# 4.2: Ratios and Proportions

- Page ID
- 7092

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Solving ratios and proportions is similar to the previous two exercises. It is simply just looking at the question a little differently. Instead of looking at it as numbers we will assign UNITS (or items) to the numbers, for example, 3 valves to 6 valves or 3 hours to 5 hours. However, the real comparisons of interest will be discussed in the next section when we compare different UNITS to each other, for example, 40 hours to 1 week, or 12 inches to 1 foot.

Example \(\PageIndex{1}\)

\[\dfrac{2}{3}=\dfrac{F}{9}\nonumber \]

You can read this as, 2 is to 3 as F is to 9.

In order to solve the above ratio or proportion is by cross multiplying.

\[9 \times 2=3 \times F \nonumber \]

This would equate to

\[18=3 F \nonumber \]

Then using the method as previously used in the Equation Section, you would divide both sides of the equation by 3. Three would then cancel on the right side of the equation isolating the variable and 18 divided by 3 would give you 6. Therefore, F equals 6.

\[\dfrac{18}{3}=\dfrac{3 F}{3} \quad \rightarrow \quad 6=F \nonumber \]

## Exercise 4.2

- \(\dfrac{D}{7}=\dfrac{27}{9}\)
- \(\dfrac{28}{L}=\dfrac{32}{8}\)
- \(\dfrac{4}{8}=\dfrac{J}{64}\)
- \(\dfrac{H}{5}=\dfrac{28}{70}\)
- \(\dfrac{K}{100}=\dfrac{1}{3}\)
- \(\dfrac{L}{12}=\dfrac{12}{3}\)
- \(\dfrac{9}{4}=\dfrac{K}{12}\)
- \(\dfrac{7}{F}=\dfrac{5}{15}\)
- \(\dfrac{H}{3}=\dfrac{20}{6}\)
- \(\dfrac{2}{5}=\dfrac{Q}{35}\)
- \(\dfrac{13}{F}=\dfrac{32}{64}\)
- \(\dfrac{1}{7}=\dfrac{11}{J}\)

Not all proportional problems are exactly as they seem. The previous problems were directly proportional and can be solved with relative ease. However, sometimes you will encounter indirectly proportional or more properly termed “inversely” proportional.

Example \(\PageIndex{2}\)

It takes 3 employees to flush 8 hydrants in 6 hours. How long would it take 5 employees to do the same job?

**Solution**

If you attempt to solve this problem as if it were directly proportional it would look like.

\[\dfrac{3}{6}=\dfrac{5}{W} \quad \rightarrow \quad W=\dfrac{30}{3} \quad \rightarrow \quad W=10 \nonumber \]

By this result it would take 5 employees 4 hours longer to do the same job.

Inversely proportional problems need to be solved as follows. It is the product of the values not the ratio that you need to equate.

3 employees × 6 hours = 5 employees x W hours

\[\begin{array}{l}

18=5W \\

W=\dfrac{18}{5} \\

W=3.6 \text { hours }

\end{array} \nonumber \]

## Exercise 4.2.1

- A safety catalog sells dust masks. They are $4.50 per dozen. How much would 4 dust masks cost?
- An operator conducted a laboratory experiment by adding 1.5 pounds of chlorine to 5 gallons of water to get a certain chlorine dosage. If the operator wanted to disinfect 1,200 gallons of water to the same dosage, how many pounds of chlorine would she need?
- Six water utility operators were able to exercise 75 valves in one work week. If 9 operators were assigned to do the same task, how long would it take them? (Assume 34 hours equates to a work week)

**In the previous problem, you ended up with a fraction (expressed as a decimal) of an hour. What would be a better way to express this answer?**

- A water utility needs to install 2,375 feet of 16” diameter pipe. The pipe costs $25.80 per foot. How much will all the pipe cost?
- In the problem above, the pipe is manufactured in 20 foot sections. How many sections would need to be purchased?
- A water storage tank needs to be recoated. A contractor gave an estimate that it would take 5 of his employees a total of 39 hours to complete the job. How sooner can the job be completed if 8 employees were to do the work?

In word problems with percentages, the first thing is to convert the percent to a decimal. Then, view the word “of” as a multiplication sign and the word “is” as an equals sign. Then, solve the problem

Example \(\PageIndex{3}\)

10% of 100 is

10% × 100 =

0.10 × 100 = 10

## Exercise 4.2.2

Solve the following percent problems.

- 30 is 20% of what number?
- 10 is 45% of what number?
- 12% of
- What percent of 75 is 225?
- 56% of
- 35 is what percent of 500? _________________
- 100% of 2,000 is ______________
- What percent of 40 is 10? ____________

Word problems give people fits! However, most of the problems that present themselves in practical situations are in fact word problems, we just don’t always think of them that way. For example, if you are buying something from the store and you want to figure out how much the tax will be on a certain item, you probably just multiply the cost of the item by the sales tax. You can though look at this problem as a word problem.

## Exercise 4.2.3

Solve the following word problems.

- A water utility executive earned $85,000 last year and received a 22% bonus. How much was her bonus?
- A worker can paint a fire hydrant in ½ hour. How many hydrants can she paint in 4 hours?
- A 5 gallon jug of bottled water is labeled 60% spring water. How many gallons in the jug is spring water?
- On a state certification treatment exam you must score at least 70% to pass. If there are 65 questions on the test how many must you get correct to pass?