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:

A ratio like $18 : 20$ and another like $34 : 40$ have different denominators, so you cannot compare them by eye. Rewrite each one with the same denominator of $100$ and the comparison falls out immediately. That is the same thing a common denominator does for two fractions: it lets 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. The percent is then just the numerator that sits over $100$. 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 examples below do.

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 place is worth ten times the next, so two place shifts are a factor of $10 \times 10 = 100$. Moving two places matches dividing or multiplying by $100$ for exactly that reason.

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. The $55$ that goes over $100$ is the percent, so the answer is $55\%$. Those two equal fractions are the two sides of the percent proportion, which all three tasks use:

Three numbers appear here (the part, the whole, and the percent), and each task hands you two of them and asks for the third. Finding a percent needs a whole that is not zero. Recovering the whole needs a percent that is not zero. Once it is set up, you finish with the cross-multiply-then-divide move from the Proportions lesson, or with a quick decimal multiplication, whichever is faster.

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 the percent 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. The percent proportion handles it directly. If $30$ is $40\%$ of some number, then $30$ is the part, $40$ is the percent, 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 $75$. 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