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How to use this calculator#
- Decide which of the three questions you are asking‘What is 15% of 240?’ finds a part. ‘30 is what percent of 150?’ finds a rate. ‘From 40 to 55’ finds a change. Picking the wrong mode is the reason most hand calculations come out wrong.
- Identify the whole and put it in the right boxThe whole is the number the percentage is measured against — the original price, the total marks, the starting value. In ‘what percent of’ it is the denominator; in percentage change it is always the older figure.
- Convert the percent to a decimal as a mental checkPercent means per hundred, so 15% is 0.15 and 7.5% is 0.075. If your answer is roughly 240 × 0.15 ≈ 36, you are in the right ballpark; if it is ten times off, you have slipped a decimal place.
- Read the multiplier, not just the answerAdding 15% is the same as multiplying by 1.15, and taking 15% off is multiplying by 0.85. Working in multipliers lets you chain several percentages together in one step instead of recalculating each time.
- Round only at the very endRounding 33.333% to 33% and then applying it to a large total introduces real money in error. Keep the full decimal through the calculation and round the final figure to the precision you actually need.
The formula#
The three percentage relationships
Part = Whole × (P ÷ 100) • P = (Part ÷ Whole) × 100 • Change % = ((New − Old) ÷ Old) × 100
- Part
- The portion you are measuring — the tip, the discount, the marks scored
- Whole
- The base the percentage is taken from — the bill, the list price, the total marks
- P
- The percentage itself, written as a number out of 100 rather than a decimal
- Old
- The starting value in a percentage change, and always the denominator
- New
- The finishing value in a percentage change
Percent means ‘per hundred’, so the word ‘of’ in ‘15% of 240’ is literally a multiplication: 240 × 15/100. All three formulas above are the same equation rearranged for a different unknown, which is why one calculator handles all of them.
The three percentage questions people ask#
‘What is X% of Y?’ multiplies Y by X/100. ‘X is what percent of Y?’ divides X by Y and multiplies by 100. ‘Percentage change’ measures the difference between an old and new value relative to the old value.
Worked examples#
Adding a tip to a bill
A $74.50 restaurant bill with an 18% tip.
- Convert the percent to a decimal: 18 ÷ 100 = 0.18
- Tip = 74.50 × 0.18 = 13.41
- Total = 74.50 + 13.41 = 87.91
- Or do it in one step with the multiplier: 74.50 × 1.18 = 87.91
$13.41 tip, $87.91 total. The multiplier route is the one to use when you also want to add sales tax.
Working backwards to the original price
A jacket costs $63 in a 30% off sale. What did it cost before the discount?
- The sale price is 100% − 30% = 70% of the original, so the multiplier was 0.70
- 63 = Original × 0.70
- Original = 63 ÷ 0.70 = 90
- Check: 90 × 0.30 = 27 discount, and 90 − 27 = 63 ✓
$90 before the discount. Adding 30% back to $63 gives $81.90, which is wrong — the 30% was taken from $90, not from $63.
Reference tables#
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.3333… | 33.33% |
| 2/3 | 0.6667… | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/6 | 0.1667… | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 1/16 | 0.0625 | 6.25% |
| 1/20 | 0.05 | 5% |
| 1/100 | 0.01 | 1% |
To go from a fraction to a percentage, divide the top by the bottom and multiply by 100: 5 ÷ 8 = 0.625 = 62.5%.
| Percentage | To add it, multiply by | To take it off, multiply by |
|---|---|---|
| 5% | 1.05 | 0.95 |
| 8% | 1.08 | 0.92 |
| 10% | 1.10 | 0.90 |
| 12.5% | 1.125 | 0.875 |
| 15% | 1.15 | 0.85 |
| 20% | 1.20 | 0.80 |
| 25% | 1.25 | 0.75 |
| 33.33% | 1.3333 | 0.6667 |
| 50% | 1.50 | 0.50 |
| 75% | 1.75 | 0.25 |
| 100% | 2.00 | 0.00 |
Two multipliers in a row combine by multiplication, not addition: a 20% discount then 8% sales tax is 0.80 × 1.08 = 0.864, so you pay 86.4% of the list price.
| What you have | Do this | Worked example |
|---|---|---|
| Price after 20% VAT was added | ÷ 1.20 | $144 ÷ 1.20 = $120 |
| Price after a 30% discount | ÷ 0.70 | $63 ÷ 0.70 = $90 |
| Value after a 15% rise | ÷ 1.15 | 230 ÷ 1.15 = 200 |
| Value after a 40% fall | ÷ 0.60 | 90 ÷ 0.60 = 150 |
| A part that is 20% of the whole | ÷ 0.20 | 30 ÷ 0.20 = 150 |
This is the calculation retailers and tax authorities call a reverse percentage. Subtracting the percentage from the result instead of dividing is the classic error.
Common mistakes#
- Dividing by the wrong baseIn ‘X is what percent of Y’, Y goes on the bottom. Swapping them turns 30 out of 150 (20%) into 150 out of 30 (500%). In percentage change the base is always the older value — dividing by the new one understates every increase and overstates every decrease.
- Confusing a percent with a percentage pointIf a savings rate moves from 4% to 5%, that is a rise of one percentage point but a 25% increase in the rate itself. Newspapers and lenders use both, and the two numbers can differ by an order of magnitude on small starting figures.
- Adding a percentage back to reverse itTaking 30% off $90 gives $63, but adding 30% to $63 gives $81.90, not $90. The percentage was applied to a different base. To undo it you divide by 0.70, which is equivalent to adding 42.86%.
- Chaining percentages by adding themA 10% rise followed by another 10% rise is 21%, not 20%, because the second rise applies to the already-increased figure. Similarly a 20% discount plus a 20% coupon is 36% off, not 40% — always multiply the multipliers.
Frequently asked questions#
How do I calculate a percentage increase?
Subtract the old value from the new value, divide by the old value, then multiply by 100. A negative result is a decrease.
How do I find what percent one number is of another?
Divide the part by the whole and multiply by 100. For example, 30 out of 150 is (30 ÷ 150) × 100 = 20%.
Key terms#
- Percent
- Literally ‘per hundred’. 15% is the fraction 15/100 and the decimal 0.15; the symbol is just shorthand for dividing by 100.
- Base (or whole)
- The number a percentage is measured against. Almost every percentage error is a wrong base rather than wrong arithmetic.
- Percentage point
- The plain arithmetic difference between two percentages. Moving from 4% to 5% is +1 percentage point and +25% at the same time.
- Reverse percentage
- Recovering the original figure from a post-change value by dividing by the multiplier, e.g. $144 ÷ 1.20 = $120 to strip 20% VAT.
- Multiplier
- The single number that applies a percentage in one operation: 1 + P/100 to add it, 1 − P/100 to remove it. Chains cleanly by multiplication.
- Basis point
- One hundredth of a percentage point, used in finance to avoid ambiguity. A 25 basis point rate cut is 0.25 percentage points.
Sources#
- Prealgebra 2e — percents — OpenStax, Rice University
- Arithmetic: percentages — Khan Academy
- Guide for the Use of the International System of Units (SP 811) — use of the percent symbol — NIST
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