Percent

The central idea

Percent gives different quantities a common comparison scale. Saying that $18$ of $25$ students submitted an assignment is useful; saying that $72\%$ submitted makes the result easier to compare with another class of a different size.

The three forms below describe the same quantity:

The form changes, but the value does not.

Moving between forms

To turn a decimal into a percent, multiply by $100$ and attach the percent sign. To turn a percent into a decimal, divide by $100$ and remove the sign.

Moving the decimal point is a shortcut for multiplying or dividing by $100$; it is not a separate rule.

Finding a percent of a quantity

Translate “of” as multiplication and write the percent as a decimal:

Finding the percent

When the part and whole are known, divide the part by the whole:

If $18$ of $24$ questions are correct, then $18/24=0.75$, so the score is $75\%$.

Finding the original whole

If the part and percent are known, rearrange the relationship:

If $27$ students represent $90\%$ of a class, the class size is $27/0.90=30$ students. The whole must be at least as large as the part when the percent is at most $100\%$.

Common mistakes

Using the new value as the denominator. Percent change is measured relative to the original value.

Treating $8\%$ as $0.8$. Eight percent is $8/100=0.08$.

Adding successive percentages. A $20\%$ decrease followed by a $20\%$ increase does not return to the start because the second percent uses a different whole. The combined multiplier is $0.8(1.2)=0.96$.

Ignoring reasonableness. If $25\%$ of $60$ is reported as $150$, the result is larger than the whole even though the percent is below $100\%$.