Choose a calculation
Select the relationship you want to solve, enter the known values, then calculate.
15% of 240 is 36.
Formula
Result = (percentage ÷ 100) × value
Calculation
- Convert 15% to decimal form: 15 ÷ 100 = 0.15.
- Multiply: 0.15 × 240 = 36.
- Final result: 36.
Worked examples & analysis
Percentage, Fraction & Ratio Examples
See how the formulas work with practical numbers, compare percentage scenarios, and test how a change in the baseline affects the result.
Worked Example: Comparing a Price Change
Suppose an item originally costs $80 and its new price is $100. Percentage change answers how much the new price changed relative to the original $80 baseline.
From $80 to $100
First calculate the dollar change by subtracting the original price from the new price.
Next divide the $20 increase by the original $80 price. The original value is the denominator because percentage change measures movement relative to the starting point.
The new price is therefore 25% higher than the original price. This does not mean that reversing the prices would produce a 25% decrease, because the baseline would then be $100 rather than $80.
Calculation Examples
These examples cover each operation available in the main calculator and show the essential arithmetic used to obtain the result.
| Calculation | Inputs | Arithmetic | Result |
|---|---|---|---|
| Percentage of a value | 15% of 240 | (15 ÷ 100) × 240 | 36 |
| What percentage is A of B? | A = 45, B = 60 | (45 ÷ 60) × 100 | 75% |
| Percentage increase | 80 → 100 | ((100 − 80) ÷ 80) × 100 | 25% increase |
| Percentage decrease | 100 → 80 | ((80 − 100) ÷ 100) × 100 | 20% decrease |
| Percentage difference | A = 80, B = 100 | 20 ÷ ((80 + 100) ÷ 2) × 100 | 22.22222222% |
| Fraction simplifier | 42/56 | GCD(42, 56) = 14; divide both by 14 | 3/4 |
| Fraction to percentage | 3/8 | (3 ÷ 8) × 100 | 37.5% |
| Ratio simplifier | 18:24 | 18 ÷ 6 : 24 ÷ 6 | 3:4 |
| Proportion | 3/4 = 12/D | 3D = 48; D = 48 ÷ 3 | D = 16 |
Scenario Comparison: Same $20 Difference, Different Percentages
A dollar difference alone does not determine percentage change. The percentage also depends on the original baseline. The same $20 increase can therefore represent very different percentage changes.
$40 → $60
50% increaseThe $20 increase equals half of the original $40 value: $20 ÷ $40 = 0.50.
$80 → $100
25% increaseThe same $20 increase equals one quarter of the original $80 value: $20 ÷ $80 = 0.25.
$200 → $220
10% increaseHere, $20 is only one tenth of the original $200 value: $20 ÷ $200 = 0.10.
Compare Percentage Change and Percentage Difference
Enter two values to compare three related measures: the absolute difference, the percentage change from A to B, and the symmetric percentage difference between A and B.
Enter comparison values
Value A is treated as the original value for percentage change. Percentage difference treats A and B symmetrically.
Comparison result
Percentage change uses 80 as its baseline, while percentage difference divides the absolute difference by the average magnitude of 80 and 100.
((100 − 80) ÷ 80) × 100 = 25%
20 ÷ 90 × 100 = 22.22222222%
Use percentage change when direction matters
Percentage change answers how much a value increased or
decreased relative to a defined starting value:
((new − original) ÷ original) × 100.
Use percentage difference for a symmetric comparison
Percentage difference compares two values without assigning
either one as the baseline:
|A − B| ÷ ((|A| + |B|) ÷ 2) × 100.
Understanding the calculation
How to Interpret Percentages, Fractions & Ratios
A correct calculation still needs the right interpretation. Percentage change, percentage difference, fractions, ratios and proportions describe related mathematical relationships, but they do not answer exactly the same questions.
Understanding the Result
Interpret the output according to the calculation selected. In particular, distinguish a directional percentage change from a symmetric percentage difference.
Percentage of a value
A result such as 36 for 15% of 240 means that 36 represents 15 parts per hundred of 240. The result has the same conceptual unit as the original value.
Percentage as a relationship
If A is 75% of B, then A = 0.75 × B. A result above 100% means A is greater than B; 100% means the values are equal; and a result below 100% means A is smaller than B when both values are positive.
Percentage change
A positive percentage change indicates an increase relative to the original value, while a negative result indicates a decrease. The original value is the baseline, so changing which value is considered "original" can change the percentage substantially.
Percentage difference
Percentage difference measures separation without assigning either value as the starting point. Under this calculator's convention, the absolute difference is compared with the average magnitude of the two values.
Fractions
Simplifying a fraction changes its written form, not its value. For example, 42/56 = 3/4. Both fractions represent the same number, and both equal 0.75.
Ratios and proportions
A ratio such as 3:4 describes relative quantities. A proportion states that two ratios are equal, such as 3/4 = 12/16. Scaling every term by the same nonzero factor preserves the relationship.
From 80 to 100, percentage change is 25% because 80 is the baseline. The percentage difference between 80 and 100 is approximately 22.2222% because it uses their average magnitude of 90 as the denominator.
Assumptions
The calculator performs arithmetic on the values supplied. Its results depend on the following assumptions about those inputs and their intended relationship.
Inputs are numerical
Each entered value represents a valid finite number rather than missing data, text, NaN or Infinity.
Compared quantities are compatible
Values used in the same comparison represent quantities that can meaningfully be compared on the same basis.
