Percents: Complete Free SAT Math Lesson
"Percentages" is the second skill listed in College Board's official Problem-Solving and Data Analysis domain, right after ratios and rates, and it's tested constantly, from straightforward conversions to multi-step discount, interest, and mixture problems. This free lesson covers everything: converting between percents, decimals, and fractions, the percent increase and percent decrease formulas, successive percent changes, finding the part, the percent, or the whole using an equation or a proportion, and percent word problems covering mixtures, interest, and combined discount-and-tax scenarios. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. Converting Between Percents, Decimals, and Fractions
The word percent means "hundredth" or "out of every hundred." To write a decimal or fraction as a percent, multiply by 100 and add the % sign. To write a percent as a decimal or fraction, multiply by 1100 and drop the % sign, then simplify.
Worked Example: Write 175% as a decimal and as a fraction.
175% = 175 × 1100 = 175100 = 1.75
175% = 175100 = 74 (Simplify the fraction)
1. Write 0.07 as a percent.
Explanation: Multiply by 100 and add the % sign: 0.07 × 100 = 7, so 0.07 = 7%.
2. Write 35 as a percent.
Explanation: 35 × 100% = 60%.
3. Write 125% as a decimal.
Explanation: Divide 125 by 100 (multiply by 1100) and drop the % sign: 125% = 1.25.
4. Write 8% as a fraction in simplest form.
Explanation: 8% = 8100. The greatest common factor of 8 and 100 is 4, so simplify to 225.
5. Which decimal is equivalent to 0.5%?
Explanation: 0.5% = 0.5 × 1100 = 0.005. It's easy to lose track of a zero here since the percent itself already contains a decimal.
2. Percent of Change
The percent a quantity increases or decreases from its original amount is the percent of change.
| Percent increase | amount of increaseoriginal amount |
|---|---|
| Percent decrease | amount of decreaseoriginal amount |
The denominator is always the original amount, never the new amount, this is the detail that trips students up most often.
Worked Example: A $450 laptop is on sale for $342. What is the percent discount?
percent discount = 450 − 342450 = 108450 = 0.24 = 24%
1. A shirt's price increased from $25 to $32. What is the percent increase?
Explanation: percent increase = 32 − 2525 = 725 = 0.28 = 28%.
2. A town's population decreased from 8,400 to 7,140. What is the percent decrease?
Explanation: percent decrease = 8,400 − 7,1408,400 = 1,2608,400 = 0.15 = 15%.
3. A $60 jacket is marked down to $45. What is the percent discount?
Explanation: percent discount = 60 − 4560 = 1560 = 0.25 = 25%.
4. A stock's value grew from $50 to $68 in one year. What is the percent increase?
Explanation: percent increase = 68 − 5050 = 1850 = 0.36 = 36%.
5. A company's revenue fell by 20% from $500,000. What is the new revenue?
Explanation: A 20% decrease means the new revenue is 80% of the original: 0.80 × $500,000 = $400,000.
6. If a price is increased by 15% and then the new price is decreased by 15%, is the final price the same as the original price?
Explanation: Starting at 100: a 15% increase gives 115. A 15% decrease of 115 is 17.25, leaving 115 − 17.25 = 97.75. The final value is lower than the original 100, because the 15% decrease was taken from a larger number than the 15% increase was added to.
3. Successive Percent Changes
When a quantity changes by one percent and then by another, each change applies to the result of the previous step, not to the original amount. This means you can't simply add or subtract the two percents to find the combined effect, you have to apply them one at a time, or multiply the corresponding decimal factors together.
Worked Example: A price of $200 is increased by 10%, and then the new price is increased by another 10%. What is the final price?
$200 × 1.10 = $220 (First increase)
$220 × 1.10 = $242 (Second increase)
Two successive 10% increases combine to a 21% increase overall, not 20%, since the second 10% was taken from the already-larger $220.
1. A price of $200 is increased by 10%, and then the new price is increased by another 10%. What is the final price?
Explanation: This is the worked example above: $200 × 1.10 = $220, then $220 × 1.10 = $242.
2. A 10% increase followed by another 10% increase is equivalent to a single percent increase of:
Explanation: Multiply the decimal factors: 1.10 × 1.10 = 1.21, which represents a 21% total increase.
3. A price is increased by 25% and then the new price is decreased by 20%. Is the final price the same as, greater than, or less than the original price?
Explanation: Starting at 100: a 25% increase gives 125. A 20% decrease of 125 is 25, leaving 125 − 25 = 100. In this specific case, the two changes exactly cancel out, since 1.25 × 0.80 = 1.00. This won't happen with just any pair of percents, it's a coincidence of these particular numbers.
4. An $80 item is discounted 30%, and then an additional 10% is taken off the sale price. What is the final price?
Explanation: $80 × 0.70 = $56 after the first discount. Then $56 × 0.90 = $50.40 after the second discount.
4. Finding the Part, the Percent, or the Whole
Every percent problem asks for one of three missing quantities: the part, the percent, or the whole. Each type can be solved either by writing a direct equation or by writing a proportion.
| Type | Example question | Equation | Proportion |
|---|---|---|---|
| Finding the part | What is 15% of 72? | n = 0.15 × 72 | 15100 = n72 |
| Finding the percent | What percent of 20 is 6? | n100 × 20 = 6 | n100 = 620 |
| Finding the whole | 17 is 25% of what number? | 17 = 0.25 × n | 25100 = 17n |
Worked Example: Translate each verbal phrase into an equation, then solve.
a. What is 40% of 85?
n = 0.40 × 85 = 34
b. What percent of 40 is 16?
n100 × 40 = 16 ⇒ n = 1,60040 = 40, so 40%
c. 18 is 30% of what number?
