Calculator 1 of 3
What is X% of Y?
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Calculator 2 of 3
X is what % of Y?
is what % of
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Calculator 3 of 3
Percentage Change
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Percentage Formulas
To find a percentage of a number, multiply the number by the percent and divide by 100: 15% of 200 = 200 × 15 ÷ 100 = 30. To find what percent one number is of another, divide the part by the whole and multiply by 100, so 30 out of 200 is 15%. The three calculators above do both, plus percentage change, and show the formula with every answer.
Percentages express a number as a fraction of 100. The word comes from the Latin per centum, "by the hundred." All three percentage questions reduce to the same underlying relationship: Part / Whole = Percent / 100.
Calculator 1
Result = Y × X / 100
15% of 200 = 200 × 15 / 100 = 30
Calculator 2
% = (X / Y) × 100
30 of 200 = (30/200) × 100 = 15%
Calculator 3
% Change = ((Y-X) / |X|) × 100
100 to 120 = ((120-100)/100) × 100 = +20%
Common Percentages Quick Reference
Each percent written as a decimal and a fraction, and applied to four common numbers.
| Percent | Decimal | Fraction | Of 50 | Of 80 | Of 200 | Of 1,000 |
|---|---|---|---|---|---|---|
| 1% | 0.01 | 1/100 | 0.5 | 0.8 | 2 | 10 |
| 5% | 0.05 | 1/20 | 2.5 | 4 | 10 | 50 |
| 10% | 0.1 | 1/10 | 5 | 8 | 20 | 100 |
| 12.5% | 0.125 | 1/8 | 6.25 | 10 | 25 | 125 |
| 15% | 0.15 | 3/20 | 7.5 | 12 | 30 | 150 |
| 20% | 0.2 | 1/5 | 10 | 16 | 40 | 200 |
| 25% | 0.25 | 1/4 | 12.5 | 20 | 50 | 250 |
| 30% | 0.3 | 3/10 | 15 | 24 | 60 | 300 |
| 33.33% | 0.3333 | 1/3 | 16.67 | 26.67 | 66.67 | 333.33 |
| 40% | 0.4 | 2/5 | 20 | 32 | 80 | 400 |
| 50% | 0.5 | 1/2 | 25 | 40 | 100 | 500 |
| 60% | 0.6 | 3/5 | 30 | 48 | 120 | 600 |
| 75% | 0.75 | 3/4 | 37.5 | 60 | 150 | 750 |
| 80% | 0.8 | 4/5 | 40 | 64 | 160 | 800 |
| 90% | 0.9 | 9/10 | 45 | 72 | 180 | 900 |
| 100% | 1 | 1/1 | 50 | 80 | 200 | 1,000 |
| 150% | 1.5 | 3/2 | 75 | 120 | 300 | 1,500 |
The 33.33% row is rounded to two decimals because one third has no exact decimal form.
Three Worked Examples
A discount, then sales tax
A $64 jacket is 25% off and your sales tax is 7%. The discount is 25% of $64 = $16, so you pay tax on $48. Tax is 7% of $48 = $3.36, and the total is $51.36. The quick route is $64 × 0.75 × 1.07, which gives the same answer. Tax always applies to the price after the discount.
A test score
You answered 43 of 52 questions correctly. Use Calculator 2: 43 ÷ 52 × 100 = 82.69%. Round only at the end; rounding 43 ÷ 52 to 0.83 first would give 83%.
How big was the raise?
Your salary goes from $52,000 to $55,120. Use Calculator 3: (55,120 − 52,000) ÷ 52,000 × 100 = 6%. The base is the old salary, never the new one.
Mental Math Shortcuts
- 10%: move the decimal point one place left. 10% of 470 is 47.
- 1%: move it two places. 1% of 470 is 4.7, so 3% is 14.1.
- 5%: half of 10%. 5% of 470 is 23.5.
- 15%: 10% plus 5%. 15% of 470 is 47 + 23.5 = 70.5.
- 25%: divide by 4. 50%: divide by 2. 12.5%: divide by 8.
- Swap trick: x% of y equals y% of x. 8% of 25 is hard, 25% of 8 is easy, and both are 2.
Edge Cases and Common Mistakes
- Percent of zero is always 0, but nothing can be a percent of 0 and there is no percentage change from 0. Both need division by zero, so the calculator reports them as undefined. For a change from 0, quote the absolute change instead.
- Negative numbers work. 20% of −50 is −10, and −10 is 20% of −50. For percentage change from a negative start, the calculator divides by the absolute value of the start, the same as =(B1−A1)/ABS(A1) in a spreadsheet.
- Results above 100% are normal. 150% of 40 is 60, and 60 is 150% of 40. A decrease of more than 100% is only possible when the value can go below zero, such as a profit turning into a loss.
- Undoing a discount needs a bigger percent. $80 less 20% is $64. Adding 20% back gives $76.80, not $80. You need to add 25%, because the base is now smaller.
- Percentages do not add. Two 10% increases make 21%, not 20%, because the second one applies to a larger number. Multiply 1.10 × 1.10 instead.
For changes over time you can also use the percentage change calculator, which adds percentage difference and a new value mode. For receipts, the sales tax calculator and tip calculator do the whole bill in one step.
