What "Percent" Actually Means
The word comes from the Latin per centum, meaning "for every hundred." So 45% simply describes 45 units out of every 100 - that's the entire concept. Hold onto that picture and percentages stop feeling like an abstract math trick.
Every percentage question, however it's phrased, reduces to a single relationship: Percentage = (Part ÷ Whole) × 100. The four categories below are just this equation rearranged to solve for a different missing piece.
1. Finding a Percentage of a Number
You'll run into this type constantly - "What is 20% of 85?" or "How much do I save on a £60 jacket with 15% off?"
What is 20% of 85?
= (20 ÷ 100) × 85
= 0.20 × 85
= 17
A fast trick: move the decimal point two places left to turn any percent into its decimal equivalent, then multiply. 20% becomes 0.20, 8.5% becomes 0.085, and this works no matter what percentage you're starting with.
More Worked Examples
| Question | Calculation | Answer |
|---|---|---|
| 15% of 200 | 0.15 × 200 | 30 |
| 7.5% of 400 | 0.075 × 400 | 30 |
| 12% of 55 | 0.12 × 55 | 6.6 |
| 25% of 180 | 0.25 × 180 | 45 |
| 8% tax on £75 | 0.08 × 75 | £6 |
2. Expressing One Number as a Percentage of Another
Here you have two numbers already and need to find the percentage linking them - "I scored 68 out of 80, what percent is that?" or "The price rose from £50 to £55, what percentage increase is that?"
68 out of 80 as a percentage:
= (68 ÷ 80) × 100
= 0.85 × 100
= 85%
This step is the one people most often get backwards when doing mental math: divide the part by the whole first, and multiply by 100 only afterward. Do it in the other order and the answer falls apart.
3. Percentage Increase and Decrease
There are really two separate questions bundled under this heading: how big was a change in percentage terms, and if you know the percentage change, what does the resulting number look like?
Calculating the Percentage Change
Price goes from £80 to £92:
= ((92 - 80) ÷ 80) × 100
= (12 ÷ 80) × 100
= 0.15 × 100
= 15% increase
A negative result simply signals a decrease rather than an increase - the formula stays exactly the same, only the sign flips.
Applying a Known Increase or Decrease
Decrease by 20%: New = Original × 0.80
£150 increased by 15%:
= 150 × 1.15 = £172.50
£90 reduced by 30% (sale price):
= 90 × 0.70 = £63
This "multiplier" approach - 1.15 for a 15% rise, 0.70 for a 30% cut - folds two separate steps into one. It takes a bit of practice to feel natural, but once it does, it's noticeably quicker than the long way.
4. Reverse Percentage - Recovering the Original Value
This is the type that catches almost everyone off guard at least once. A jacket costs £48 after a 20% discount, and you want to know its price before the sale. Adding 20% back onto £48 feels like the obvious move - it isn't, though.
After 20% off, price is £48. Original price?
= 48 ÷ 0.80
= £60
Check: 60 × 0.80 = 48 ✓
Why doesn't adding 20% of £48 get you there? Because 20% of £48 is just £9.60, landing you at £57.60 instead of £60. The 20% discount was calculated off the original price, not the discounted one, so addition can't reverse it. Dividing by 0.80 is what correctly undoes the discount.
Mental Math Shortcuts Worth Learning
A few of these will spare you from reaching for a calculator most of the time:
| Percentage | Shortcut | Example (of 80) |
|---|---|---|
| 50% | Divide by 2 | 80 ÷ 2 = 40 |
| 25% | Divide by 4 | 80 ÷ 4 = 20 |
| 10% | Divide by 10 | 80 ÷ 10 = 8 |
| 5% | Find 10%, halve it | 8 ÷ 2 = 4 |
| 1% | Divide by 100 | 80 ÷ 100 = 0.8 |
| 15% | 10% + 5% | 8 + 4 = 12 |
| 75% | 50% + 25% | 40 + 20 = 60 |
Percentage vs. Percentage Points
Even experienced writers mix these two up. If an interest rate climbs from 2% to 5%, that's a jump of 3 percentage points - but expressed as a percentage change, it's actually a 150% increase, since 5 is 150% larger than 2. Neither statement is wrong; they're just answering two different questions. Keep an eye out for this in news coverage and financial reports, where the two get conflated more often than they should be.
When a Calculator Makes More Sense
Mental shortcuts work well until the numbers stop cooperating, at which point a calculator saves time and cuts down on careless errors. SolverCalc's percentage calculator handles all four cases covered here - finding a percent of a number, expressing one number as a percent of another, working out a percentage change, and reversing a percentage back to its starting value. Give it what you know, and it fills in the rest.
Where That Leaves You
Almost every percentage problem comes back to that one formula, Part ÷ Whole × 100, just rearranged depending on which piece you're solving for. Three habits will get you through nearly all of them: convert the percentage to a decimal before multiplying, use a multiplier like 1.15 or 0.70 for increases and decreases instead of doing it in two steps, and divide - never add or subtract - when reversing a percentage back to its original value. Get those three down and percentage problems stop being something to dread.