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%=p100p\% = \frac{p}{100} (to convert a percent to a fraction or decimal, divide by 100)

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

Quick Example

25%=25100=0.25=1425\% = \frac{25}{100} = 0.25 = \frac{1}{4} and 50%=50100=0.5=1250\% = \frac{50}{100} = 0.5 = \frac{1}{2}

Notation

p%p\% means pp per hundred; equivalently p100\frac{p}{100} or 0.0p0.0p (for single/double-digit pp)

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%25\% means 25 out of every 100.

Formal View

p%=p100p\% = \frac{p}{100}, so p%p\% of xx is p100⋅x\frac{p}{100} \cdot x

Worked Examples

Example 1

easy
What is 25%25\% of $80\$80?

Answer

$20\$20

First step

1
Recall that "percent" means per hundred, so 25%=25100=0.2525\% = \frac{25}{100} = 0.25 as a decimal.

Full solution

  1. 2
    Multiply the decimal by the whole amount: 0.25×800.25 \times 80.
  2. 3
    Calculate: 0.25×80=200.25 \times 80 = 20
Finding a percentage of a number means multiplying the number by the percentage expressed as a decimal (or fraction). 25%25\% is the same as 14\frac{1}{4}.

Example 2

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

Example 3

easy
Convert 710\dfrac{7}{10} to a percent.

Common Mistakes

  • Using the percent number directly without dividing by 100 - 20%20\% means 0.200.20, not 20.
  • Forgetting what the percent is of - 30%30\% off $80 and 30%30\% off $50 are different dollar amounts.
  • Adding percents of different wholes as if they share a scale - 50%50\% of one thing plus 50%50\% of another is not 100%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%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%25\% means 25 out of every 100.

What do the symbols mean in the Percentages formula?

p%p\% means pp per hundred; equivalently p100\frac{p}{100} or 0.0p0.0p (for single/double-digit pp)

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%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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