📖 Guide

Percentages, Mastered: Every Common Problem Explained

What is X% of Y, X is what percent of Y, percentage change, and reverse percentage. Five formulas that cover almost every real-world percentage question.

The Five Percentage Problems That Cover Almost Everything

Most questions about percentages reduce to one of five patterns. Recognize the pattern, pick the formula, get the answer. The rest of this guide walks through each one with worked examples and the everyday situations where you will meet them.

What is P% of X?
X is what percent of Y?
What is the percent change from A to B?
Increase or decrease X by P%
X is the post-discount price at P% off. What was the original?

1. What is P% of X?

Formula: X × P / 100

The most common percentage question. A 7% sales tax on a $60 dinner: 60 × 7 / 100 = $4.20. An 18% tip on a $42 bill: 42 × 18 / 100 = $7.56. A 4.5% mortgage rate applied to a $20,000 monthly balance for interest: 20,000 × 4.5 / 100 / 12 = $75 of interest this month.

Mental shortcut: 10% of any number is that number with the decimal moved one place left. 10% of 847 = 84.7. Once you have 10%, you can build other percentages: 5% is half of that, 20% is double, 15% is 10% plus half of 10%.

2. X is What Percent of Y?

Formula: (X / Y) × 100

You scored 27 out of 240 on a test. The percentage: 27 / 240 × 100 = 11.25%. Your $500 grocery bill out of a $4,000 monthly budget: 500 / 4000 × 100 = 12.5%. Your $80,000 down payment on a $400,000 home: 80,000 / 400,000 × 100 = 20% down.

This is the formula every fraction-to-percentage conversion uses. Three-eighths as a percent: 3 / 8 × 100 = 37.5%. Two-thirds: 2 / 3 × 100 = 66.67%.

3. Percent Change from A to B

Formula: ((B − A) / |A|) × 100

A stock went from $80 to $92. The percent change: (92 − 80) / 80 × 100 = 15%. Your salary went from $52,000 to $58,500. The raise: (58,500 − 52,000) / 52,000 × 100 = 12.5%. A negative number means the value dropped.

The trap most people fall into: percent change is not symmetric. A stock that loses 50% does not need a 50% gain to break even. It needs a 100% gain. Drop from $100 to $50. Now $50 needs to double to reach $100 again.

Worked Example: The Compounding-Percentages Trap

Why a 20% raise plus a 20% cut leaves you below where you started

Your salary goes from $50,000 to $60,000. That's a 20% raise. The next year, the company gives a 20% pay cut. New salary: 60,000 × 0.80 = $48,000. Not $50,000.

The reason: the second 20% was applied to a different (larger) base than the first 20%. To undo a 20% increase, you need a 16.67% decrease. To undo a 50% increase, you need a 33.3% decrease. The bigger the original move, the wider the gap.

Same effect shows up in investment losses ("I lost 50% but a 50% gain will get me back to even", does not work), restaurant tipping math, discount stacking, and political claims about budget changes. Work with the actual numbers, not the percentages.

4. Increase or Decrease X by P%

Formula: X × (1 ± P/100)

A $200 item with 7% sales tax: 200 × 1.07 = $214. A $80 shirt at 30% off: 80 × 0.70 = $56. A $1,000 investment up 8.5% in a year: 1,000 × 1.085 = $1,085.

This formula is the engine behind compound growth. Run it n times, and you get $1,000 × 1.085ⁿ. After 20 years at 8.5%, that's $1,000 × 5.11 = $5,112. Twenty multiplications by 1.085, nothing more exotic than that.

5. Reverse Percentage: What Was the Original Price?

Formula: X / (1 − P/100) for discounts, X / (1 + P/100) for markups.

You paid $80 for a shirt at 20% off. Original price: 80 / (1 − 0.20) = 80 / 0.80 = $100. You paid $107 for a $100 product including 7% sales tax: 107 / 1.07 = $100.

Common mistake: assuming "20% off then add back 20%" returns the original. It does not. After 20% off, you are at 80% of the original. Adding 20% to 80% gives 80 × 1.20 = 96, not 100.

Percent vs Percentage Points

The most confused distinction in news and economics writing.

Unemployment rises from 5% to 7%. That is a 2 percentage point increase. The same change is a 40 percent increase, because 2 / 5 = 40%.

A headline that says "unemployment rose 40%" sounds catastrophic. A headline that says "unemployment rose 2 percentage points from 5% to 7%" sounds bad but contained. Both describe the same change. The framing decides which one feels worse.

The same trap appears in tax debates ("the top rate dropped 5 percentage points" vs "the top rate dropped 14%"), interest rate news, and election polling. Watch for which one the writer chose. The choice is rarely neutral.

Mental Shortcuts Worth Remembering

10% of X: move the decimal one place left.
5%: 10% divided by 2.
15%: 10% plus half of 10%. For $80: 8 + 4 = $12.
20%: 10% times 2. Restaurant tipping math in 3 seconds.
1%: move the decimal two places left.
X% of Y equals Y% of X. 8% of 50 is the same as 50% of 8. Both are 4. Use whichever is easier to do in your head.
Doubling equals +100%. Halving equals −50%.

Real-World Use Cases by Pattern

Scenario	Pattern	Formula
Tip on a restaurant bill	P% of X	Bill × tip / 100
Sales tax	Increase X by P%	Price × (1 + tax/100)
Quiz score	X is what % of Y	(Correct / Total) × 100
Stock return	% change A → B	((End − Start) / Start) × 100
Discount to original price	Reverse decrease	Sale / (1 − discount/100)
Markup to wholesale price	Reverse increase	Retail / (1 + markup/100)
Inflation adjustment over years	Compound increase	Old × (1 + inflation/100)^years

Common Mistakes to Avoid

Adding percentages from different bases. A 10% increase followed by a 10% increase is +21%, not +20%.
Confusing decrease percentages with the after number. "30% off" does not mean the price is 30%. You save 30% and pay 70%.
Mixing percentages and percentage points in headlines and arguments. The two say different things.
Forgetting that percentage changes are not symmetric. Lose 50% then gain 50% leaves you at 75% of where you started.
Quoting percentages of tiny bases. "Sales grew 200%" sounds great until you learn the move was from 1 unit to 3 units.

Frequently Asked Questions

What is 0% of something?

Zero. 0% of $1,000,000 is zero dollars. 0% of one molecule is zero molecules. The base doesn't matter.

Can a percentage be greater than 100%?

Yes, when describing an increase or a quantity that exceeds the reference. A stock that returned 150% over five years means the new value is 2.5 times the original. A 200% efficiency gain means the new process produces three times as much output as the old one.

Can a percentage be negative?

Yes, for losses or decreases. A −20% change means the value dropped 20% from the starting point.

What's the difference between "percent" and "percentage"?

"Percent" goes with a number (5 percent). "Percentage" is a noun for the concept (a small percentage of voters). The math is the same.

How do I turn a decimal into a percentage?

Multiply by 100. 0.75 = 75%. 0.043 = 4.3%. 1.20 = 120%.

How do I turn a fraction into a percentage?

Divide numerator by denominator, then multiply by 100. 3/8 = 0.375 = 37.5%. See the fraction calculator for help with the division.

Why is "20% off then 20% off again" not 40% off?

The second 20% applies to the already-reduced price. 100 × 0.80 = 80, then 80 × 0.80 = 64. Total reduction: 36%, not 40%. Two stacked discounts at 20% give the same result as one 36% discount.