Compute a percentage of a number, percent increase and decrease, successive and reverse percentages, and simple interest in real contexts (Number and Quantity, Integrating Essential Skills).

What this topic is asking

Percentages are everywhere on the ACT: discounts, taxes, tips, interest, population change and data. The area asks you to compute a percent of a number, find a percent change, chain successive percentages, and run a percentage backwards to recover an original amount. The reliable approach treats a percent as a multiplier.

Percent as a multiplier

The single most useful idea is to turn a percentage into a decimal multiplier.

Working with multipliers makes discounts, taxes and successive changes a matter of multiplying decimals.

Percent increase and decrease

Successive percentages

When two percentage changes act one after another, multiply their factors. A 10% increase followed by a 10% decrease is $\times 1.10 \times 0.90 = \times 0.99$, a net 1% decrease, not zero. This is why "a 50% rise then a 50% fall" does not return to the start: $1.50 \times 0.50 = 0.75$, a 25% net loss. The order does not change the product, but each percentage acts on the running amount, not the original.

Reverse percentages

A reverse-percentage question gives the result of a change and asks for the original. Because the change is a multiplication, you divide to undo it.

If a price after a 20% increase is \72$, then$1.20 \times \text{original} = 72$, so original$= \frac{72}{1.20} = 60$. The classic trap is to take 20$72$(giving$57.60$), which is wrong because 20% of the original differs from 20% of the larger result. Always set up the multiplier equation and divide.

Simple interest

Simple interest is a percentage of the principal earned each period: $I = Prt$, where $P$ is the principal, $r$ the rate (as a decimal) per period, and $t$ the number of periods. After $t$ periods the balance is $P + Prt = P(1 + rt)$. For example, \500$at 4$I = 500 \times 0.04 \times 3 = 60$, for a balance of$\$560$. (Compound interest, which the ACT may also ask, multiplies by $(1 + r)$ each period; see the exponential functions topic.)

Why multipliers beat memorised steps

Treating every percentage as a multiplier unifies discounts, markups, taxes, tips, successive changes and reversals into one method: identify the multiplier, then multiply forward or divide backward. It also prevents the most common errors, such as adding a discount and a tax as if they were a single 22% change, or subtracting to reverse a change that was made by multiplying. A quick estimate ("a 30% discount on $120 should be around$84") guards against slips.

Try this

Q1. What is 15% of 240? [1 point]

• Cue. $0.15 \times 240 = 36$.

Q2. A population falls from 500 to 460. What is the percent decrease? [1 point]

• Cue. $\frac{500 - 460}{500} = \frac{40}{500} = 0.08 = 8\%$.