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Hướng dẫn học tập > Prealgebra

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: x63=47\frac{x}{63}=\frac{4}{7}. Solution
x63=47\frac{x}{63}=\frac{4}{7}
To isolate xx , multiply both sides by the LCD, 6363. 63(x63)=63(47)\color{red}{63}(\frac{x}{63})=\color{red}{63}(\frac{4}{7})
Simplify. x=9⋅7⋅47x=\frac{9\cdot\color{red}{7}\cdot4}{\color{red}{7}}
Divide the common factors. x=36x=36
Check: To check our answer, we substitute into the original proportion.
x63=47\frac{x}{63}=\frac{4}{7}
Substitute x=36x=\color{red}{36} 3663=?47\frac{\color{red}{36}}{63}\stackrel{?}{=}\frac{4}{7}
Show common factors. 4⋅97⋅9=?47\frac{4\cdot9}{7\cdot9}\stackrel{?}{=}\frac{4}{7}
Simplify. 47=47\frac{4}{7}=\frac{4}{7}
 

try it

[ohm_question]146811[/ohm_question]
In the next video we show another example of how to solve a proportion equation using the LCD. https://youtu.be/pXvzpSU4DyU When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions. We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.  

example

Solve: 144a=94\frac{144}{a}=\frac{9}{4}.

Answer: Solution Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

.
Find the cross products and set them equal. 4⋅144=a⋅94\cdot144=a\cdot9
Simplify. 576=9a576=9a
Divide both sides by 99. 5769=9a9\frac{576}{9}=\frac{9a}{9}
Simplify. 64=a64=a
Check your answer.
144a=94\frac{144}{a}=\frac{9}{4}
Substitute a=64a=\color{red}{64} 14464=?94\frac{144}{\color{red}{64}}\stackrel{?}{=}\frac{9}{4}
Show common factors.. 9⋅164⋅16=?94\frac{9\cdot16}{4\cdot16}\stackrel{?}{=}\frac{9}{4}
Simplify. 94=94✓\frac{9}{4}=\frac{9}{4}\quad\checkmark

  Another method to solve this would be to multiply both sides by the LCD, 4a4a. Try it and verify that you get the same solution. The following video shows an example of how to solve a similar problem by using the LCD. https://youtu.be/zrgLddU8pFU

try it

[ohm_question]146813[/ohm_question]
   

example

Solve: 5291=−4y\frac{52}{91}=\frac{-4}{y}.

Answer: Solution

Find the cross products and set them equal. .
y⋅52=91(−4)y\cdot52=91(-4)
Simplify. 52y=−36452y=-364
Divide both sides by 5252. 52y52=−36452\frac{52y}{52}=\frac{-364}{52}
Simplify. y=−7y=-7
Check:
5291=−4y\frac{52}{91}=\frac{-4}{y}
Substitute y=−7y=\color{red}{-7} 5291=?−4−7\frac{52}{91}\stackrel{?}{=}\frac{-4}{\color{red}{-7}}
Show common factors. 13⋅413⋅4=?−4−7\frac{13\cdot4}{13\cdot4}\stackrel{?}{=}\frac{-4}{\color{red}{-7}}
Simplify. 47=47✓\frac{4}{7}=\frac{4}{7}\quad\checkmark

 

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 55 milliliters (ml) of acetaminophen for every 2525 pounds of the child’s weight. If Zoe weighs 8080 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 a=a= ml of acetaminophen.
Write a sentence that gives the information to find it. If 55 ml is prescribed for every 2525 pounds, how much will be prescribed for 8080 pounds?
Translate into a proportion. .
Substitute given values—be careful of the units. 525=a80\frac{5}{25}=\frac{a}{80}
Multiply both sides by 8080. 80⋅525=80⋅a8080\cdot\frac{5}{25}=80\cdot\frac{a}{80}
Multiply and show common factors. 16⋅5⋅55⋅5=80a80\frac{16\cdot5\cdot5}{5\cdot5}=\frac{80a}{80}
Simplify. 16=a16=a
Check if the answer is reasonable.
Yes. Since 8080 is about 33 times 2525, the medicine should be about 33 times 55.
Write a complete sentence. The pediatrician would prescribe 1616 ml of acetaminophen to Zoe.

  You could also solve this proportion by setting the cross products equal.

try it

[ohm_question]146816[/ohm_question]
   

example

One brand of microwave popcorn has 120120 calories per serving. A whole bag of this popcorn has 3.53.5 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 c=c= number of calories.
Write a sentence that gives the information to find it. If there are 120120 calories per serving, how many calories are in a whole bag with 3.53.5 servings?
Translate into a proportion. .
Substitute given values. 1201=c3.5\frac{120}{1}=\frac{c}{3.5}
Multiply both sides by 3.53.5. (3.5)(1201)=(3.5)(c3.5)(3.5)(\frac{120}{1})=(3.5)(\frac{c}{3.5})
Multiply. 420=c420=c
Check if the answer is reasonable.
Yes. Since 3.53.5 is between 33 and 44, the total calories should be between 360(3⋅120)360 (3⋅120) and 480(4⋅120)480 (4⋅120).
Write a complete sentence. The whole bag of microwave popcorn has 420420 calories.

 

try it

[ohm_question]146817[/ohm_question]
   

example

Josiah went to Mexico for spring break and changed $325325 dollars into Mexican pesos. At that time, the exchange rate had $11 U.S. is equal to 12.5412.54 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 p=p= number of pesos.
Write a sentence that gives the information to find it. If $1\text{\$1} U.S. is equal to 12.5412.54 Mexican pesos, then $325\text{\$325} is how many pesos?
Translate into a proportion. .
Substitute given values. 112.54=325p\frac{1}{12.54}=\frac{325}{p}
The variable is in the denominator, so find the cross products and set them equal. p⋅1=12.54(325)p\cdot{1}=12.54(325)
Simplify. c=4,075.5c=4,075.5
Check if the answer is reasonable.
Yes, $100\text{\$100} would be $1,254\text{\$1,254} pesos. $325\text{\$325} is a little more than 33 times this amount.
Write a complete sentence. Josiah has 4075.54075.5 pesos for his spring break trip.

 

try it

[ohm_question]146819[/ohm_question]
In the following video we show another example of how to solve an application that involves proportions. https://youtu.be/vnB1mh5X5cA

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