Percent

Percent means “out of 100”

A percent is a ratio whose second term is fixed at $100$. Writing $25\%$ is just another way of writing the ratio $25 : 100$, or the fraction $\frac{25}{100}$:

Read it as “twenty-five per hundred,” or “twenty-five out of every hundred.” If $25\%$ of a class walks to school, then for every $100$ students, $25$ of them walk. The symbol $\%$ is doing one job and one job only: it stands for “divide by $100$.” So whenever you see a percent, you can replace the $\%$ sign with $\div\, 100$ and nothing changes:

The same idea explains why $18 : 20$ and $34 : 40$ could not be compared until they were rescaled above: two ratios with different denominators cannot be compared by eye. Rewriting each one with a denominator of $100$ does for a ratio what a common denominator does for two fractions, letting you see which is larger. Percent is just the agreement that the common denominator will always be $100$.

The grid below is one whole, cut into $100$ equal squares. Change how many are shaded and read the four names the amount goes by, all of them counting the same squares.

Shade $25$ squares and you have filled two and a half rows: $25\%$, or $\frac{25}{100}$, or $0.25$. Shade $50$ and you have filled exactly half the grid, which is why $50\%$ and one half are the same amount. Shade all $100$ and you have one whole, so $100\%$ is the whole amount. The grid stops at $100$ squares, but a percent does not have to.

Converting between a percent and a fraction

Turning a percent into a fraction takes no new rule at all, just the definition: drop the $\%$ sign and write the number over $100$. After that, simplify the fraction to lowest terms with the greatest-common-factor method from the Fractions chapter.

Going the other way, from a fraction to a percent, you want the fraction rewritten with a denominator of $100$. The reason is that a percent is a count out of $100$. When the denominator divides into $100$, scale the fraction up:

This is exactly the proportion idea from the last lesson: $\frac{3}{4}$ and $\frac{75}{100}$ are equal ratios. When the denominator does not divide evenly into $100$, you can still find the percent by treating it as a proportion. Solve that proportion for the numerator, which is what the worked example below does.

Converting between a percent and a decimal

The link between percents and decimals comes straight from the meaning of the two place columns after the decimal point. From the Decimals chapter, the first two decimal places are tenths and hundredths, so a two-place decimal is already a count of hundredths. Since a percent is also a count of hundredths, the two are the same idea in different clothing:

To go from a percent to a decimal, divide by $100$, because $\%$ means “divide by $100$.” That division moves the decimal point two places to the left:

To go from a decimal to a percent, do the reverse: multiply by $100$, which moves the decimal point two places to the right. Then attach the $\%$ sign:

Each row below is one number written as a fraction, a decimal, and a percent.

The last row shows that $100\%$ is the whole thing, because $\frac{100}{100} = 1$. So $50\%$ is half, $25\%$ is a quarter, and a percent above $100$, like $150\%$, means more than the whole ($\frac{150}{100} = 1.5$).

The three percent tasks

Almost every percent question is one of three closely related tasks, and all three come from the same relationship. Take a bag of $20$ marbles where $11$ of them are blue. Scaling that ratio to a denominator of $100$ multiplies both terms by $5$, so $55$ marbles in every $100$ would be blue:

The $11$ is the part and the $20$ is the whole. $55$ is the number before the percent sign, so the percent is $55\%$. Those two equal fractions are the two sides of the percent proportion, which all three tasks use. Let $p$ stand for the number written before the percent sign, so the percent itself is $p\%$:

Three numbers appear here (the part, the whole, and $p$), and each task hands you two of them and asks for the third. Once it is set up, cross-multiply and divide to solve for whichever one is missing, the same move from the Proportions lesson. The division needs a nonzero number to divide by: finding $p$ divides by the whole, so the whole must be nonzero; finding the whole divides by $p$, so $p$ must be nonzero. Every problem in this lesson already satisfies whichever one its own task needs.

Task A: find a percent of a number

To find $20\%$ of $80$, the cleanest route is to turn the percent into a decimal and multiply, because “of” with percents means multiply. Since $20\% = 0.2$,

Why does multiplying work? The percent proportion says $\frac{\text{part}}{80} = \frac{20}{100}$, and solving it for the part means multiplying $80$ by $\frac{20}{100} = 0.2$. The decimal method is just that proportion with the arithmetic already done. You can also reason in fraction form: $20\%$ is $\frac{1}{5}$, and $\frac{1}{5}$ of $80$ is $80 \div 5 = 16$, the same answer.

Task B: find what percent one number is of another

Here you know the part and the whole and you want the percent. Make the ratio of part to whole, then rewrite it over $100$. To ask what percent $18$ is of $24$, start with the ratio $\frac{18}{24}$ and turn it into a percent by writing it over $100$:

This is a proportion with $p$ as the missing numerator. Cross-multiply the known diagonal and divide by the number diagonally opposite the blank:

A shortcut that gives the same thing: divide the part by the whole to get a decimal, then convert that decimal to a percent. Here $18 \div 24 = 0.75 = 75\%$.

Task C: find the whole from a part and its percent

The last task gives you a part and the percent it represents, and asks for the whole. If $30$ is $40\%$ of some number, here is a quick way to see the answer before any algebra: $40\%$ is four tenths, so one tenth of the whole is $30 \div 4 = 7.5$, and the whole is ten of those tenths, $10 \times 7.5 = 75$.

The percent proportion gets there directly too. $30$ is the part, $p = 40$, and the whole is the missing piece:

The whole sits in a denominator this time, but cross-multiplication does not care where the blank is. Multiply the diagonal you fully know, $30$ and $100$, then divide by the $40$ diagonally opposite the blank:

So $30$ is $40\%$ of $75$, matching the tenths estimate. The check is to run it forward: $40\%$ of $75$ is $0.4 \times 75 = 30$, which matches the part you started with.

Common mistakes