Percent

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A percent is a type of ratio where something is compared to a hundred. Probabilities, scores, and success rates are commonly written in percents.

Conversion to Fractions and Decimals

By definition, $n \%$ means $\frac{n}{100}$, which means $n$ hundredths. In other words, converting percents to decimals means moving two decimal places to the left, and converting decimals to percents means moving two decimal places to the right. For instance, $25 \% = 0.25$, $0.1 \% = 0.001$, and $125 \% = 1.25$.

Percent of a Number

Let $p$ be the percentage, $n$ be the number we want to take the percentage of, and $x$ be the wanted quantity. By definition, $\frac{x}{n} = p \% = \frac{p}{100}$, so $x = n \cdot \frac{p}{100}$. In other words, to find the percent of the number, we convert the percent to the fraction (or decimal) and then multiply it with the number we want to take the percentage of. For example, $50 \%$ of $60$ equals $0.5 \cdot 60 = 30$.

Percent Change

The percent change equals the ratio of the positive difference between the original quantity and the new quantity to the original quantity when written as a percent. For example, the percent change from 10 to 15 equals $\frac{15-10}{10} = 0.5 = 50 \%$.

Percent Increase and Decrease

If the new quantity is greater than the original quantity, then the percent change is called a percent increase. If the new quantity is smaller than the original quantity, then the percent change is called a percent decrease.

Let $p$ be the percentage, $y$ be the new quantity, and $x$ be the old quantity. From the definition of percent change, a percent increase is written in the form $\frac{y-x}{x} = \frac{p}{100}$, and solving for $y$ results in $y = x(1 + \frac{p}{100})$. Additionally, a percent decrease is written in the form $\frac{x-y}{x} = \frac{p}{100}$, and solving for $y$ results in $y = x(1 - \frac{p}{100})$.

For a written example, a $20 \%$ increase from $30$ is $1.2 \cdot 30 = 36$, and a $20 \%$ decrease from $30$ is $0.8 \cdot 30 = 24$.

Problems

Introductory Problems

Intermediate Problems

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