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Section 3.2 Ratio and Proportion

Subsection Ratios

A ratio is a type of quotient used to compare two numerical quantities.

For example, suppose the ratio of pupils to computers at a local elementary school is 10 to 3. This means that for every 10 pupils in the school there are 3 computers. We often display a ratio as a fraction, like this:

\begin{equation*} \dfrac{\text{Number of pupils}}{\text{Number of computers}}=\dfrac{10}{3} \end{equation*}
Caution 3.4.

This ratio does not mean that there are exactly 10 pupils and 3 computers at the school. There could be 40 pupils and 12 computers or 200 pupils and 60 computers (or many other combinations) because

\begin{equation*} \dfrac{40}{12}=\dfrac{200}{60}=\dfrac{10}{3} \end{equation*}
Example 3.5.

In a survey of 100 of his classmates, Greg found that 68 students supported a strict gun control law, while 32 students did not. What is the ratio of survey respondents who support gun control to those opposed?

Solution.

The ratio of those supporting gun control to those opposed is 68 to 32, or \(\dfrac{68}{32}\text{.}\) This ratio reduces to \(\dfrac{17}{8}\text{,}\) or 17 to 8.

Caution 3.6.

When we write a ratio, it is important to notice which number is the numerator, and which number is the denominator. When we write "the ratio of \(a\) to \(b\)" as a fraction, \(a\) is the numerator and \(b\) is the denominator.

In Example 3.5, we could also compute the ratio of opponents to supporters of the gun control law. In that case, the answer would be \(\dfrac{32}{68}\) or \(\dfrac{8}{17}\text{,}\) the reciprocal of the answer to Example 1.

Ratio.

A ratio is a type of quotient used to compare two numerical quantities. The ratio of \(a\) to \(b\) is written \(\dfrac{a}{b}\text{.}\)

Look Closer.

A ratio can be expressed as a decimal instead of a common fraction. Because \(\dfrac{17}{8}\) is equal to 2.125, we can write the ratio in Example 3.5 as 2.125 to 1. We usually express a ratio as a decimal fraction if the numbers being compared are decimal numbers.

Example 3.7.

A small circuit board measures 4.2 centimeters long by 2.4 centimeters wide. What is the ratio of its length to its width?

Solution.

The ratio of length to width is \(\dfrac{4.2~ \text{cm}}{2.4~ \text{cm}}\text{,}\) or 1.75, as you can verify with your calculator. (Can you express this ratio as a common fraction?)

Reading Questions Reading Questions

1.

What is a ratio?

Answer.

A type of quotient comparing two quantities

2.

When do we express a ratio as a decimal fraction?

Answer.

When the numbers being compared are decimal numbers

Subsection Rates

Rate.

A rate is a ratio that compares two quantities with different units.

You are already familiar with many rates. If you say that your average speed was 50 miles per hour, or that apples cost 89 cents per pound, you are using rates. Because per usually indicates division,

\begin{gather*} \text{50 miles per hour}~~~~~~\text{means}~~~~~~\dfrac{50~ \text{miles}}{1~ \text{hour}}\\ \text{89 cents per pound}~~~~~~\text{means}~~~~~~\dfrac{89~ \text{cents}}{1~ \text{pound}} \end{gather*}
Example 3.8.

Sarah traveled 390 miles on 15 gallons of gas. Express her rate of fuel consumption as a ratio, and then as a rate.

Solution.

As a ratio, Sarah's rate of fuel consumption was \(\dfrac{390~ \text{miles}}{15~ \text{gallons}}\text{.}\)

We simplify by dividing the denominator into the numerator, to get \(\dfrac{26~ \text{miles}}{1~ \text{gallon}}\text{,}\) or 26 miles per gallon.

Subsection Proportions

Proportion.

A proportion is a statement that two ratios are equal.

