Percents: Learn It 1

  • Write percents
  • Determine unit rate using percentages
  • Find both the relative and absolute change

Percents

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

percent

A ratio is a comparison of two numbers by division. 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 percentages, we write the percent as a decimal or fraction.

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.

How to: Convert a Percent to a Fraction

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

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

How to: 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 [latex]10[/latex], [latex]100[/latex], [latex]1000[/latex], 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 [latex]\%[/latex], 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]
Write each as a percent:

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