Percentages, Percent Change & Markup: The Math of Everyday Money
A percent is just a fraction out of 100. Master three moves — finding a percent, finding percent change, and reversing a markup — and you handle taxes, tips, discounts, and a whole genre of SAT, ACT, and SSAT word problems.
The Short Version
- A percent is a fraction out of 100: 25% = 25/100 = 0.25.
- "Percent of" means multiply: 20% of 80 = 0.20 × 80 = 16.
- Percent change = (new − old) / old × 100 — always divide by the original.
- A 20% increase then a 20% decrease does not return the original — a favorite SAT/ACT trap.
Percents are everywhere money is: sales tax, restaurant tips, store discounts, interest, population growth. The word "percent" literally means "per hundred," so every percent is just a fraction with a denominator of 100. Once you can translate a percent into a decimal and remember that "of" means multiply, the entire topic reduces to careful arithmetic.
This guide covers the three core moves, the percent-change formula students misremember most, and the compounding trap the tests love, with worked and practice problems matched to real test difficulty at Northside Tutoring.
Why Percentages Matter
Percent problems are among the most common on every math section because they connect arithmetic to the real world. They also reward precision: a single misplaced decimal or a division by the wrong number turns a right answer into a tempting wrong one. Students who internalize the formulas move through these quickly and confidently.
What a Percent Really Is
A percent is a fraction out of 100. To convert, slide the decimal two places: 45% = 0.45, and 0.08 = 8%. To turn a fraction into a percent, divide and multiply by 100. This fluency — percent ↔ decimal ↔ fraction — is the foundation for everything else.
Finding a Percent of a Number
The word "of" signals multiplication. To find a percent of a number, convert the percent to a decimal and multiply:
This single translation — "of means times" — unlocks most percent word problems.
Percent Increase & Decrease
Percent change always compares the size of the change to the original amount:
If a price rises from $50 to $60, the change is $10 over the original $50: 10/50 = 0.20 = a 20% increase. Dividing by the new value instead of the old is the most common error here.
The multiplier shortcut
A 15% increase multiplies by 1.15; a 15% decrease multiplies by 0.85. Thinking in multipliers makes multi-step problems (and the compounding trap below) far faster than adding and subtracting separately.
Markup, Discount & Tax
Markup, tax, and tip all add a percent; discounts subtract one. Use multipliers: a $40 item with 8% tax costs 40 × 1.08 = $43.20. A $40 item at 25% off costs 40 × 0.75 = $30. Stack them by multiplying the multipliers.
Working Backward From a Percent
Sometimes the test gives you the result and asks for the original. If $60 is the price after a 20% discount, then $60 represents 80% of the original: original = 60 / 0.80 = $75. The key is identifying what percent the known amount represents, then dividing.
The compounding trap
A 20% raise followed by a 20% cut does not return you to the start. Start at 100: ×1.20 = 120, then ×0.80 = 96 — a net 4% loss. Percents compound on a changing base, and the test always offers "no change" as the trap answer.
Where You'll See This — Test by Test
Percentages need no reference sheet — just translation and care. They are pervasive on all three tests, especially in data and word-problem contexts, and the percent-change and compounding ideas appear as their own conceptual questions.
Digital SAT
Heavily tested across data analysis and word problems: percent of, percent change, and reverse-percent questions are all common.
Explore SAT Tutoring → College AdmissionsACT
Frequent percent and percent-change problems, often involving discounts, tax, or population growth.
Explore ACT Tutoring → Independent School AdmissionsSSAT
Core Middle- and Upper-Level content. Usually "percent of" and simple percent-change questions.
Explore SSAT Tutoring → K-12 CurriculumSchool Math & Stats
A foundational life-and-school skill underpinning statistics, finance, and science.
Explore Math Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Percentages — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
For the developer / editor
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Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
What is 35% of 240?
Show solution
Convert and multiply: 0.35 × 240 = 84.
A price increases from $80 to $92. What is the percent increase?
Show solution
Change = 92 − 80 = 12, over the original 80.
12/80 = 0.15 = 15%.
A $50 jacket is marked 30% off. What is the sale price?
Show solution
30% off means you pay 70%: 50 × 0.70 = 35.
After a 20% discount, a item costs $48. What was the original price?
Show solution
$48 is 80% of the original: original = 48 / 0.80 = 60.
A stock rises 10% one year, then falls 10% the next. If it started at $200, what is its final value?
Show solution
200 × 1.10 = 220, then 220 × 0.90 = 198.
Net result is below the start — not $200.
Common Mistakes to Avoid
Three traps that catch students every year
- Dividing by the new value for percent change. Percent change is always over the original amount, not the new one.
- Assuming a raise and an equal cut cancel. Percents compound on a changing base, so +20% then −20% leaves you below where you started.
- Forgetting to convert the percent. 25% is 0.25, not 25, in a multiplication. A missing decimal shift is an instant 100× error.
Practice Problems — You Try
Three problems below. Work each before checking the solution.
What is 12% of 150?
Show solution
0.12 × 150 = 18.
A $25 meal gets an 18% tip. What is the total?
Show solution
25 × 1.18 = 29.50.
A population grows 25% to reach 5,000. What was the original population?
Show solution
5,000 is 125% of the original: original = 5000 / 1.25 = 4,000.
The Northside Method — How We Teach This 1-on-1
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Every Northside student works through a four-step framework:
- Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
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