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Hướng dẫn học tập > Prealgebra

Writing Proportions

Learning Outcomes

  • Given a statement, write a proportion
  • Given an equation, determine whether it is a 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 ab=cd\frac{a}{b}=\frac{c}{d}, where b≠0,d≠0b\ne 0,d\ne 0. The proportion states two ratios or rates are equal. The proportion is read "a\text{"}a is to bb, as cc is to d".d\text{".}
  The equation 12=48\frac{1}{2}=\frac{4}{8} is a proportion because the two fractions are equal. The proportion 12=48\frac{1}{2}=\frac{4}{8} is read "1\text{"}1 is to 22 as 44 is to 8".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 20 students1 teacher=60 students3 teachers\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: ⓐ 33 is to 77 as 1515 is to 3535. ⓑ 55 hits in 88 at bats is the same as 3030 hits in 4848 at-bats. ⓒ $1.50\text{\$1.50} for 66 ounces is equivalent to $2.25\text{\$2.25} for 99 ounces. Solution
ⓐ
33 is to 77 as 1515 is to 3535.
Write as a proportion. 37=1535\frac{3}{7}=\frac{15}{35}
ⓑ
55 hits in 88 at-bats is the same as 3030 hits in 4848 at-bats.
Write each fraction to compare hits to at-bats. hitsat-bats=hitsat-bats\frac{\text{hits}}{\text{at-bats}}=\frac{\text{hits}}{\text{at-bats}}
Write as a proportion. 58=3048\frac{5}{8}=\frac{30}{48}
ⓒ
$1.50\text{\$1.50} for 66 ounces is equivalent to $2.25\text{\$2.25} for 99 ounces.
Write each fraction to compare dollars to ounces. \frac{$}{\text{ounces}}=\frac{$}{\text{ounces}}
Write as a proportion. 1.506=2.259\frac{1.50}{6}=\frac{2.25}{9}
 

try it

  [ohm_question]146798[/ohm_question] [ohm_question]146799[/ohm_question]
  Look at the proportions 12=48\frac{1}{2}=\frac{4}{8} and 23=69\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. 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 ab=cd\frac{a}{b}=\frac{c}{d}, where b≠0,d≠0b\ne 0,d\ne 0, its cross products are equal. No Alt Text 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: ⓐ 49=1228\frac{4}{9}=\frac{12}{28} ⓑ 17.537.5=715\frac{17.5}{37.5}=\frac{7}{15}

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

ⓐ
49=1228\frac{4}{9}=\frac{12}{28}
Find the cross products. 28⋅4=1129⋅12=10828\cdot 4=1129\cdot 12=108 .
Since the cross products are not equal, 28⋅4≠9⋅1228\cdot 4\ne 9\cdot 12, the equation is not a proportion.
ⓑ
17.537.5=715\frac{17.5}{37.5}=\frac{7}{15}
Find the cross products. 15⋅17.5=262.537.5⋅7=262.515\cdot 17.5=262.537.5\cdot 7=262.5 .
Since the cross products are equal, 15⋅17.5=37.5⋅715\cdot 17.5=37.5\cdot 7, the equation is a proportion.

 

try it

[ohm_question]146809[/ohm_question]
 

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