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By the end of this section, you will be able to:

  • Use the definition of proportion
  • Solve proportions
  • Solve applications using proportions
  • Write percent equations as proportions
  • Translate and solve percent proportions

Before you get started, take this readiness quiz.

  1. Simplify: 1 3 4 .
    If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .
  2. Solve: x 4 = 20 .
    If you missed this problem, review Solve Equations with Fractions .
  3. Write as a rate: Sale rode his bike 24 miles in 2 hours.
    If you missed this problem, review Ratios and Rate .

Use the definition of proportion

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    .

Proportion

A proportion is an equation of the form a b = c d , where b 0 , d 0 .

The proportion states two ratios or rates are equal. The proportion is read a is to b , as c is to d ”.

The equation 1 2 = 4 8 is a proportion because the two fractions are equal. The proportion 1 2 = 4 8 is read 1 is to 2 as 4 is to 8 ”.

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

Write each sentence as a proportion:

  1. 3 is to 7 as 15 is to 35 .
  2. 5 hits in 8 at bats is the same as 30 hits in 48 at-bats.
  3. $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.

Solution

3 is to 7 as 15 is to 35.
Write as a proportion. 3 7 = 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. hits at-bats = hits at-bats
Write as a proportion. 5 8 = 30 48
$1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.
Write each fraction to compare dollars to ounces. $ ounces = $ ounces
Write as a proportion. 1.50 6 = 2.25 9
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Write each sentence as a proportion:

  1. 5 is to 9 as 20 is to 36 .
  2. 7 hits in 11 at-bats is the same as 28 hits in 44 at-bats.
  3. $2.50 for 8 ounces is equivalent to $3.75 for 12 ounces.

  1. 5 9 = 20 36
  2. 7 11 = 28 44
  3. 2.50 8 = 3.75 12

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Write each sentence as a proportion:

  1. 6 is to 7 as 36 is to 42 .
  2. 8 adults for 36 children is the same as 12 adults for 54 children.
  3. $3.75 for 6 ounces is equivalent to $2.50 for 4 ounces.

  1. 6 7 = 36 42
  2. 8 36 = 12 54
  3. 3.75 6 = 2.50 4

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Look at the proportions 1 2 = 4 8 and 2 3 = 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.

The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 · 1 = 8 and 2 · 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 · 2 = 18 and 3 · 6 = 18.

Cross products of a proportion

For any proportion of the form a b = c d , where b 0 , d 0 , its cross products are equal.

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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.

Practice Key Terms 3

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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