Percentage Calculator

A percentage calculator answers three common percent questions: what is a percent of a number, what percent is one value of another, and what is the percent change from one value to another. Percent means per 100, so the calculator converts between ratios, decimals, and hundred-based language used in discounts, taxes, grades, finance, science, and everyday comparisons.

Percent as a ratio out of 100

Percent notation is a standardized way to compare quantities with different sizes. Saying 18 out of 20 and 90 out of 100 are both 90 percent makes the relationship easy to compare. The symbol is compact, but it can hide the denominator if you are not careful. Every percent statement needs a base: 20 percent of what, or 20 is what percent of what whole?

The calculator separates three tasks because each asks for a different unknown. In percent-of mode, the percent and whole are known, and the part is unknown. In “X is what percent of Y” mode, the part and whole are known, and the percent is unknown. In percent-change mode, the starting and ending values are known, and the relative change is unknown. Keeping those roles clear prevents most percent mistakes.

Formulas and variable definitions

To find a part from a percent and a whole:

$\text{part} = \frac{\text{percent}}{100} \times \text{whole}$

To find what percent a part is of a whole:

$\text{percent} = \frac{\text{part}}{\text{whole}} \times 100$

To find percent change from a starting value to an ending value:

$\text{percent change} = \frac{\text{ending value} - \text{starting value}}{\left|\text{starting value}\right|} \times 100$

where:

• $\text{part}$ is the amount being compared with a whole.
• $\text{whole}$ is the base amount in the comparison and cannot be zero for a percent-of-whole calculation.
• $\text{starting value}$ is the baseline for change and cannot be zero in this calculator.
• $\text{ending value}$ is the new amount after the change.

The absolute value in the percent-change denominator follows the formula above. It gives a positive denominator even if the starting value is negative, and the sign of the numerator still determines increase or decrease.

Examples: solving percentage questions

The default mode is percent of a number with 20 percent and an of value of 80. The calculation is

$\frac{20}{100} \times 80 = 16$

The primary label reads 20% of 80, and the displayed result is 16.

In the second mode, use part 20 and whole 80. The calculation is

$\frac{20}{80} \times 100 = 25$

The calculator displays 25%, meaning 20 is one quarter of 80.

In percent-change mode, the defaults are from 80 to 100. The difference is

$100 - 80 = 20$

Then compare the difference with the starting magnitude:

$\frac{20}{\left|80\right|} \times 100 = 25$

Because the change is positive, the calculator labels the result as an increase and displays 25%. It also lists the difference, from value, and to value. If the ending value were 60 instead, the numerator would be negative, and the calculator would label a decrease of 25%.

Interpretation and applications

Percentages are useful because they normalize comparisons. A 5-point score improvement means something different on a 20-point quiz than on a 500-point exam, but the percent change or percent correct puts both on a common scale. Retail discounts, sales tax, interest rates, error rates, completion progress, nutritional labels, and laboratory concentrations all use percent language.

Choose the specialized calculator when the task is narrower. Use the percent off calculator for sale prices and discounts, the percent to goal calculator for progress toward a target, and the average percentage calculator when combining several percent values. If you are specifically comparing a before-and-after value, the percentage change calculator provides a focused version of the third mode.

Edge cases and common mistakes

The denominator cannot be zero when asking what percent a part is of a whole. There is no finite answer to “20 is what percent of 0” because no multiple of zero produces 20. Similarly, the calculator rejects percent change from a starting value of zero. Moving from zero to a positive number is important in real life, but it is not a finite percentage increase because there is no nonzero baseline to divide by.

Percentages can be negative or greater than 100 depending on context. Negative percent change represents a decrease. A negative percent-of-number input can model reductions or signed rates. A result above 100 means the part exceeds the whole or the change exceeds the original magnitude.

Two wording mistakes cause many errors. First, “20 percent more than 80” is not the same as “20 percent of 80”; it means 80 plus 16, or 96. Second, a change from 10 percent to 15 percent is 5 percentage points, not simply 5 percent. Relative to 10 percent, the increase is 50 percent. State the baseline before calculating.

Sources

• Wolfram MathWorld, Percent — definition of percent notation.
• Wolfram MathWorld, Percentage — related mathematical terminology.
• Khan Academy, Percent word problem examples — educational examples of percent reasoning.
• OpenStax, Solve General Applications of Percent — textbook treatment of percent equations.