Knowing how to calculate percentages is one of the most useful everyday maths skills — you need it for discounts, test scores, tips, taxes, and statistics. A percentage expresses a number as a fraction of 100, which makes comparing values easy. This guide covers the percentage formula, the three main percentage problems, and worked examples for each. For instant results, use the free percentage calculator.
What Is a Percentage?
A percentage (the term comes from the Latin per centum, “by the hundred”) shows how many parts you have out of one hundred. The symbol % means “divided by 100”, so 25% equals 25/100 or 0.25. Because everything is scaled to a hundred, percentages let you compare proportions of different-sized groups — for example, saying 40% of students passed, regardless of class size.
The Percentage Formula
The core formula for how to calculate percentages is:
Percentage = (Part ÷ Whole) × 100
- The part is the portion you are interested in.
- The whole is the total or original amount.
- Multiplying by 100 converts the decimal into a percentage.
How to Calculate Percentages: Three Common Problems
1. Finding what percentage one number is of another
Example: You scored 42 out of 50 on a test. What is your percentage score?
- Percentage = (42 ÷ 50) × 100
- = 0.84 × 100
- = 84%
2. Finding a percentage OF a number
To find, say, 15% of 80, convert the percentage to a decimal and multiply:
- 15% of 80 = 0.15 × 80 = 12
This is the classic shopping calculation: a $60 jacket with a 25% discount saves 0.25 × 60 = $15.
3. Finding the whole when you know a percentage
If 30% of a number is 12, what is the number?
- Number = 12 ÷ 0.30 = 40
Percentage Change: Increase and Decrease
To compare an old value to a new value, use the percentage change formula:
Percentage Change = ((New − Old) ÷ Old) × 100
A positive result is a percentage increase; a negative result is a percentage decrease.
Example: A gym membership rose from $50 to $65.
- Change = (65 − 50) ÷ 50 × 100 = 0.3 × 100 = 30% increase
See our detailed guides on percentage increase for more worked examples.
Percentage Points vs Percent Change
Do not confuse percentage points with percent change. If an interest rate rises from 4% to 5%, that is a 1 percentage point increase, but a 25% increase in the rate itself ((5−4)÷4 × 100). News headlines mix these up constantly, so knowing the difference keeps your maths — and your finances — accurate.
Quick Mental Maths Tricks for Percentages
- 10% trick: move the decimal point one place left (10% of 250 = 25). Build other percentages from it: 5% is half of 10%, 30% is 10% × 3, 15% is 10% + 5%.
- 1% trick: move the decimal two places (1% of 250 = 2.5).
- Swapping works: 8% of 50 = 50% of 8 = 4. Use whichever is easier.
- 25% is dividing by 4; 50% is dividing by 2; 75% is 3 × 25%.
Percentage Examples in Real Life
| Situation | Calculation | Result |
|---|---|---|
| Tip on a $48 meal (18%) | 0.18 × 48 | $8.64 |
| Test score 36/40 | (36 ÷ 40) × 100 | 90% |
| Sale price $80 after 20% off | 80 × (1 − 0.20) | $64 |
| Tax on $120 at 8.5% | 0.085 × 120 | $10.20 |
Common Percentage Mistakes
- Dividing the wrong way (always divide the part by the whole).
- Forgetting to multiply by 100 at the end.
- Adding percentages of different wholes (a 20% discount then 10% tax is not 30%).
- Confusing percentage points with percent change.
How to Convert Between Percentages, Decimals, and Fractions
Percentages, decimals, and fractions are three ways to write the same value. Converting between them is easy:
- Percentage to decimal: divide by 100 and drop the % sign (45% = 0.45).
- Decimal to percentage: multiply by 100 (0.62 = 62%).
- Percentage to fraction: write the number over 100 and simplify (35% = 35/100 = 7/20).
- Fraction to percentage: divide the numerator by the denominator, then multiply by 100 (3/8 = 0.375 = 37.5%).
Knowing all three forms helps you pick the easiest method for a given question. Fractions are exact; decimals are easy to multiply; percentages are easy to compare.
More Worked Percentage Problems
Finding a percentage (test score)
A student answers 31 of 40 questions correctly. What percentage is that?
- 31 ÷ 40 = 0.775
- 0.775 × 100 = 77.5%
Finding a percentage of a number (budget)
You budget 22% of a $2,600 monthly income for groceries.
- 0.22 × 2,600 = $572
Finding the whole (sale stock)
A shop sold 168 chairs. That is 35% of its stock. How many chairs did it start with?
- 168 ÷ 0.35 = 480 chairs
Percentage greater than 100
Percentages can exceed 100 when the part is bigger than the whole. If a factory plans 400 units and builds 500, output is 125% of plan. The formula does not change: (500 ÷ 400) × 100 = 125%.
