How to solve ACT percent problems

Percent means per hundred, so every percent question turns into ordinary arithmetic the moment you rewrite the percent as a decimal and decide what counts as the whole. Deciding what the whole is turns out to be the entire game. Nearly every wrong answer on this topic comes from dividing by the wrong number.

Percent means per hundred

Forty percent is 40 per hundred, which is 40/100, which is 0.40. Moving between those three forms should be automatic: divide by 100 to go from percent to decimal, multiply by 100 to go the other way. Then two small translations handle most questions. The word of means multiply, and the word is means equals.

Percentages sit inside Integrating Essential Skills, which is 20 percent of the Math section and covers rates, percentages, proportional relationships, area, and similar ideas applied in more complex ways. That means you will rarely see a bare percent computation. You will see one buried in a story about prices, populations, or survey results.

Finding a percent of a number

Convert and multiply. 40 percent of 60 = 0.40 times 60 = 24. Fifteen percent of 240 = 0.15 times 240 = 36.

Friendly percents are worth doing in your head. Ten percent is just the number with the decimal point moved one place left, so 10 percent of 240 is 24. Five percent is half of that, or 12. Twenty percent is double it, or 48. Build 15 percent as 24 plus 12 = 36, which matches the decimal method and takes less time.

Finding what percent one number is of another

Divide the part by the whole, then convert to a percent. If a question asks what percent 18 is of 45, compute 18 divided by 45 = 0.4, which is 40 percent. If it asks what percent 63 is of 180, compute 63 divided by 180 = 0.35, which is 35 percent.

The whole is the number that follows the word of. Reversing the two, and computing 45 divided by 18, gives 2.5, or 250 percent, and that wrong answer will be sitting in the choices.

Percent increase and decrease

Both use the same formula: the change divided by the original amount. Find the difference, then divide by where you started, never by where you ended.

• Increase. A jacket goes from 80 dollars to 92 dollars. The change is 92 minus 80 = 12, and 12 divided by 80 = 0.15, so the price rose 15 percent.
• Decrease. A club shrinks from 250 members to 200. The change is 50, and 50 divided by 250 = 0.20, so membership fell 20 percent.
• Working backward. A shirt costs 60 dollars after a 25 percent discount. Paying 75 percent of the original means 0.75 times the original = 60, so the original is 60 divided by 0.75 = 80 dollars. Check: 25 percent of 80 is 20, and 80 minus 20 is 60.

That last pattern is worth practicing. Multiplying by 1 plus the percent handles increases and 1 minus the percent handles decreases, and dividing by that multiplier undoes the change.

Successive percent changes do not add

Apply changes one at a time, or multiply the factors. They never simply add, because the second change acts on a new base.

Start at 100 and raise it 20 percent: 100 times 1.20 = 120. Now drop that by 20 percent: 120 times 0.80 = 96. You are not back at 100. You are 4 percent below where you started, because the 20 percent decrease was taken from the larger 120.

Increases behave the same way. A 200 dollar item goes up 10 percent to 220, then up another 25 percent to 220 times 1.25 = 275. The combined multiplier is 1.10 times 1.25 = 1.375, so the total increase is 37.5 percent, not 35 percent.

Common traps

• Dividing by the new value. In the 250 to 200 example, 50 divided by 200 gives 25 percent, which is wrong. The denominator is always the original.
• Adding successive percents. A 20 percent rise followed by a 20 percent drop is a net loss, not a wash.
• Answering the wrong question. Some questions want the percent change, others want the new amount. Underline which.
• Confusing percent with percentage points. Going from 20 percent to 25 percent is a rise of 5 percentage points, but a 25 percent increase in the rate itself.
• Flipping part and whole. The whole follows the word of.

A short drill

Run these four in under two minutes: 30 percent of 90, what percent 24 is of 96, the percent change from 45 to 54, and the result of raising 500 by 10 percent and then cutting it by 10 percent. The answers are 27, then 25 percent, then 9 divided by 45 which is 20 percent, then 500 times 1.10 = 550 and 550 times 0.90 = 495. thirty-six quizzes mix these into word problems the way the test does, which is where the real difficulty lives.

Percent change is also the computation most often requested by chart and table questions, and it shares its proportional reasoning with rate, distance, and work problems. For where all of this fits in the section as a whole, see the full list of ACT Math topics.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.