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Percentages: SAT Math notes
These questions test whether you can find a percent of a number, work out what percent one number is of another, and handle percent increases, decreases and chains of percents.
Study it with worked examples and 8 practice questionsPractise Percentages questions
What this topic tests
Percent means per hundred, so 35% is just 35/100=0.35. Almost every percentage question gets easier once you turn the percent into a decimal and multiply.
The SAT loves to test the language around percents: of, greater than, less than, increased by, and a percent of a percent. Each phrase turns into a specific multiplication, and the questions reward careful translating over guessing.
You'll see everyday settings like tips, sales tax, discounts and survey tables, plus harder questions that use letters instead of numbers or stack two or three percents on top of each other.
Key ideas
- Part = percent × whole. Write the percent as a decimal first: 4% is 0.04, and 0.08% is 0.0008.
- To find what percent a part is of a whole, divide part by whole, then multiply by 100. The whole is whatever comes after the word of.
- Increasing by p% means multiplying by 1+p/100. A 9% increase is × 1.09, and a 131% increase is × 2.31.
- Decreasing by p% means multiplying by 1-p/100. A 20% discount is × 0.80, and 70% less is × 0.30.
- Percent change = (new-old)/old, times 100. Always divide by the original amount.
- Percent changes in a row multiply; they don't add. A 10% rise then a 10% drop is 1.1×0.9=0.99, which is a 1% loss overall.
- To undo a percent change, divide by the multiplier. If 1.06n=318, then n=318÷1.06.
The 4 kinds of Percentages question
Sorted from every Percentages question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Find a percent of an amount
How to spot it. You're given a percent and a whole and asked for the part: what is 4% of 350, how many minutes is 15% of a workday, how big is the tip. Sometimes it's turned around: you know a part and its percent and must find the whole.
How to answer it:
- Turn the percent into a decimal by dividing by 100 (move the decimal point two places left).
- Make sure the whole is in the units the question wants. Convert first if needed, for example hours to minutes.
- Multiply: part = decimal × whole.
- If the whole is the unknown, divide instead: whole = part ÷ decimal.
The trap. Misplacing the decimal point with tiny or unusual percents: 0.001% is 0.00001, not 0.001. Also watch the units. 15% of an 8-hour day is 1.2 hours, but the question may want the answer in minutes.
2. What percent is it? (including percent change)
How to spot it. You know two amounts and need a percent: what percent of the beads are silver, what was the percent increase from 1960 to 1970, group C is p% of group A. Look for the words what percent, percent increase or percent decrease.
How to answer it:
- Identify the whole: the amount you're comparing against. It comes after of, or it's the original (starting) value in a change.
- For a plain percent, divide the part by the whole.
- For a percent change, subtract first (new minus old), then divide by the old value.
- Multiply by 100 to turn the decimal into a percent.
The trap. Dividing by the wrong number. Percent change is always measured from the starting value: going from 75.74 dollars to 79.86 dollars, divide by 75.74, not 79.86. And if the new amount is 349% of the old one, the increase is 249%, not 349%.
3. Increase or decrease by a percent
How to spot it. Something goes up or down by a percent: a price plus sales tax, a sale price, a 29% projected increase, a number 36% greater than 50. Some questions run it backward (after a 6% increase the result is 318, so what was the number?) or repeat the change over several years.
How to answer it:
- Build the multiplier: 1+p/100 for an increase, 1-p/100 for a decrease.
- Multiply the starting amount by the multiplier to get the new amount.
- If the same change happens several times, raise the multiplier to that power.
- If you know the new amount and need the original, divide by the multiplier.
The trap. Undoing an increase by taking the percent off the new number. If 318 is 6% more than n, then n is 318÷1.06=300, not 318 minus 6% of 318 (about 299). Another trap is finding only the change, like 0.09x, when the question wants the new total, 1.09x.
4. Chains of percents
How to spot it. Two or more percents are stacked: 30% of the 40% that are red, a price that's 270% of cost and then 70% off, a is 110% greater than b and b is 90% less than 47. The answer comes from linking the steps together.
How to answer it:
- Translate each sentence into its own multiplication. Of means ×. p% greater means × (1+p/100). p% less means × (1-p/100).
- Start from the number you know and work through the chain one step at a time, or substitute one equation into the next.
- Multiply the decimals together. Never add or subtract the percents themselves.
- Re-read what's being asked: the final amount, a multiple of the original, or the percent that is NOT in the group.
The trap. Adding or subtracting the percents directly. 80% more than a number that is 70% less than b is not 10% more than b; it's 1.8×0.3=0.54 times b. Also, 110% greater means 210% of the number, not 110% of it.
Practise Percentages
On One for All SAT every question has a step-by-step solution, worked both by hand and in Desmos, the calculator built into the digital SAT, and the notes have worked examples of each kind of question, then practice questions you can check as you go. It's free, and there's nothing to sign up for.
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