# 8.7 Solve proportion and similar figure applications

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By the end of this section, you will be able to:
• Solve proportions
• Solve similar figure applications

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

1. Solve $\frac{n}{3}=30$ .
If you missed this problem, review [link] .
2. The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side?
If you missed this problem, review [link] .

## Solve proportions

When two rational expressions are equal, the equation relating them is called a proportion .

## Proportion

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

The proportion is read “ $a$ is to $b$ , as $c$ is to $d$ .”

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 “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:

$\frac{1\phantom{\rule{0.2em}{0ex}}\text{teacher}}{20\phantom{\rule{0.2em}{0ex}}\text{students}}=\frac{x\phantom{\rule{0.2em}{0ex}}\text{teachers}}{60\phantom{\rule{0.2em}{0ex}}\text{students}}$

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations . We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

 Multiply both sides by the LCD, 60. Simplify. The principal needs 3 teachers for 60 students.

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

Solve the proportion: $\frac{x}{63}=\frac{4}{7}.$

## Solution

 To isolate $x$ , multiply both sides by the LCD, 63. Simplify. Divide the common factors. Check. To check our answer, we substitute into the original proportion. Show common factors. Simplify.

Solve the proportion: $\frac{n}{84}=\frac{11}{12}.$

$77$

Solve the proportion: $\frac{y}{96}=\frac{13}{12}.$

$104$

When we work with proportion    s, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

Solve the proportion: $\frac{144}{a}=\frac{9}{4}.$

## Solution

 Multiply both sides by the LCD. Remove common factors on each side. Simplify. Divide both sides by 9. Simplify. Check. Show common factors. Simplify.

Solve the proportion: $\frac{91}{b}=\frac{7}{5}.$

$65$

Solve the proportion: $\frac{39}{c}=\frac{13}{8}.$

$24$

Solve the proportion: $\frac{n}{n+14}=\frac{5}{7}.$

## Solution

 Multiply both sides by the LCD. Remove common factors on each side. Simplify. Solve for $n$ . Check. Simplify. Show common factors. Simplify.

Solve the proportion: $\frac{y}{y+55}=\frac{3}{8}.$

$33$

Solve the proportion: $\frac{z}{z-84}=-\frac{1}{5}.$

$14$

#### Questions & Answers

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be? 74.25 × .16 then get the total and that will be your tip David$74.25 x 0.16 = $11.88 total bill:$74.25 + $11.88 =$86.13
ericka
yes and tip 16% will be $11.88 David what is the shorter way to do it Cesar Reply Priam has dimes and pennies in a cup holder in his car. The total value of the coins is$4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup? Cecilia Reply Arnold invested$64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year Heidi Reply List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often. Elbert Reply Avery and Caden have saved$27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in$45 and $25 stock shares. The number of$25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy? Jawad Reply let # of$25 shares be (x) and # of $45 shares be (y) we start with$25x + $45y=475, right? we are told the number of$25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910. David Reply . A cashier has 54 bills, all of which are$10 or $20 bills. The total value of the money is$910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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