Thus, a proportion is an equation in which both sides are ratios. Here are some proportions:

\begin{equation*} \dfrac{6}{9}=\dfrac{8}{12}~~~~~~\text{and}~~~~~~\dfrac{3.2}{8}=\dfrac{1}{2.5} \end{equation*}

A proportion may involve variables. How can we solve a proportion such as this one?

\begin{equation*} \dfrac{7}{5}=\dfrac{x}{6} \end{equation*}

Because \(x\) is divided by 6, we could multiply both sides of the equation by 6 to clear the fraction. But we can clear both fractions at the same time if we multiply by the LCD (Lowest Common Denominator) of the two fractions. In this case, the LCD is \(5 \cdot 6 = 30\text{.}\) Multiplying both sides by 30 gives us

\begin{align*} \blert{30}\left(\dfrac{7}{5}\right) \amp =\left(\dfrac{x}{6}\right)\blert{30}\\ 42 \amp = 5x \end{align*}

Now we can divide both sides by 5 to get the answer,

\begin{equation*} x=\dfrac{42}{5}=8.4 \end{equation*}
Look Ahead.

There is a short-cut we can use that avoids calculating an LCD. We can obtain the equation \(42=5x\) by cross-multiplying:

cross-multiplying
Then we complete the solution as before.

Property of Proportions.

We can clear the fractions from a proportion by cross-multiplying.

\begin{equation*} \blert{\text{If}~~\dfrac{a}{b}=\dfrac{c}{d},~~~\text{then}~~~ad=bc.} \end{equation*}
Example 3.9.

Solve \(~~\dfrac{2.4}{1.5}=\dfrac{8.4}{x}\)

Solution.

We apply the property of proportions and cross-multiply to get

\begin{align*} 2.4x \amp = 1.5(8.4) \amp \amp \blert{\text{Divide both sides by 2.4.}}\\ x \amp = \dfrac{1.5(8.4)}{2.4} \amp \amp \blert{\text{Simplify the right side.}}\\ x \amp = 5.25 \end{align*}

We can check the solution by substituting \(x=5.25\) into the original proportion.

Caution 3.10.

Cross-mutiplying works only for solving proportions! Do not try to use cross-multiplying on other types of equations, or for other operations on fractions. In particular, do not use cross-multiplying when multiplying fractions or adding fractions.

Reading Questions Reading Questions

3.

What is a rate?

Answer.

A ratio that compares quantities with different units

4.

What is a proportion?

Answer.

A statement that two ratios are equal

5.

What is the short-cut for solving a proportion called?

Answer.

Cross-multiplying

Subsection Proportional Variables

Suppose that grape juice costs 80 cents per quart, and complete the table.

Number of Quarts Total Price (cents) \(\dfrac{\text{Total Price}}{\text{Number of Quarts}}\)
\(1\) \(80\) \(\dfrac{80}{1}=80\)
\(2\) \(160\) \(\dfrac{160}{2}=\)
\(3\) \(240\) \(\hphantom{0000}\)
\(4\) \(320\) \(\hphantom{0000}\)

Did you find that the ratio \(\dfrac{\text{total price}}{\text{number of quarts}}\text{,}\) or price per quart, is the same for each row of the table? This agrees with common sense: The price per quart of grape juice is the same no matter how many quarts you buy.

Proportional Variables.

Two variables are proportional if their ratio is always the same.

Example 3.11.

One quart of grape juice costs 80 cents. The price of grape juice is proportional to the number of quarts you buy. How much grape juice can you buy for $10.00?

Solution.

Let \(x\) represent the number of quarts you can buy for $10.00, or 1000 cents. Because the variables total price and number of quarts are proportional, we know that their ratio is constant. Thus, the ratio

\begin{gather*} \dfrac{\text{1000 cents}}{x~\text{quarts}}~~~~~~\text{is equal to the ratio}~~~~~~\dfrac{\text{80 cents}}{\text{1 quart}}\\ \dfrac{1000}{x}=\dfrac{80}{1} \end{gather*}

We solve the proportion by cross-multiplying to get

\begin{align*} 1000(1) \amp = 80x \amp \amp \blert{\text{Divide both sides by 80.}}\\ x \amp = \dfrac{1000}{80}=12.5 \end{align*}

You can buy 12.5 quarts of grape juice for $10.00. (Or, if you can only buy quart bottles, then you can buy 12 quarts and have 40 cents left over.)

