How To Calculate Percentage

To calculate a percentage, divide the part by the whole and multiply by $100$, as long as the whole is not $0$. If you are finding a percent of a number instead, convert the percent to a decimal and multiply.

That one split solves most percentage questions:

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

The key is choosing the right whole. If the base changes, the percentage changes.

What Percentage Means

A percentage is a ratio written on a scale of $100$. So $75\%$ means $75$ out of $100$, or an equivalent fraction such as $\frac{3}{4}$.

This is why percentages are useful for comparison. They put different situations on the same scale, whether you are looking at grades, discounts, or survey results.

Worked Example: What Percent Is $18$ Out Of $24$?

Suppose a student gets $18$ questions correct out of $24$. The part is $18$, and the whole is $24$.

$$\frac{18}{24} \times 100\%$$

Divide first:

$$\frac{18}{24} = 0.75$$

Then multiply by $100$:

$$0.75 \times 100\% = 75\%$$

So $18$ out of $24$ is $75\%$. A quick check helps: $18$ is three quarters of $24$, and three quarters is $75\%$.

How To Find A Percent Of A Number

Sometimes you already know the percentage and want the amount. In that case, convert the percent to a decimal and multiply.

For example, to find $15\%$ of $80$:

$$0.15 \times 80 = 12$$

So $15\%$ of $80$ is $12$. This is different from asking what percent $12$ is of $80$, which would use division first.

Common Mistakes When Calculating Percentage

Using The Wrong Whole

The denominator should be the full amount you are comparing against. If the question changes the base, the percentage changes too.

Forgetting To Convert The Percent

$25\%$ means $0.25$, not $25$. This mistake makes answers too large by a factor of $100$.

Mixing Up Percentage And Percent Change

A plain percentage compares part to whole. Percent change compares the change to the original value, so the base is the starting number.

$$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$$

This formula only applies when the original value is not $0$.

When Percentages Are Useful

Percentages appear in grades, discounts, tax, tips, interest, survey results, and data summaries. They help because they make different-sized situations easier to compare on the same scale.

Try A Similar Problem

Try your own version with a sale price or a test score. First name the whole, then decide whether you need to divide to find a percentage or multiply to find a percent of a number.