Solving Proportions
Learning Outcomes
- Solve a proportion equation
- Solve a proportion application
To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.
example
Solve: [latex]\frac{x}{63}=\frac{4}{7}[/latex]. Solution| [latex]\frac{x}{63}=\frac{4}{7}[/latex] | ||
| To isolate [latex]x[/latex] , multiply both sides by the LCD, [latex]63[/latex]. | [latex]\color{red}{63}(\frac{x}{63})=\color{red}{63}(\frac{4}{7})[/latex] | |
| Simplify. | [latex]x=\frac{9\cdot\color{red}{7}\cdot4}{\color{red}{7}}[/latex] | |
| Divide the common factors. | [latex]x=36[/latex] | |
| Check: To check our answer, we substitute into the original proportion. | ||
| [latex]\frac{x}{63}=\frac{4}{7}[/latex] | ||
| Substitute [latex]x=\color{red}{36}[/latex] | [latex]\frac{\color{red}{36}}{63}\stackrel{?}{=}\frac{4}{7}[/latex] | |
| Show common factors. | [latex]\frac{4\cdot9}{7\cdot9}\stackrel{?}{=}\frac{4}{7}[/latex] | |
| Simplify. | [latex]\frac{4}{7}=\frac{4}{7}[/latex] | |
try it
[ohm_question]146811[/ohm_question]example
Solve: [latex]\frac{144}{a}=\frac{9}{4}[/latex].Answer: Solution Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.
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| Find the cross products and set them equal. | [latex]4\cdot144=a\cdot9[/latex] | |
| Simplify. | [latex]576=9a[/latex] | |
| Divide both sides by [latex]9[/latex]. | [latex]\frac{576}{9}=\frac{9a}{9}[/latex] | |
| Simplify. | [latex]64=a[/latex] | |
| Check your answer. | ||
| [latex]\frac{144}{a}=\frac{9}{4}[/latex] | ||
| Substitute [latex]a=\color{red}{64}[/latex] | [latex]\frac{144}{\color{red}{64}}\stackrel{?}{=}\frac{9}{4}[/latex] | |
| Show common factors.. | [latex]\frac{9\cdot16}{4\cdot16}\stackrel{?}{=}\frac{9}{4}[/latex] | |
| Simplify. | [latex]\frac{9}{4}=\frac{9}{4}\quad\checkmark[/latex] | |
try it
[ohm_question]146813[/ohm_question]example
Solve: [latex]\frac{52}{91}=\frac{-4}{y}[/latex].Answer: Solution
| Find the cross products and set them equal. |  |
|
| [latex]y\cdot52=91(-4)[/latex] | ||
| Simplify. | [latex]52y=-364[/latex] | |
| Divide both sides by [latex]52[/latex]. | [latex]\frac{52y}{52}=\frac{-364}{52}[/latex] | |
| Simplify. | [latex]y=-7[/latex] | |
| Check: | ||
| [latex]\frac{52}{91}=\frac{-4}{y}[/latex] | ||
| Substitute [latex]y=\color{red}{-7}[/latex] | [latex]\frac{52}{91}\stackrel{?}{=}\frac{-4}{\color{red}{-7}}[/latex] | |
| Show common factors. | [latex]\frac{13\cdot4}{13\cdot4}\stackrel{?}{=}\frac{-4}{\color{red}{-7}}[/latex] | |
| Simplify. | [latex]\frac{4}{7}=\frac{4}{7}\quad\checkmark[/latex] | |
try it
[ohm_question]146814[/ohm_question]Solve Applications Using Proportions
The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.example
When pediatricians prescribe acetaminophen to children, they prescribe [latex]5[/latex] milliliters (ml) of acetaminophen for every [latex]25[/latex] pounds of the child’s weight. If Zoe weighs [latex]80[/latex] pounds, how many milliliters of acetaminophen will her doctor prescribe?Answer: Solution
| Identify what you are asked to find. | How many ml of acetaminophen the doctor will prescribe |
| Choose a variable to represent it. | Let [latex]a=[/latex] ml of acetaminophen. |
| Write a sentence that gives the information to find it. | If [latex]5[/latex] ml is prescribed for every [latex]25[/latex] pounds, how much will be prescribed for [latex]80[/latex] pounds? |
| Translate into a proportion. |  |
| Substitute given values—be careful of the units. | [latex]\frac{5}{25}=\frac{a}{80}[/latex] |
| Multiply both sides by [latex]80[/latex]. | [latex]80\cdot\frac{5}{25}=80\cdot\frac{a}{80}[/latex] |