Method Summary
Every percentage question comes down to one of three forms:
| You know | You want | Method |
|---|---|---|
| Part and whole | The percentage | (part ÷ whole) × 100 |
| Percentage and whole | The part | percentage ÷ 100 × whole |
| Part and percentage | The whole | part ÷ (percentage ÷ 100) |
Write down which two numbers the question gives you, match them to the table, and apply the method. With practice you will spot the form in seconds.
How to Work With Percentages Greater Than 100%
Percentages are not limited to values between 0 and 100. When the part exceeds the whole, the percentage climbs past 100% — a situation that appears whenever actual production overshoots a target, or whenever this year’s sales dwarf last year’s baseline. If a factory planned 400 units and delivered 500, then production reached (500 ÷ 400) × 100 = 125% of plan. The formula never changes; only the interpretation does.
Finding the Whole From a Percentage: Reverse Percentage Problems
Reverse percentage questions give you the part and the percentage, then ask for the original whole — the “finding the whole” problem that appears constantly in discount and tax scenarios. Because a sale price already includes the discount, you cannot simply add the percentage back on; you must divide instead. Consider an item that sells for $76 after a 20% reduction: the sale price represents 80% of the original, so the original price equals 76 ÷ 0.80 = $95. Writing the relationship down in these terms (part = percentage × whole) before touching a calculator prevents nearly every reverse-percentage error, and the same careful method applies to questions about pre-tax prices and gross salaries.
Percentage of a Percentage
Multiplying percentages together requires converting both to decimals first. To find 20% of 30% of a quantity, calculate 0.20 × 0.30 = 0.06, which is 6% of the original number. This pattern arises in finance (a fee applied to a taxed amount), in statistics (a proportion of a subgroup), and in probability (two independent events both occurring). The answer to any chained-percentage question is always smaller than either percentage alone.
Percentages in Statistics and Data
Statistics leans on percentages because raw counts across differently sized groups resist direct comparison. A survey reporting that 62% of 450 respondents prefer one option communicates more than the raw figure of 279 people, because the percentage automatically scales for the sample size. Percentage change, percentage points, and relative frequency tables all descend from the same basic formula taught on this page. When a report presents numbers in percentage terms, always check what the whole is — the denominator defines the meaning of the result.
How Percentages Are Written: Notation and Symbols
The % symbol is simply shorthand for “divided by 100”, so 40% and 0.40 and 40/100 are three ways of writing exactly the same value. Historical merchants wrote terms like “per cento” (by the hundred) in commercial arithmetic long before the modern symbol evolved from a stylised abbreviation of that phrase. In modern usage, percentages may be written with decimals (12.5%), as fractions (1/8), or as ratios (1 in 8), and fluency converting between these written forms is what makes percentage calculations feel easy rather than mysterious.
Worked Exam-Style Questions
Question 1. A laptop priced at $899 is discounted 15%, and an additional 5% student discount applies to the reduced price. What is the final amount payable?
- First discount: 899 × 0.85 = $764.15
- Second discount: 764.15 × 0.95 = $725.94
- The two discounts do not combine into a single 20% reduction — sequential percentages always multiply.
Question 2. After a 12% pay rise, a salary is $67,200. What was the original salary?
- The new salary is 112% of the original: original = 67,200 ÷ 1.12 = $60,000
Question 3. A solution of 250 ml contains 15% salt by volume. How much pure water must be added to reduce the concentration to 10%?
- Salt present: 0.15 × 250 = 37.5 ml
- Final volume for 10%: 37.5 ÷ 0.10 = 375 ml
- Water to add: 375 − 250 = 125 ml
Teaching Percentages to Others
When explaining percentages to children or students, begin with the meaning of the words (per = “for each”, cent = “hundred”) and then anchor every calculation in the same three-part relationship: part, whole, and percentage. Once a learner can identify which two of those three quantities a question supplies, choosing the correct method becomes mechanical, and finding the answer is a matter of arithmetic rather than memorisation. Concrete materials — hundred squares, base-ten blocks, or even pocket money — make the abstract numbers tangible before introducing the formula in its written form.
FAQ
How do I calculate a percentage of a number?
Convert the percentage to a decimal (divide by 100) and multiply by the number. For example, 25% of 200 = 0.25 × 200 = 50. The percentage calculator does this instantly.
How do I calculate what percentage a number is of another number?
Divide the part by the whole and multiply by 100: (part ÷ whole) × 100. If you have 18 correct answers out of 24, that is (18 ÷ 24) × 100 = 75%.
How do I calculate a discount percentage?
Multiply the original price by the decimal form of the discount and subtract it from the original: price × (1 − discount%). A $90 item at 30% off costs 90 × 0.70 = $63.
How do I convert a percentage to a fraction or decimal?
Drop the % sign and divide by 100: 45% = 45/100 = 0.45. To simplify as a fraction, 45/100 reduces to 9/20.
How do I add two percentages together?
You can only add percentages directly when they share the same whole. Otherwise, calculate each percentage of the base number separately, then add the resulting amounts.