Understand Percent

Use the definition of percent

How many cents are in one dollar? There are $100$ cents in a dollar. How many years are in a century? There are $100$ years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is $100$. We use the percent symbol, %, to show percent.

According to data from the American Association of Community Colleges, about $57\%$ of community college students are female. This means $57$ out of every $100$ community college students are female. Out of $100$ equal squares in a grid, $57$ would be shaded, which we write as the ratio $\tfrac{57}{100}$.

Similarly, $25\%$ means a ratio of $\tfrac{25}{100}$, $3\%$ means a ratio of $\tfrac{3}{100}$, and $100\%$ means a ratio of $\tfrac{100}{100}$. In words, “one hundred percent” means the total $100\%$ is $\tfrac{100}{100}$, and since $\tfrac{100}{100} = 1$, we see that $100\%$ means one whole.

Example. According to the Public Policy Institute of California, $44\%$ of parents of public school children would like their youngest child to earn a graduate degree. Write this percent as a ratio.

The amount we want to convert is $44\%$. Since percent means per $100$, $44\% = \tfrac{44}{100}$.

Example. In $2007$, according to a U.S. Department of Education report, $21$ out of every $100$ first-time freshmen college students at $4$-year public institutions took at least one remedial course. Write this as a ratio and then as a percent.

The amount we want to convert is $21$ out of $100$, so the ratio is $\tfrac{21}{100}$. Converting $21$ per $100$ to percent gives $21\%$.

Convert percents to fractions and decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per $100$, so the denominator of the fraction is $100$.

Example. Convert each percent to a fraction: (a) $36\%$ (b) $125\%$.

(a) Write $36\%$ as a ratio with denominator $100$: $\tfrac{36}{100}$. Simplify: $\tfrac{9}{25}$.

(b) Write $125\%$ as a ratio with denominator $100$: $\tfrac{125}{100}$. Simplify: $\tfrac{5}{4}$.

This example shows that a percent can be greater than $1$ — $125\%$ means $\tfrac{125}{100}$, or $\tfrac{5}{4}$. This is an improper fraction, and its value is greater than one.

Example. Convert each percent to a fraction: (a) $24.5\%$ (b) $33\tfrac{1}{3}\%$.

(a) Write $24.5\%$ as a ratio with denominator $100$: $\tfrac{24.5}{100}$. Clear the decimal by multiplying the numerator and denominator by $10$: $\tfrac{245}{1000}$. Rewrite showing common factors and simplify: $\tfrac{5 \cdot 49}{5 \cdot 200} = \tfrac{49}{200}$.

(b) Write $33\tfrac{1}{3}\%$ as a ratio with denominator $100$: $\tfrac{33\frac{1}{3}}{100}$. Write the numerator as an improper fraction: $\tfrac{\frac{100}{3}}{100}$. Rewrite as fraction division, replacing $100$ with $\tfrac{100}{1}$: $\tfrac{100}{3} \div \tfrac{100}{1}$. Multiply by the reciprocal: $\tfrac{100}{3} \cdot \tfrac{1}{100}$. Simplify: $\tfrac{1}{3}$.

In Decimals, we learned how to convert fractions to decimals. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

Example. Convert each percent to a decimal: (a) $6\%$ (b) $78\%$.

Because we want to change to a decimal, we leave the fractions with denominator $100$ instead of removing common factors.

(a) Write $6\%$ as $\tfrac{6}{100}$. Divide $6$ by $100$: $0.06$.

(b) Write $78\%$ as $\tfrac{78}{100}$. Divide $78$ by $100$: $0.78$.

Example. Convert each percent to a decimal: (a) $135\%$ (b) $12.5\%$.

(a) Write $135\%$ as $\tfrac{135}{100}$. Divide $135$ by $100$: $1.35$.

(b) Write $12.5\%$ as $\tfrac{12.5}{100}$. Divide $12.5$ by $100$: $0.125$.

Let’s summarize the results so we can look for a pattern.

Do you see the pattern? To convert a percent number to a decimal number, we move the decimal point two places to the left and remove the % sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer $6$ as $6.0$, we can think of $6\%$ as $6.0\%$.) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

Convert decimals and fractions to percents

To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is $100$, it is easy to change that fraction to a percent.

Example. Convert each decimal to a percent: (a) $0.05$ (b) $0.83$.

(a) Write $0.05$ as a fraction: $\tfrac{5}{100}$. Write this ratio as a percent: $5\%$.

(b) Write $0.83$ as a fraction: $\tfrac{83}{100}$. Write this ratio as a percent: $83\%$.

To convert a mixed number or a decimal greater than $1$ to a percent, we follow the same steps.

Example. Convert each decimal to a percent: (a) $1.05$ (b) $0.075$.

(a) Write $1.05$ as a fraction: $1\tfrac{5}{100}$. Write as an improper fraction with denominator $100$: $\tfrac{105}{100}$. Write this ratio as a percent: $105\%$. Notice that since $1.05 > 1$, the result is more than $100\%$.

(b) Write $0.075$ as a fraction: $\tfrac{75}{1{,}000}$. Divide the numerator and denominator by $10$, so that the denominator is $100$: $\tfrac{7.5}{100}$. Write this ratio as a percent: $7.5\%$.

Do you see the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.

In Decimals, we also learned how to convert fractions to decimals. So to convert a fraction to a percent, we first change it to a decimal and then convert that decimal to a percent.

Example. Convert each fraction or mixed number to a percent: (a) $\tfrac{3}{4}$ (b) $\tfrac{11}{8}$ (c) $2\tfrac{1}{5}$.

To convert a fraction to a decimal, divide the numerator by the denominator.

(a) Change $\tfrac{3}{4}$ to a decimal: $0.75$. Write as a percent by moving the decimal two places: $75\%$.

(b) Change $\tfrac{11}{8}$ to a decimal: $1.375$. Write as a percent by moving the decimal two places: $137.5\%$.

(c) Write $2\tfrac{1}{5}$ as an improper fraction: $\tfrac{11}{5}$. Change to a decimal: $2.2$. Write as a percent: $220\%$.

Notice that we needed to add zeros at the end of the number when moving the decimal two places to the right.

Sometimes when changing a fraction to a decimal, the division continues for many decimal places and we round off the quotient. The number of decimal places we round to will depend on the situation. If the decimal involves money, we round to the hundredths place. For most other cases we round the number to the nearest thousandth, so the percent will be rounded to the nearest tenth.

Example. Convert $\tfrac{5}{7}$ to a percent.

Change $\tfrac{5}{7}$ to a decimal, rounding to the nearest thousandth: $0.714$. Write as a percent: $71.4\%$.

When we first looked at fractions and decimals, we saw that some fractions convert to a repeating decimal. We use the same notation — a line over the repeating digits — as well as fraction notation, when we convert fractions to percents.

Example. An article in a medical journal claimed that approximately $\tfrac{1}{3}$ of American adults are obese. Convert the fraction $\tfrac{1}{3}$ to a percent.

Change $\tfrac{1}{3}$ to a decimal by dividing $1$ by $3$: $0.333\ldots$, a repeating decimal. Write as a percent: $33\tfrac{1}{3}\%$. We could also write the percent as $33.\overline{3}\%$.

Key terms

percent — a ratio whose denominator is $100$; written with the symbol %.

Practice

Use the definition of percent

Convert percents to fractions and decimals

Convert decimals and fractions to percents

This section is adapted from Prealgebra 2e, Section 6.1: Understand Percent by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the percent grid as an accessible inline graphic and the percent/decimal pattern table as a markdown table; omitted the Be Prepared quiz, the playing-card photo, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, with each multipart exercise expanded into one question per part.