## Using the Definition of Proportion

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In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a **proportion**.

### Definition: Proportion

A proportion is an equation of the form \(\frac{a}{b}=\frac{c}{d},\) where \(b\ne 0,d\ne 0.\)

The proportion states two ratios or rates are equal. The proportion is read \(\text{“}a\) is to \(b,\) as \(c\) is to \(d\text{”.}\)

The equation \(\frac{1}{2}=\frac{4}{8}\) is a proportion because the two fractions are equal. The proportion \(\frac{1}{2}=\frac{4}{8}\) is read \(\text{“}1\) is to \(2\) as \(4\) is to \(8\text{”.}\)

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion \(\frac{\text{20 students}}{\text{1 teacher}}=\frac{\text{60 students}}{\text{3 teachers}}\) we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

## Example

Write each sentence as a proportion:

- \(\phantom{\rule{0.2em}{0ex}}3\) is to \(7\) as \(15\) is to \(35.\)
- \(\phantom{\rule{0.2em}{0ex}}5\) hits in \(8\) at bats is the same as \(30\) hits in \(48\) at-bats.
- \(\phantom{\rule{0.2em}{0ex}}\text{\$1.50}\) for \(6\) ounces is equivalent to \(\text{\$2.25}\) for \(9\) ounces.

### Solution

3 is to 7 as 15 is to 35. | |

Write as a proportion. | \(\frac{3}{7}=\frac{15}{35}\) |

5 hits in 8 at-bats is the same as 30 hits in 48 at-bats. | |

Write each fraction to compare hits to at-bats. | \(\frac{\text{hits}}{\text{at-bats}}=\frac{\text{hits}}{\text{at-bats}}\) |

Write as a proportion. | \(\frac{5}{8}=\frac{30}{48}\) |

\$1.50 for 6 ounces is equivalent to \$2.25 for 9 ounces. | |

Write each fraction to compare dollars to ounces. | \(\frac{\$}{\text{ounces}}=\frac{\$}{\text{ounces}}\) |

Write as a proportion. | \(\frac{1.50}{6}=\frac{2.25}{9}\) |

Look at the proportions \(\frac{1}{2}=\frac{4}{8}\) and \(\frac{2}{3}=\frac{6}{9}.\) From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the **cross products** of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.

### Definition: Cross Products of a Proportion

For any proportion of the form \(\frac{a}{b}=\frac{c}{d},\) where \(b\ne 0,d\ne 0,\) its cross products are equal.

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.

## Example

Determine whether each equation is a proportion:

- \(\phantom{\rule{0.2em}{0ex}}\frac{4}{9}=\frac{12}{28}\)
- \(\frac{17.5}{37.5}=\frac{7}{15}\)

### Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

Find the cross products. | \(28\cdot 4=112\phantom{\rule{2em}{0ex}}9\cdot 12=108\) |

Since the cross products are not equal, \(28·4\ne 9·12,\) the equation is not a proportion.

Find the cross products. | \(15\cdot 17.5=262.5\phantom{\rule{2em}{0ex}}37.5\cdot 7=262.5\) |

Since the cross products are equal, \(15·17.5=37.5·7,\) the equation is a proportion.