Caution 3.12.

When writing a proportion, we must be careful that both ratios have the same units in their numerators, and the same units in their denominators. In the example above, it would not be correct to equate \(\dfrac{1000}{x}\) and \(\dfrac{1}{80}\text{,}\) because the ratios do not have the same units.

For the grape juice example, the ratio \(\dfrac{\text{total price}}{\text{number of quarts}}\) was always equal to 80. The number 80 is called the constant of proportionality.

Look Closer.

If we let \(P\) stand for the price of the grape juice and \(q\) stand for the number of quarts purchased, we have the equation

\begin{equation*} \begin{aligned} \dfrac{P}{q} \amp = 80\\ \text{or}~~~~P \amp = 80q \end{aligned} \end{equation*}

The graph of this equation is a straight line that passes through the origin, as shown below.

graph of line
Equation for Proportional Variables.

If two variables are proportional, they are related by the equation

\begin{equation*} y=kx \end{equation*}

where \(k\) is the constant of proportionality.

Reading Questions Reading Questions

6.

What are proportional variables?

Answer.

Variables whose ratio is always the same

7.

If and are proportional, what equation do they satisfy?

Answer.

\(y=kx\)

Subsection Skills Warm-Up

Exercises Exercises

Exercise Group.

Convert each common fraction to a decimal, and each decimal to a common fraction.

1.
\(\dfrac{3}{8}\)
2.
\(\dfrac{7}{5}\)
3.
\(\dfrac{4}{3}\)
4.
\(\dfrac{5}{12}\)
5.
\(0.04\)
6.
\(0.\overline{6}\)
7.
\(1.875\)
8.
\(2.2\)

Subsubsection Answers to Skills Warm-Up

  1. \(\displaystyle 0.375\)

  2. \(\displaystyle 1.4\)

  3. \(\displaystyle 1.\overline{3}\)

  4. \(\displaystyle 0.41\overline{6}\)

  5. \(\displaystyle \dfrac{1}{25}\)

  6. \(\displaystyle \dfrac{2}{3}\)

  7. \(\displaystyle \dfrac{15}{8}\)

  8. \(\displaystyle \dfrac{11}{5}\)

Subsection Lesson

Subsubsection Activity 1: Ratios and Rates

Exercises Exercises
1.

On a Saturday evening, a restaurant found that 84 of its customers asked for the nonsmoking section, and 35 customers preferred the smoking section.

  1. What was the ratio of smokers to nonsmokers?

  2. Write the ratio in simplest form.

2.

Major Motors budgeted $5.6 million for research and development (R&D) next year and $3.5 million for advertising. What is the ratio of the amount budgeted for advertising to the amount budgeted for R&D

3.

Bita made $240 for 25 hours of work last week. Express her rate of pay as a ratio, and then as a rate.

Subsubsection Activity 2: Proportions

Exercises Exercises
1.

Solve \(~~\dfrac{q}{32} = \dfrac{1.25}{4}\)

2.

If Sarah can drive 390 miles on 15 gallons of gas, how much gas will she need to travel 800 miles?

\(\blert{\text{Assign a variable:}}\)

\(\blert{\text{Write a proportion:}}\)

\(\blert{\text{Solve the proportion:}}\)

\(\blert{\text{Solution:}}\)

Subsubsection Activity 3: Proportional Variables

Exercises Exercises
1.

The table below shows the price, \(p\text{,}\) for \(q\) quarts of SereniTea. Plot the data on the grid. Then compute the ratio \(\dfrac{\text{Price}}{\text{Quarts}}\) for each data point.

Quarts Total Price \(\dfrac{\text{Price}}{\text{Quarts}}\)
\(4\) $6.00 \(\dfrac{6.00}{4} = ?\)
\(6\) $9.00 \(\dfrac{9.00}{6} = ?\)
\(9\) $13.50 \(\hphantom{0000}\)
\(12\) $18.00 \(\hphantom{0000}\)
\(15\) $22.50 \(\hphantom{0000}\)
triangle
2.