| Multiply and show common factors. | [latex]\frac{16\cdot5\cdot5}{5\cdot5}=\frac{80a}{80}[/latex] |
| Simplify. | [latex]16=a[/latex] |
| Check if the answer is reasonable. | |
| Yes. Since [latex]80[/latex] is about [latex]3[/latex] times [latex]25[/latex], the medicine should be about [latex]3[/latex] times [latex]5[/latex]. | |
| Write a complete sentence. | The pediatrician would prescribe [latex]16[/latex] ml of acetaminophen to Zoe. |
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[ohm_question]146816[/ohm_question]example
One brand of microwave popcorn has [latex]120[/latex] calories per serving. A whole bag of this popcorn has [latex]3.5[/latex] servings. How many calories are in a whole bag of this microwave popcorn?Answer: Solution
| Identify what you are asked to find. | How many calories are in a whole bag of microwave popcorn? |
| Choose a variable to represent it. | Let [latex]c=[/latex] number of calories. |
| Write a sentence that gives the information to find it. | If there are [latex]120[/latex] calories per serving, how many calories are in a whole bag with [latex]3.5[/latex] servings? |
| Translate into a proportion. |  |
| Substitute given values. | [latex]\frac{120}{1}=\frac{c}{3.5}[/latex] |
| Multiply both sides by [latex]3.5[/latex]. | [latex](3.5)(\frac{120}{1})=(3.5)(\frac{c}{3.5})[/latex] |
| Multiply. | [latex]420=c[/latex] |
| Check if the answer is reasonable. | |
| Yes. Since [latex]3.5[/latex] is between [latex]3[/latex] and [latex]4[/latex], the total calories should be between [latex]360 (3⋅120)[/latex] and [latex]480 (4⋅120)[/latex]. | |
| Write a complete sentence. | The whole bag of microwave popcorn has [latex]420[/latex] calories. |
try it
[ohm_question]146817[/ohm_question]example
Josiah went to Mexico for spring break and changed $[latex]325[/latex] dollars into Mexican pesos. At that time, the exchange rate had $[latex]1[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos. How many Mexican pesos did he get for his trip?Answer: Solution
| Identify what you are asked to find. | How many Mexican pesos did Josiah get? |
| Choose a variable to represent it. | Let [latex]p=[/latex] number of pesos. |
| Write a sentence that gives the information to find it. | If [latex]\text{\$1}[/latex] U.S. is equal to [latex]12.54[/latex] Mexican pesos, then [latex]\text{\$325}[/latex] is how many pesos? |
| Translate into a proportion. |  |
| Substitute given values. | [latex]\frac{1}{12.54}=\frac{325}{p}[/latex] |
| The variable is in the denominator, so find the cross products and set them equal. | [latex]p\cdot{1}=12.54(325)[/latex] |
| Simplify. | [latex]c=4,075.5[/latex] |
| Check if the answer is reasonable. | |
| Yes, [latex]\text{\$100}[/latex] would be [latex]\text{\$1,254}[/latex] pesos. [latex]\text{\$325}[/latex] is a little more than [latex]3[/latex] times this amount. | |
| Write a complete sentence. | Josiah has [latex]4075.5[/latex] pesos for his spring break trip. |
try it
[ohm_question]146819[/ohm_question]Licenses & Attributions
CC licensed content, Original
- Question ID 146819, 146818, 146817. Authored by: Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Ex: Solve a Proportion by Clearing Fractions (x/a=b/c, Whole Num Solution). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Ex: Solve a Proportion by Clearing Fractions ((a/x=b/c, Fraction Solution). Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Examples: Applications Using Proportions. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].
