PSAT/NMSQT Math: Percentages
Percentages appear constantly on the PSAT/NMSQT, from simple "what is x% of y" questions to multi-step problems involving discounts, tax, and compound growth. This free, complete lesson covers basic percentage calculations, percent increase and decrease, reverse percentage problems and sales tax, and sequential and compound percentage changes. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quizzes below.
1. Basic Percentage Calculations
To find a percentage of a number, convert the percentage to a decimal (divide by 100) and multiply. To express one number as a percentage of another, divide the part by the whole and multiply by 100.
Worked Example: What is 25% of 80?
0.25 × 80 = 20
1. What is 25% of 80? (see worked example above)
Explanation: 25% = 0.25. 0.25 × 80 = 20.
2. A jar contains 240 beads. How many beads represent 15% of the total?
Explanation: 0.15 × 240 = 36.
3. A student answered 35 out of 50 survey questions correctly. What percentage did they answer correctly?
Explanation: (35/50) × 100 = 0.70 × 100 = 70%.
2. Percent Increase and Decrease
Percent change is calculated as (change in value / original value) × 100. Percent increase uses a positive change (new value larger); percent decrease uses the drop in value, still divided by the original value.
Worked Example: A jacket's price rises from $50 to $65. Find the percent increase.
Change = 65 − 50 = 15.
(15/50) × 100 = 30%
1. A jacket's price rises from $50 to $65. Find the percent increase. (see worked example above)
Explanation: Change = 15. (15/50) × 100 = 30%, using the original value (50) as the base.
2. A phone's price falls from $400 to $300. Find the percent decrease.
Explanation: Change = 400 − 300 = 100. (100/400) × 100 = 25%, using the original value (400) as the base.
3. A district projects a 34% increase in jobs over ten years from a baseline of 180,000 positions. How many new positions does this represent?
Explanation: 0.34 × 180,000 = 61,200 new positions.
3. Reverse Percentage Problems and Sales Tax
A reverse percentage problem gives you the result after a percentage change and asks for the original value; divide (rather than multiply) using the appropriate decimal factor. Sales tax problems work the same way as percent increase: multiply the price by (1 + tax rate) to find the total.
Worked Example: A shirt is on sale for $40 after a 20% discount. Find its original price.
$40 represents 80% (100% − 20%) of the original price P:
0.80P = 40 → P = 50, so the original price was $50
1. A shirt is on sale for $40 after a 20% discount. Find its original price. (see worked example above)
Explanation: $40 is 80% of the original: 0.80P = 40 → P = 50.
2. Calculate the total cost of a $120 item after a 7.5% sales tax.
Explanation: Total = 120 × (1 + 0.075) = 120 × 1.075 = $129.
3. Out of 750,000 total beads, 697,500 are gold. What percentage of the beads are gold?
Explanation: (697,500/750,000) × 100 = 93%.
4. Sequential and Compound Percentage Changes
When a value undergoes multiple percentage changes in sequence, apply each one to the result of the previous step, never to the original value alone. This means a 10% increase followed by a 10% decrease does not return to the original value.
Worked Example: Apply a 25% discount, then a 20% discount, to an $80 jacket. Find the final price.
After 25% off: 80 × 0.75 = $60. After 20% off that: 60 × 0.80 = $48.
Final price = $48
1. Apply a 25% discount, then a 20% discount, to an $80 jacket. Find the final price. (see worked example above)
Explanation: First discount: 80 × 0.75 = 60. Second discount, applied to 60 (not 80): 60 × 0.80 = 48.
2. A value increases by 10%, then decreases by 10%. What is the net percent change from the original value?
Explanation: Starting from 100: after +10%, 110; after −10% of 110 (11), 110 − 11 = 99, a net 1% decrease from the original 100, since the second change applies to the larger, already-increased value.
3. $2,500 is deposited at 4% annual compound interest. Find the balance after 3 years.
Explanation: A = 2500(1.04)³ = 2500 × 1.124864 ≈ $2,812.16, compounding each year on the previous year's total.
Common Mistakes to Avoid
- Using the wrong base value for percent change. Percent increase or decrease is always calculated relative to the original (starting) value, not the new value.
- Applying multiple discounts or increases to the same original value instead of sequentially. Each successive percentage change applies to the result of the previous step.
- Assuming an increase and an equal-percentage decrease cancel out. They don't, since the decrease applies to a different (larger or smaller) base value than the increase did.
- Multiplying instead of dividing on a reverse percentage problem. If you're given the result and need the original, set up an equation and solve for the unknown, don't just multiply by the percentage directly.
- Forgetting to add 1 to the decimal rate when calculating a total after tax or a markup. The multiplier for a 7.5% increase is 1.075, not just 0.075.
Frequently Asked Questions
Why doesn't a 10% increase followed by a 10% decrease return to the original value?
Because the second percentage is applied to the new, larger amount rather than the original. A 10% decrease off a bigger number removes more than a 10% increase originally added, resulting in a small net loss.
What's the fastest way to solve a reverse percentage problem?
Write an equation where the known result equals the decimal multiplier times the unknown original value, then solve for that unknown by dividing, rather than trying to work backward informally.
Where can I practice more problems like these?
The Percentages quizzes in the PSAT/NMSQT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.