 8, so there are 40 girls. This unit method also handles three-part ratios without any new machinery.
Successive percent changes do not cancel and do not add. Up 10 percent then down 10 percent lands below the start, because the second change is taken from a bigger number. Multiply the factors: 1.10 times 0.90 is 0.99.
Proportion problems reward speed. Map scales, recipe scaling, and unit conversions are all one cross multiplication with a calculator. Write the setup so the same units sit on top of both fractions, and you cannot invert by accident.
Strategy note: percent problems with no starting number given are begging you to pick 100 as the starting value. Every percentage becomes a plain number, and the answer choices check themselves.