Percentages Formula

Percentages are a way of expressing a quantity as a fraction of 100, written with the symbol % to mean 'per hundred.'

The Formula

p%=p100 (to convert a percent to a fraction or decimal, divide by 100)

When to use: Percent means 'per hundred.' 25% means 25 out of every 100.

Quick Example

25%=25100=0.25=14 and 50%=50100=0.5=12

Notation

p% means p per hundred; equivalently p100 or 0.0p (for single/double-digit p)

What This Formula Means

A way of expressing a quantity as a fraction of 100, written with the symbol % to mean 'per hundred.'

Percent means 'per hundred.' 25% means 25 out of every 100.

Formal View

p%=p100, so p% of x is p100⋅x

Worked Examples

Example 1

easy
What is 25% of $80?

Answer

$20

First step

1
Recall that "percent" means per hundred, so 25%=25100=0.25 as a decimal.

Full solution

  1. 2
    Multiply the decimal by the whole amount: 0.25×80.
  2. 3
    Calculate: 0.25×80=20
Finding a percentage of a number means multiplying the number by the percentage expressed as a decimal (or fraction). 25% is the same as 14.

Example 2

medium
A student scored 42 out of 60 on a test. What is the percentage score?

Example 3

easy
Convert 710 to a percent.

Common Mistakes

  • Using the percent number directly without dividing by 100 - 20% means 0.20, not 20.
  • Forgetting what the percent is of - 30% off $80 and 30% off $50 are different dollar amounts.
  • Adding percents of different wholes as if they share a scale - 50% of one thing plus 50% of another is not 100% of anything.

Why This Formula Matters

Percents put unlike comparisons on one ruler — a test score, a sales tax, and a discount all become 'out of 100' — which is why they run grades, prices, statistics, and probability. Miss that % means per hundred and you turn 50% into the number 50. Recognizing it by "Is the quantity being measured against a scale of 100?" — rather than by familiar numbers — is what lets a student tell it apart from decimal and fraction and percent change in a mixed problem set.

Frequently Asked Questions

What is the Percentages formula?

A way of expressing a quantity as a fraction of 100, written with the symbol % to mean 'per hundred.'

How do you use the Percentages formula?

Percent means 'per hundred.' 25% means 25 out of every 100.

What do the symbols mean in the Percentages formula?

p% means p per hundred; equivalently p100 or 0.0p (for single/double-digit p)

Why is the Percentages formula important in Math?

Percents put unlike comparisons on one ruler — a test score, a sales tax, and a discount all become 'out of 100' — which is why they run grades, prices, statistics, and probability. Miss that % means per hundred and you turn 50% into the number 50. Recognizing it by "Is the quantity being measured against a scale of 100?" — rather than by familiar numbers — is what lets a student tell it apart from decimal and fraction and percent change in a mixed problem set.

What do students get wrong about Percentages?

The procedure for percentages is the easy part; the trap is using the percent number directly without dividing by 100. Asking "Is the quantity being measured against a scale of 100?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Percentages formula?

Before studying the Percentages formula, you should understand: fractions, decimals.

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