PSAT 8/9 Math: Percentages
Percent problems appear constantly on the PSAT 8/9, in both pure computations and real-world word problems. This free, complete lesson covers finding a percent of a number, percent of a total in word problems, finding the whole from a part and percent, and finding the percent from a part and whole. 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 quiz below.
1. Finding a Percent of a Number
To find a percent of a number, convert the percent to a decimal (divide by 100) and multiply.
Worked Example: Find 18% of 200.
Convert 18% to a decimal: 0.18.
0.18 × 200 = 36
1. Find 18% of 200. (see worked example above)
Explanation: 18% = 0.18. Then 0.18 × 200 = 36.
2. Find 25% of 360.
Explanation: 25% = 0.25. Then 0.25 × 360 = 90.
3. Find 75% of 960.
Explanation: 75% = 0.75. Then 0.75 × 960 = 720.
2. Percent of a Total in Word Problems
Word problems often ask for a specific count that represents a given percent of a total group. The method is the same as a basic percent calculation: convert to a decimal and multiply by the total.
Worked Example: A library has 800 books, and 12% of them are science books. How many science books are there?
0.12 × 800 = 96 science books
1. A library has 800 books, and 12% of them are science books. How many science books are there? (see worked example above)
Explanation: 12% = 0.12. Then 0.12 × 800 = 96 science books.
2. Out of 450 students, 20% play a sport. How many students play a sport?
Explanation: 20% = 0.20. Then 0.20 × 450 = 90 students.
3. Liam has a \$200 allowance and saves 15% of it. How much does he save?
Explanation: 15% = 0.15. Then 0.15 × 200 = \$30.
3. Finding the Whole from a Part and Percent
When you know a part and its percent of some unknown total, divide the part by the decimal form of the percent to find the whole.
Worked Example: 12% of a number is 36. Find the number.
Let the number be n: 0.12n = 36.
Divide by 0.12: n = 300
1. 12% of a number is 36. Find the number. (see worked example above)
Explanation: 0.12n = 36. Divide by 0.12: n = 300.
2. 40% of a number is 52. Find the number.
Explanation: 0.40n = 52. Divide by 0.40: n = 130.
3. 8% of a school's students, which is 48 students, are in the chess club. How many students does the school have in total?
Explanation: 0.08n = 48. Divide by 0.08: n = 600 students.
4. Finding the Percent from a Part and Whole
When you know both a part and the whole, divide the part by the whole, then multiply by 100 to express the result as a percent.
Worked Example: What percent is 80 of 400?
Divide: 80/400 = 0.2.
Convert to a percent: 0.2 × 100 = 20%
1. What percent is 80 of 400? (see worked example above)
Explanation: 80/400 = 0.2, which is 20%.
2. Out of 500 warehouse items, 40 are damaged. What percent of the items are damaged?
Explanation: 40/500 = 0.08, which is 8%.
3. A student answers 42 questions correctly out of 60 total questions. What percent did they answer correctly?
Explanation: 42/60 = 0.7, which is 70%.
Common Mistakes to Avoid
- Forgetting to convert a percent to a decimal before multiplying, multiplying directly by "18" instead of "0.18" gives an answer 100 times too large.
- Dividing instead of multiplying when finding a percent of a number, remember: percent of a number always means decimal × number.
- Mixing up "find the whole" and "find the percent of a number" problems. If the total is unknown, divide the part by the decimal; if the part is unknown, multiply the decimal by the total.
- Forgetting to multiply by 100 after dividing a part by a whole, the raw decimal (like 0.2) isn't the final percent (20%) until converted.
- Misreading which number is the "part" and which is the "whole" in a word problem, re-read carefully to identify which quantity is a subset of the other.
Frequently Asked Questions
What's the quickest way to convert a percent to a decimal?
Move the decimal point two places to the left (which is the same as dividing by 100). For example, 18% becomes 0.18, and 7% becomes 0.07.
How do I know which percent formula to use?
Ask which value is unknown. If the part is unknown, multiply: part = percent (as a decimal) × whole. If the whole is unknown, divide: whole = part / percent (as a decimal). If the percent is unknown, divide and convert: percent = (part / whole) × 100.
Where can I practice more problems like these?
The Percentages quizzes in the PSAT 8/9 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 8/9. You can also browse the full PSAT 8/9 Math Question Bank.