The table below shows the population, \(P\text{,}\) of a new suburb \(t\) years after it was built. Plot the data on the grid. Then compute the ratio \(\dfrac{\text{People}}{\text{Year}}\) for each data point.

Years Population \(\dfrac{\text{People}}{\text{Year}}\)
\(1\) 10 \(\dfrac{10}{1} = ?\)
\(2\) 20 \(\dfrac{20}{2} = ?\)
\(3\) 40 \(\hphantom{0000}\)
\(4\) 80 \(\hphantom{0000}\)
\(5\) 160 \(\hphantom{0000}\)
triangle
3.

At this point, can you make a conjecture (educated guess) about the graphs of proportional variables? To help you decide if your conjecture is true, continue with the graphs in parts 4 and 5.

\(\blert{\text{Conjecture:}}\)

4.

Tuition at Woodrow University is $400 plus $30 per unit.

  1. Write an equation for tuition, \(T\text{,}\) in terms of the number of units, \(u\text{.}\)

    \begin{equation*} T=\hphantom{0000} \end{equation*}

  2. Use your equation to fill in the second column of the table.

    Units Tuition \(\dfrac{\text{Tuition}}{\text{Units}}\)
    \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(5\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(8\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(10\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(12\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. Graph the equation on the grid.

  4. Are the variables proportional? Compute their ratios to decide.

grid
5.

Anouk is traveling by train across Alaska at 60 miles per hour.

  1. Write an equation for the distance, \(D\text{,}\) Anouk has traveled in terms of hours, \(h\text{.}\)

    \begin{equation*} D=\hphantom{0000} \end{equation*}

  2. Use your equation to fill in the second column of the table.

    Hours Distance \(\dfrac{\text{Distance}}{\text{Hours}}\)
    \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(5\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(8\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(10\) \(\hphantom{0000}\) \(\hphantom{0000}\)
    \(16\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. Graph the equation on the grid.

  4. Are the variables proportional? Compute their ratios to decide.

grid
6.

Revise your conjecture about the graphs of proportional variables so that it applies to the graphs on this page as well.

Hint: Look at the two graphs of proportional variables. What is the \(y\)-intercept of both graphs?

\(\blert{\text{Revised Conjecture:}}\)

Subsubsection Wrap-Up

Objectives.

In this Lesson we practiced the following skills:

  • Comparing two quantities with a ratio or rate

  • Solving a proportion

  • Writing a proportion to model a problem

  • Recognizing proportional variables

Questions.

  1. In Activity 1, Problem 2, did you express the ratio as a common fraction or as a decimal fraction? Why?

  2. In Activity 2, Problem 3, did you use miles or gallons in the numerators of your ratios? Could you have used the other unit?

  3. In Activity 3, what did you learn about the graphs of proportional variables?

Subsection Homework Preview

Exercises Exercises

1.

A cake recipe asks for \(2\frac{1}{2}\) cups of flour and \(\frac{3}{4}\) cup of brown sugar. What is the ratio of flour to brown sugar?

2.

Decide whether the two variables are proportional:

\(x\) \(2\) \(5\) \(10\) \(16\)
\(y\) \(3.2\) \(8\) \(16\) \(25.6\)
3.

Write and solve a proportion to answer the question:

On a map of Chesterfield County, 1 inch represents \(4\frac{1}{2}\) miles. If Richmond and Petersburg are \(6\frac{2}{3}\) inches apart on the map, what is the actual distance between the two towns?

Subsubsection Answers to Homework Preview

  1. \(\displaystyle \dfrac{10}{3}\)

  2. Yes

  3. \(\dfrac{1}{9/2} = \dfrac{20/3}{x};~~\) 30 mi

Exercises Homework 3.2

Exercise Group.

For Problems 1–4, write ratios or rates.

1.

In a survey of 300 employees at a large company, 125 used public transportation to commute to work. What is the ratio of employees who use public transportation to those who do not?

2.

Erica tutored in the math lab for 6.5 hours last week and made $56.68. What is Erica's rate of pay in dollars per hour?

3.

