Percentage Formulas: Increase, Decrease, Discounts and Percentage Points
Almost every percentage problem is one of three questions: what is X% of Y, what percent is X of Y, and how much did something change. Each has one formula. The mistakes come from stacked discounts, removing tax, and confusing percent with percentage points.
The three questions every percentage problem asks
"Percent" means "per hundred", so 15% is 15 ÷ 100 = 0.15. Almost every percentage problem is one of three questions, and each has one formula. These are the three modes in the Percentage Calculator.
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | X ÷ 100 × Y | 15% of 240 = 0.15 × 240 = 36 |
| X is what percent of Y? | X ÷ Y × 100 | 30 of 120 = 30 ÷ 120 × 100 = 25% |
| Percentage change from old to new | (new − old) ÷ old × 100 | 80 → 100 = 20 ÷ 80 × 100 = +25% |
A fourth question is the first one backwards: "45 is 15% of what?" Divide the part by the percentage as a decimal: 45 ÷ 0.15 = 300.
Increase and decrease aren't mirror images
The base of a percentage change is always the starting value. That's why going up and coming back down use different percentages:
- 80 → 100 is an increase of 20 ÷ 80 = 25%.
- 100 → 80 is a decrease of 20 ÷ 100 = 20%.
The same trap catches investors and shoppers: a price that rises 50% and then falls 50% does not return to where it started. 100 × 1.5 = 150, then 150 × 0.5 = 75, a net loss of 25%.
Stacked discounts don't add up
"20% off, plus an extra 10% at checkout" is not 30% off. The second discount applies to the already-reduced price:
- 100 × 0.80 = 80 after the first discount.
- 80 × 0.90 = 72 after the second.
You pay 72, a total discount of 28%. The shortcut: multiply the "remaining" fractions (0.80 × 0.90 = 0.72) and subtract from 1.
Removing tax or VAT from a price
If a price of 115 includes 15% VAT, the pre-tax price is 115 ÷ 1.15 = 100, and the tax is 15. A common mistake is to take 15% off the total instead: 115 × 0.85 = 97.75, which is wrong by 2.25. The tax was 15% of the pre-tax price, not of the total, so you divide rather than subtract.
Adding tax is the easy direction: 100 × 1.15 = 115.
Percent vs percentage points
When a rate moves from 4% to 5%, it has risen by 1 percentage point but by 25 percent (1 ÷ 4 × 100). News reports, loan offers and exam statistics often mix the two up. Use "percentage points" for the difference between two percentages, and "percent" for relative change.
Quick mental shortcuts
- 10%: move the decimal point one place left. 10% of 86 = 8.6.
- 5%: half of 10%. 5% of 86 = 4.3.
- 15%: 10% plus 5%. 15% of 86 = 12.9.
- 1%: move the decimal point two places. 1% of 86 = 0.86.
- Swap trick: X% of Y equals Y% of X. 8% of 50 is the same as 50% of 8, which is 4.
Everyday examples
| Situation | Calculation | Answer |
|---|---|---|
| Exam: 42 of 60 correct | 42 ÷ 60 × 100 | 70% |
| Salary 3,000 → 3,240 | 240 ÷ 3,000 × 100 | +8% |
| Phone 899 with 12% off | 899 × 0.88 | 791.12 |
| Savings goal: 1,350 of 5,000 | 1,350 ÷ 5,000 × 100 | 27% |
| Bill 64 plus 18% tip | 64 × 1.18 | 75.52 |
For tips and bill splitting, see How Much to Tip and How to Split a Bill Fairly.
Common mistakes
- Using the wrong base. "Sales fell from 500 to 400": the base is 500, so it's a 20% fall, not 25%.
- Averaging percentages. 50% of 10 people and 100% of 2 people is not a 75% average; it's 7 of 12, or 58%.
- Rounding too early. Keep full precision until the last step, especially with money.
Frequently asked questions
How do I calculate a percentage increase?
Subtract the old value from the new one, divide by the old value, and multiply by 100. From 80 to 100: 20 ÷ 80 × 100 = 25%.
Can a percentage be more than 100%?
Yes. A value that goes from 50 to 150 has increased by 200%. A share of a whole, like marks on a test, can't exceed 100%.
How do I find the original price before a discount?
Divide the sale price by the remaining fraction. If 60 is the price after 25% off, the original was 60 ÷ 0.75 = 80.
Why is my percentage change negative?
A negative result means the value went down. −20% is a 20% decrease.