Percents

Learning Outcomes

  • Given the part and the whole, write a percent
  • Evaluate changes in amounts with percent calculations

 

A percent is a fraction

Recall that a fraction is written [latex]\dfrac{a}{b},[/latex] where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b \neq 0[/latex]. In a fraction, [latex]a[/latex] is called the numerator and [latex]b[/latex] is called the denominator.

A percent can be expressed as a fraction, that is a ratio, of some part of a quantity out of the whole quantity,  [latex]\dfrac{\text{part}}{\text{whole}}[/latex].

Ex. Suppose you take an informal poll of your classmates to find out how many of them like pizza. You find that, out of 25 classmates, 20 of them like pizza. You can represent your findings as a ratio of how many like pizza out of how many classmates you asked.

[latex]\dfrac{20}{25}[/latex] represents the 20 out of 25 classmates who like pizza.

To find out what percent of the 25 asked said they like pizza, divide the numerator by the denominator, then multiply by 100.

[latex]\dfrac{20}{25} = 20 \div 25 = 0.80 = 80 \%[/latex]

Percent literally means “per 100,” or “parts per hundred.” When we write 40%, this is equivalent to the fraction [latex]\displaystyle\frac{40}{100}[/latex] or the decimal 0.40. Notice that 80 out of 200 and 10 out of 25 are also 40%, since [latex]\displaystyle\frac{80}{200}=\frac{10}{25}=\frac{40}{100}[/latex].

Rounded rectangle divided into ten vertical sections. The left four are shaded yellow, while the right 6 are empty.

A visual depiction of 40%

convert a percent to a  decimal or fraction

To do mathematical calculations with a given percent, we must first write it in numerical form. A percent may be represented as a percent, a fraction, or a decimal.

Convert a percent to a fraction

  1. Write the percent over a denominator of [latex]100[/latex] and drop the percent symbol %.
  2. Reduce the resulting fraction as needed.

Ex. [latex]80 \% =\dfrac{80}{100}=\dfrac{4\cdot 20}{5\cdot 20}=\dfrac{4}{5}[/latex]

Convert a percent to a decimal

There are three methods for writing a percent as a decimal.

  • You can write the percent as a fraction, simply fully, then divide the numerator by the denominator.
    • [latex]80 \% =\dfrac{80}{100}=\dfrac{4\cdot 20}{5\cdot 20}=\dfrac{4}{5}=0.8[/latex]
  • You can write the percent as a fraction, simplify to a denominator of 10, 100, 1000, etc., then rewrite as a decimal.
    • [latex]80 \% =\dfrac{80}{100}=\dfrac{8\cdot 10}{10\cdot 10}=\dfrac{8}{10}=\text{ eight tenths }=0.8[/latex]
  • Write the percent without the percent symbol %, then place a decimal after the ones place and move it two places to the left.
    • [latex]80 \% =80.0=0.80=0.8[/latex]

 

Percent

If we have a part that is some percent of a whole, then [latex]\displaystyle\text{percent}=\frac{\text{part}}{\text{whole}}[/latex], or equivalently, [latex]\text{percent}\cdot\text{whole}=\text{part}[/latex].

To do calculations using percents, we write the percent as a decimal or fraction.

The video and following few examples demonstrate how to convert between percent, fraction, and decimal representations.

Example

In a survey, 243 out of 400 people state that they like dogs. What percent is this?

 

Example

Write each as a percent:

  1. [latex]\displaystyle\frac{1}{4}[/latex]
  2. 0.02
  3. 2.35

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