The instructions for mixing a rose fertilizer call for \(\dfrac{3}{4}\) cup of potash and \(1\dfrac{3}{8}\) cups of nitrogen. What is the ratio of nitrogen to potash?

4.

An orange contains 0.14 milligram of thiamin and 0.6 milligram of niacin. What is the ratio of niacin to thiamin?

5.

Which of the following expressions are proportions?

  1. \(\displaystyle \dfrac{3}{x}=\dfrac{8}{15}\)

  2. \(\displaystyle \dfrac{v}{9}-\dfrac{12}{5}\)

  3. \(\displaystyle \dfrac{w}{6}+\dfrac{2}{5}=\dfrac{3}{2}\)

  4. \(\displaystyle \dfrac{2.6}{m}=\dfrac{m}{1.6}\)

Exercise Group.

For Problems 6–9, solve the proportion.

6.
\(\dfrac{x}{16}=\dfrac{9}{24}\)
7.
\(\dfrac{182}{65}=\dfrac{21}{w}\)
8.
\(\dfrac{7}{b}=\dfrac{5}{9}\)
9.
\(\dfrac{7/3}{3}=\dfrac{z}{9/2}\)
Exercise Group.

For Problems 10–12, decide whether the two variables are proportional.

10.

Time Distance
1 45
2 90
4 180
5 225

11.

Length Area
3 9
4 16
8 64
10 100

12.

Rate Time
20 40
40 20
50 16
80 10

Exercise Group.

For Problems 13–16, decide whether the two quantities are proportional.

13.

A 9-inch pizza costs $6, and a 12-inch pizza costs $10.

14.

A commuter train travels 10 miles in 20 minutes, and 15 miles in 30 minutes.

15.

It takes \(4\dfrac{1}{2}\) cups of flour to make two loaves of bread, and 18 cups to make eight loaves.

16.

An 18-foot sailboat sleeps four, and a 32-foot sailboat sleeps six.

Exercise Group.

For Problems 17–22,

  1. Write a proportion.

  2. Solve the proportion and answer the question.

17.

If 3 pounds of coffee makes 225 cups, how many pounds of coffee is needed to make 3000 cups of coffee?

18.

Gunther's car uses 32 liters of gas to travel 184 kilometers. How many liters will he need to travel 575 kilometers?

19.

On a map of Fairfield County, \(\frac{3}{4} \) centimeter represents a distance of 10 kilometers. If Eastlake and Kenwood are 6 centimeters apart on the map, what is the actual distance between the two towns?

20.

In a survey of 1200 voters, 863 favored the construction of a light rail system. If 8000 people vote, about how many will vote for the light rail system?

21.

If 1 inch equals 2.54 centimeters, what is the length in inches of a wire 35 centimeters long?

22.

A cinnamon bread recipe calls for 1\(\frac{1}{4} \) tablespoons of cinnamon and 5 cups of flour. How much cinnamon would be needed with 8 cups of flour?

Exercise Group.

For Problems 23–24,

  1. Find the constant of proportionality and write an equation relating the variables.

  2. Make a table of values and graph the equation.

  3. Use your equation to answer the question.

23.

Megan's wages, \(w\text{,}\) are proportional to the time, \(h\text{,}\) she works. Megan made $60 for 8 hours work as a lab assistant. How long will it take her to make $500?

\(h\) \(w\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
grid
24.

The amount of pure copper, \(c\text{,}\) produced by Copperfield Mine is proportional to the amount of ore extracted, \(g\text{.}\) A new lode produced 24 grams of copper from 800 grams of ore. How many grams of copper will one kilogram of ore yield?

\(g\) \(c\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
\(\hphantom{0000}\) \(\hphantom{0000}\)
grid
25.

Trish's real-estate commission is proportional to the amount of the sale. She made $3200 commission on a sale of $80,000.

  1. Write an equation relating Trish's sales and her commission.

  2. What would her commission be on the sale of a $200,000 property?

26.

Everett can bicycle 16 miles in 2 hours and 24 miles in 3 hours. The distance \(d\) that Everett travels is proportional to the time \(t\) that he bicycles.