18 = 0.30 × n ⇒ n = 180.30 = 60
1. What is 40% of 85?
Explanation: This is the worked example above: n = 0.40 × 85 = 34.
2. What percent of 40 is 16?
Explanation: n100 × 40 = 16, so n = 1,60040 = 40. The answer is 40%.
3. 18 is 30% of what number?
Explanation: This is the worked example above: 18 = 0.30n, so n = 60.
4. What is 0.75% of 400?
Explanation: n = 0.0075 × 400 = 3. Remember that 0.75% converts to the decimal 0.0075, not 0.075.
5. 45 is what percent of 180?
Explanation: 45180 = 0.25 = 25%.
6. 72 is 120% of what number?
Explanation: 72 = 1.20n, so n = 721.20 = 60. A percent over 100%, like 120%, is completely valid, it just means "more than the whole."
5. Percent Word Problems
Three types of word problems account for most of the multi-step percent questions on the SAT: mixing two solutions of different concentrations, splitting an investment between two interest rates, and combining a discount with a tax. Each one is solved by writing an equation that balances a total quantity before and after the percent is applied.
Mixtures of Two Solutions
Worked Example: How many milliliters of 70% acid solution must be added to 50 milliliters of a 30% acid solution to make a 50% acid solution?
Let x = the amount of 70% solution added.
0.30(50) + 0.70x = 0.50(50 + x)
15 + 0.7x = 25 + 0.5x
0.2x = 10, so x = 50
50 milliliters of the 70% solution must be added.
1. How many liters of a 60% saline solution must be added to 20 liters of a 20% saline solution to make a 40% saline solution?
Explanation: Let x = liters of 60% solution added. 0.20(20) + 0.60x = 0.40(20 + x), so 4 + 0.6x = 8 + 0.4x. Then 0.2x = 4, and x = 20.
Interest and Investments
Worked Example: Maria invested $9,000 in stocks and bonds. The stocks pay 4% interest a year and the bonds pay 7% interest a year. Her interest income is $480 this year. How much money was invested in stocks?
Let x = the amount invested in stocks, so 9,000 − x is invested in bonds.
0.04x + 0.07(9,000 − x) = 480
0.04x + 630 − 0.07x = 480
−0.03x = −150, so x = 5,000
$5,000 was invested in stocks.
2. James invested $6,000 in two accounts. One pays 3% interest and the other pays 6% interest. His total interest income for the year was $270. How much was invested at 6%?
Explanation: Let y = the amount invested at 6%, so 6,000 − y is invested at 3%. 0.03(6,000 − y) + 0.06y = 270, so 180 − 0.03y + 0.06y = 270. Then 0.03y = 90, and y = 3,000.
Discounts and Tax
Worked Example: The sale price of a bicycle is $246.24 after a 20% discount and 8% additional tax. What was the original price of the bicycle before discount and tax?
Let x = the original price.
x − 0.20x = 0.80x (Price after the 20% discount)
0.80x(1.08) = 0.864x (Price after 8% tax)
0.864x = 246.24, so x = 285
The original price of the bicycle was $285.
3. The sale price of a television is $481.50 after a 25% discount and 7% additional tax. What was the original price before discount and tax?
Explanation: Let x = the original price. x(0.75)(1.07) = 481.50, so 0.8025x = 481.50, and x = 600. Check: $600 × 0.75 = $450 after the discount, and $450 × 1.07 = $481.50 after tax.
4. A jacket costs $84 before an 8% sales tax. What is the total cost including tax?
Explanation: $84 × 1.08 = $90.72.
Common Mistakes to Avoid
- Dividing by the new amount instead of the original amount. Percent of change always uses the original, starting amount in the denominator, never the final amount.
- Adding or subtracting successive percents directly. A 10% increase followed by a 10% decrease is not a net 0% change. Each percent applies to a different base amount, so you must apply them one step at a time or multiply the decimal factors.
- Losing track of a decimal place when converting a small percent. Percents under 1%, like 0.5% or 0.75%, are easy to convert incorrectly. Write out the multiplication by 1100 carefully rather than estimating.
- Confusing "percent of" with "percent more/less than." "20% of 80" means 0.20 × 80 = 16. "20% more than 80" means 80 + 0.20(80) = 96. These are very different calculations that share similar wording.
- Setting up a mixture or investment equation with the wrong total. The two amounts on the left side of a mixture or interest equation must add up to the actual total described in the problem, double-check that the "60 + x" or "9,000 − x" style term matches what the problem states.
- Applying tax before discount, or forgetting to apply one of them entirely. When a problem combines a discount and a tax, apply the discount first, then apply the tax to the discounted price, not the original price, unless the problem specifically states otherwise.
Frequently Asked Questions
Should I use the equation method or the proportion method for percent problems?
Both give the same answer, it's a matter of preference. The equation method (like n = 0.15 × 72) is usually faster for simple problems. The proportion method can feel more intuitive for "what percent" questions, since it directly compares part to whole.
Why doesn't a percent increase and an equal percent decrease cancel out?
Because the two percents are calculated from different base amounts. The increase is a percent of the smaller, original amount, while the decrease is the same percent of the larger, already-increased amount, so the decrease removes more than the increase added.
What's the fastest way to set up a mixture or investment word problem?
Build a small table with three columns: the amount, the percent (or rate), and the resulting part. Fill in what you know, represent the unknown amount with a variable, and set the sum of the individual parts equal to the total part described in the problem.
Where can I practice more problems like these?
The Percentages quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.