Method and sources. Calculator 1 uses part = whole × percent ÷ 100. Calculator 2 uses percent = part ÷ whole × 100. Calculator 3 uses percentage change = (new − old) ÷ |old| × 100. Every value in the table and examples above was computed with these formulas. Percent means per hundred (1% = 0.01), the standard definition used in the NIST Guide for the Use of the International System of Units (SP 811).
Percentage Questions & Formulas
Multiply Y by X, then divide by 100. Formula: Result = Y × X / 100. Example: What is 15% of 200? Calculation: 200 × 15 / 100 = 30. A mental shortcut: first find 10% (move the decimal left one place), then add half of that for 15%. 10% of 200 = 20. Half of 20 = 10. Total = 30.
Divide X by Y and multiply by 100. Formula: Percentage = (X / Y) × 100. Example: 30 is what percent of 200? Calculation: (30 / 200) × 100 = 15%. This is the reverse of Calculator 1. You can check your answer using Calculator 1: 15% of 200 = 30.
Percentage change = ((New Value − Old Value) / |Old Value|) × 100. A positive result is an increase; negative is a decrease. Examples: Price rose from $100 to $120: ((120 − 100) / 100) × 100 = +20% increase. Price fell from $200 to $150: ((150 − 200) / 200) × 100 = −25% decrease. Note: use the original (old) value as the denominator, not the new value.
Use Calculator 1 above. For a 20% tip on a $75 bill: What is 20% of 75? = 75 × 20 / 100 = $15. Mental shortcut for 20%: divide by 10 ($7.50), then double ($15). For 15% tip: find 10% ($7.50), add half ($3.75), total = $11.25. For an 18% tip: 10% is $7.50 and 8% is $6 (10% minus a fifth of it), so the tip is $13.50.
Two approaches: (1) Find the discount amount and subtract. Discount amount = Original price × Discount % / 100. Sale price = Original − Discount. Example: 30% off $80 = $80 × 30 / 100 = $24 discount. Sale price = $80 − $24 = $56. (2) Multiply by the complement. Sale price = Original × (1 − Discount%/100) = $80 × 0.70 = $56. Both methods give the same answer. Calculator 1 gives you the discount amount; subtract manually for the sale price.
Formula: Final price = Original × (1 + Percentage / 100). Example: $45 item with 8% sales tax: $45 × 1.08 = $48.60. For a 7.5% tax on $120: $120 × 1.075 = $129. You can also use Calculator 1 to find the tax amount (8% of $45 = $3.60) and add it manually ($45 + $3.60 = $48.60). The multiplication method is faster for round tax rates.
A percentage point is an absolute difference between two percentages. A percent change is a relative difference. Example: interest rates rise from 2% to 3%. The increase is 1 percentage point (3 − 2 = 1). But the percent change is 50% ((3 − 2) / 2 × 100 = 50%). Headlines mix the two up often, so check which one a figure uses.
If you know the result of a percentage calculation but want the original, divide by the decimal form. Example: After a 20% increase, a salary is $60,000. What was the original? $60,000 / 1.20 = $50,000. Example: A shirt costs $35 after a 30% discount. Original price? $35 / 0.70 = $50. The key: 20% increase means the result is 120% of original (multiply by 1.20). 30% decrease means the result is 70% of original (multiply by 0.70). To reverse, divide by that decimal.
You cannot simply add percentage changes. A 10% increase followed by a 10% decrease does NOT return to the original. Correct method: multiply the multipliers. 10% increase = ×1.10. 10% decrease = ×0.90. Combined: 1.10 × 0.90 = 0.99 = a 1% decrease. Example: Investment gains 50% then loses 50%: $1,000 × 1.50 = $1,500, then $1,500 × 0.50 = $750. You end up with $750, not $1,000. For multiple changes, multiply all the decimal multipliers together, then use Calculator 3 to find the overall percentage change.
Grade percentage = (Points earned / Total possible points) × 100. Use Calculator 2: "X is what percent of Y?" Enter your points as X and the total as Y. Example: scored 42 out of 50 questions: (42 / 50) × 100 = 84%. For weighted grades: multiply each component score by its weight, sum all weighted scores. Example: midterm 75% (weight 40%) + final 88% (weight 60%) = (75 × 0.40) + (88 × 0.60) = 30 + 52.8 = 82.8% overall grade.
Multiply a decimal by 100 and add the percent sign: 0.375 = 37.5%. For a fraction, divide the top by the bottom first, then multiply by 100: 3/8 = 0.375 = 37.5%. To go back, divide by 100: 62.5% = 0.625. Calculator 2 does the fraction step for you: enter 3 as X and 8 as Y.
Yes. A number larger than the whole is more than 100% of it: 60 is 150% of 40. An increase can also exceed 100%: going from 5 to 50 is a 900% increase. Only a decrease is limited to 100% when the value cannot drop below zero, as with a price or a population.
Percentage change divides by the starting value, and division by zero has no answer. Going from 0 to 25 is not a 100% or an infinite percent increase; it is simply an increase of 25. In reports, write the absolute change or say the item is new. The same applies to "X is what percent of 0", which the calculator also marks as undefined.