  1. Write an equation for \(d\) in terms of \(t\text{.}\)

  2. Graph your equation.
    grid
  3. Complete the tables, and use them to answer the question below.
    \(t\) \(d\)
    \(2\) \(\hphantom{0000}\)
    \(3\) \(\hphantom{0000}\)
    \(5\) \(\hphantom{0000}\)
    \(t\) \(d\)
    \(4\) \(\hphantom{0000}\)
    \(6\) \(\hphantom{0000}\)
    \(10\) \(\hphantom{0000}\)
    What happens to the distance Everett travels when he doubles the time he bicycles?
27.

The sales tax on $12 is 72 cents, and the sales tax on $15 is 90 cents. The tax \(T\) is proportional to the price \(P\text{.}\)

  1. Write an equation for \(T\) in terms of \(P\text{.}\)

  2. Graph your equation.

    grid

  3. Complete the tables, and use them to answer the question below.

    \(P\) \(T\)
    \(12\) \(\hphantom{0000}\)
    \(20\) \(\hphantom{0000}\)
    \(30\) \(\hphantom{0000}\)
    \(P\) \(T\)
    \(6\) \(\hphantom{0000}\)
    \(10\) \(\hphantom{0000}\)
    \(15\) \(\hphantom{0000}\)

    What happens to the sales tax when the price is cut in half?

28.

The cost of tuition at Walden College is given by the formula \(T=500+40u\text{,}\) where \(u\) is the number of units you take.

  1. Fill in the tables.

    Units Tuition
    \(3\) \(\hphantom{0000}\)
    \(5\) \(\hphantom{0000}\)
    \(8\) \(\hphantom{0000}\)
    Units Tuition
    \(6\) \(\hphantom{0000}\)
    \(10\) \(\hphantom{0000}\)
    \(16\) \(\hphantom{0000}\)

  2. Does doubling the number of units you take double your tuition?

  3. Is \(T\) proportional to \(u\text{?}\)

29.

  1. State a formula for the circumference of a circle in terms of its radius.

  2. Complete the table.

    \(r\) \(2\) \(5\) \(7\) \(10\)
    \(C\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. Graph your equation, using the values in your table.

    grid

  4. Is the circumference of a circle proportional to its radius?

30.

  1. State a formula for the area of a square in terms of the length of its side.

  2. Complete the table.

    \(s\) \(2\) \(5\) \(6\) \(8\)
    \(A\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(\hphantom{0000}\)

  3. Graph your equation, using the values in your table.

    grid

  4. Is the area of a square proportional to the length of its side?

31.

The following recipe for Cadet Mess Sloppy Joes comes from the West Point Officers Wives' Club Cookbook.

For 1000 servings:

310 lb. ground beef \(\hphantom{0000}\) 2 cans diced red peppers
100 lb. ground turkey \(\hphantom{0000}\) 20 lb. onions, sliced
\(5\dfrac{1}{2}\) cans mushrooms \(\hphantom{0000}\) 3 lb. chili powder
4 lb. green peppers, sliced \(\hphantom{0000}\) 1 lb. garlic powder
18 cans tomato puree \(\hphantom{0000}\) \(3\dfrac{1}{2}\) pounds sugar
12 cans tomato catsup \(\hphantom{0000}\) 1 qt. light flavor glow
1 qt cider vinegar \(\hphantom{0000}\) 28 oz. salt and pepper
Cook off turkey and beef in oven; place in pots. Braise off chopped peppers and diced onions (turkey will absorb liquid). Add stock to turkey to moisten well before adding other products. Stir together. Heat till piping hot. Serve on rolls.

  1. What is the ratio of tomato puree to tomato catsup? How many servings of sloppy joes can you make if you have only 5 cans of tomato catsup? How many cans of tomato puree will you need?

  2. How much chili powder (in ounces) will you need to serve sloppy joes to 50 people? How many pounds of ground beef?

  3. The cookbook also lists the ingredients needed for 4500 servings. How many cans of mushrooms are required for 4500 servings? How many pounds of onions?