Percent input means percent
An input of 15 in a percentage field means 15%, not 0.15%. The calculator converts 15% to 0.15 internally where needed.
The baseline is intentional
For percentage change, the value identified as original is intentionally being used as the reference denominator.
Fractions use a nonzero denominator
A denominator of zero is mathematically undefined and cannot form a valid fraction for these calculations.
Displayed rounding is not new data
A shortened decimal is a formatted representation of the calculated value, not an increase in the precision of the original inputs.
Limitations
These formulas describe numerical relationships. They do not determine whether the relationship is meaningful for a particular business, financial, scientific or statistical decision.
What the calculator does not model
- It does not determine why a percentage increased or decreased.
- It does not adjust values for inflation, taxes, fees, interest, time or purchasing power.
- It does not determine whether two observed values are statistically significant.
- It does not infer uncertainty, measurement error or confidence intervals.
- It does not automatically convert physical measurement units before comparing values.
Mathematical boundary conditions
- Division by zero is undefined.
- Conventional percentage change is undefined when the original baseline is zero.
- Percentage difference under the selected formula is undefined when both values are zero.
- A proportion is invalid if a denominator in the stated ratio is zero.
- Extremely large or small floating-point values are subject to normal computer-number precision limits.
Common Percentage, Fraction & Ratio Errors
Many incorrect results come from choosing the wrong reference value or interpreting percentage notation incorrectly rather than from the arithmetic itself.
Percentage change divides by the original value, not the new value. From 80 to 100 the increase is 20 ÷ 80 = 25%, not 20 ÷ 100 = 20%.
In a field labeled percentage, enter 15 for 15%. Entering 0.15 would represent 0.15% under that input convention.
Percentage change is directional and requires a baseline. Percentage difference is symmetric. Select the operation according to the question being asked.
A 25% increase followed by a 25% decrease does not return to the starting value. For example, 80 increased by 25% becomes 100; decreasing 100 by 25% gives 75.
Convert measured quantities to a common unit before comparing them. For example, convert feet and inches to the same unit before forming a meaningful ratio.
Repeatedly rounding intermediate values can accumulate error. Keep the available precision during the calculation and round the final displayed result instead.
Equivalent ratios require the same nonzero scale factor on both terms. For example, 3:4 = 6:8, but 3:4 is not equivalent to 6:4.
Percentage Reference Table
Common percentages can be expressed as decimals and fractions. These equivalent forms are useful for quick mental calculations.
| Percentage | Decimal | Fraction | Quick interpretation |
|---|---|---|---|
| 1% | 0.01 | 1/100 | 1 part per 100 |
| 5% | 0.05 | 1/20 | 5 parts per 100 |
| 10% | 0.1 | 1/10 | One tenth |
| 12.5% | 0.125 | 1/8 | One eighth |
| 20% | 0.2 | 1/5 | One fifth |
| 25% | 0.25 | 1/4 | One quarter |
| 33â…“% | 0.333... | 1/3 | Approximately one third |
| 50% | 0.5 | 1/2 | One half |
| 66â…”% | 0.666... | 2/3 | Approximately two thirds |
| 75% | 0.75 | 3/4 | Three quarters |
| 100% | 1 | 1/1 | The entire reference amount |
| 200% | 2 | 2/1 | Twice the reference amount |
Formula Selection Reference
Choose the formula according to the relationship you need to calculate.
| Question | Use | Formula / method |
|---|---|---|
| What is P% of V? | Percentage of a value | (P ÷ 100) × V |
| What percentage is A of B? | Part-to-whole percentage | (A ÷ B) × 100 |
| How much did a value increase or decrease? | Percentage change | ((N − O) ÷ O) × 100 |
| How different are two values without a baseline? | Percentage difference | |A − B| ÷ ((|A| + |B|) ÷ 2) × 100 |
| Can a fraction be reduced? | Fraction simplification | Divide numerator and denominator by their GCD |
| What percentage does a fraction represent? | Fraction to percentage | (n ÷ d) × 100 |
| Can a ratio be written in lower terms? | Ratio simplification | Divide both terms by the same greatest common factor |
| What value makes two ratios equivalent? | Proportion solver | A ÷ B = C ÷ D |
Percentage Increase and Decrease Factors
A percentage change can also be represented by a multiplication factor. For an increase, use 1 + the decimal rate. For a decrease, use 1 − the decimal rate.
| Change | Multiplier | Example starting at 100 |
|---|---|---|
| 10% decrease | × 0.90 | 100 × 0.90 = 90 |
| 5% decrease | × 0.95 | 100 × 0.95 = 95 |
| No change | × 1.00 | 100 × 1.00 = 100 |
| 5% increase | × 1.05 | 100 × 1.05 = 105 |
| 10% increase | × 1.10 | 100 × 1.10 = 110 |
| 25% increase | × 1.25 | 100 × 1.25 = 125 |
| 50% increase | × 1.50 | 100 × 1.50 = 150 |
| 100% increase | × 2.00 | 100 × 2.00 = 200 |
Why Percentages Use 100
The word percent expresses the idea of "per hundred." Writing a proportion relative to 100 provides a common scale, making quantities of different sizes easier to compare.
The percent sign (%) is the modern shorthand for this per-hundred relationship. Mathematically, a percentage is therefore a fraction with an implied denominator of 100: 25% means 25/100, which simplifies to 1/4.