Percentages, Ratios, and Proportions

The Language of Comparison

Percentages, ratios, and proportions all answer the same underlying question: how do two quantities compare? They are among the most common ACT Math topics because they hide inside word problems about money, mixtures, maps, and rates. Treat them as one connected toolkit rather than three separate rules.

Percentages

A percent is a fraction out of $100$: $37\% = \tfrac{37}{100} = 0.37$. To take a percent of a number, convert the percent to a decimal and multiply.

$\text{part} = \text{percent (as a decimal)} \times \text{whole}$

So $15\%$ of $80$ is $0.15 \times 80 = 12$. To go the other way — "what percent is one number of another?" — divide the part by the whole and multiply by $100$: $18$ out of $24$ is $\tfrac{18}{24} = 0.75 = 75\%$.

Percent Change

Percent change compares how much a quantity grew or shrank relative to where it started:

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

The denominator is always the original value. If a price rises from $40 to $52, the change is $\tfrac{52 - 40}{40} = \tfrac{12}{40} = 0.30 = 30\%$ increase. A negative result means a decrease.

The Multiplier Shortcut

Rather than computing the change and adding it back, multiply directly:

An item costing $80 with $25\%$ off becomes $80 \times 0.75 = \dollar 60$. This shortcut is essential for successive percents, where you simply chain the multipliers. A $80 item at $25\%$ off, then charged $8\%$ sales tax, costs $80 \times 0.75 \times 1.08 = \dollar 64.80$. Crucially, a $20\%$ discount followed by a $20\%$ tax does not return the original price, because the two percents act on different amounts.

Reverse Percent

When the final amount is known and you need the original, divide by the multiplier. If a price after a $20\%$ increase is $84, then $\text{original} \times 1.20 = 84$, so the original is $\tfrac{84}{1.20} = \dollar 70$. A frequent trap is subtracting $20\%$ of $84 instead — that uses the wrong base.

Ratios

A ratio compares quantities part-to-part, such as $3 : 2$ for red to blue. The key move is to recognize the total number of parts. A ratio of $3 : 2$ has $3 + 2 = 5$ parts, so each part is $\tfrac{1}{5}$ of the whole. To split $150 in the ratio $2 : 3$, there are $5$ parts worth $\tfrac{150}{5} = \dollar 30$ each, giving $60 and $90.

Watch the wording: a part-to-part ratio ($3 : 2$) is different from a part-to-whole ratio ($3 : 5$). Read carefully to see whether "3 to 2" or "3 out of 5" is intended.

Proportions

A proportion sets two ratios equal, and you solve it by cross-multiplying:

$\frac{a}{b} = \frac{c}{d} \quad \Longrightarrow \quad a \cdot d = b \cdot c$

Proportions model any situation where two quantities scale together: recipe amounts, map distances, and similar figures. If a map uses a scale of $1$ inch to $25$ miles, then $4.5$ inches represents $\tfrac{25}{1} = \tfrac{x}{4.5}$, so $x = 25 \times 4.5 = 112.5$ miles.

Unit Rates

A unit rate expresses "how much per one" — miles per gallon, dollars per ounce, words per minute. Reduce any rate so the second quantity is $1$, which makes different offers directly comparable. If $5$ notebooks cost $8.75, the unit price is $\tfrac{8.75}{5} = \dollar 1.75$ each, and you can now compare it fairly to a pack priced differently. When two options are given, computing the unit rate for each is the fastest way to see which is the better value.

A Reliable Game Plan

Decide which tool the question needs: for a percent of a change, use $\tfrac{\text{new} - \text{old}}{\text{old}}$; for discounts and taxes, chain multipliers; for splitting a total, find the value of one ratio "part"; for scaling quantities, set up and cross-multiply a proportion. Above all, identify the correct base — most percent errors come from dividing by the wrong number.