At the beginning of the school year, the 7^{th} grade students decided to keep a tally of the books they read. At the end of the year, the students had read a total of 544 books and organized them into categories.

History: 12 books

Adventure: 250 books

Romance: 100 books

Mystery: 120 books

Nature/Science: 62 books

What is the ratio of mystery books to adventure books? Do these ratios form a proportion? Why or why not?

In this concept, you will learn how to solve problems involving ratios and proportions.

### Solving Problems Involving Ratios and Proportions

A **ratio ** compares two quantities.

Ratios can be written as fractions, with a colon or with the word “to.”

\begin{align*}\frac{2}{3}\end{align*}

A **proportion ** is created when two ratios are found to be equivalent or equal. A proportion can be written

**Proportional reasoning,**** **or examining the relationship between two numbers, can be used to determine an unknown value.

A proportion can also be written with colons.

\begin{align*}\frac{a}{b}=\frac{c}{d} \ \ or \ \ a:b=c:d\end{align*}

When a proportion is written with colons, in the form a:b = c:d, the two terms closest together, b and c, are called the “means.” The two terms on the ends, a and d, are called the “extremes.”

The **Cross Products Property of Proportions** states that the product of the means is equal to the product of the extremes.

\begin{align*}\frac{a}{b}&=\frac{c}{d}\\
b\cdot c&=a\cdot d\end{align*}

Let's look at this example.

If two ratios form a proportion, the means will be equal to the extremes.

Some problems will be easier to solve using proportional reasoning. Others will be easier using cross multiplication.

### Examples

#### Example 1

Earlier, you were given a problem about the 7^{th} grade students and a tally of the books they read for a year.

At the end of the year, the students had read a total of 544 books and organized them into categories.

History: 12 books

Adventure: 250 books

Romance: 100 books

Mystery: 120 books

Nature/Science: 62 books** **

What is the ratio of mystery books to adventure books? What is the ratio of History to Romance? Do these ratios form a proportion? Why or why not?

First, write the ratios.

Next, write a proportion.

\begin{align*}\frac{120}{250}=\frac{12}{100}\end{align*}

Then, cross multiply.

The answer is that the ratios do not form a proportion because they are not equal.

#### Example 2

Solve two ways:

\begin{align*}\frac{a}{4}=\frac{6}{8}\end{align*}

First, use proportional reasoning. Ask what can be done to 8 to make it equal to 4.

Next, divide both numerator and denominator of the given ratio by 2.

The answer is 3.

\begin{align*}\frac{6}{8}=\frac{3}{4}\end{align*}

Then, solve the problem using cross products.

The answer is the same, a = 3

#### Example 3

Sonia measured the straight-line distance between Baltimore, MD, and Washington, D.C., to be 2 centimeters on a map. The scale on the map shows that 1 centimeter = 28 kilometers. What is the actual straight-line distance between Baltimore and Washington D.C.?

First, choose a method.

This problem involves a map, which is a type of scale drawing. It makes sense to use proportions to solve it.

Next, write the ratios for the unit scale and the scale distance compared to actual distance.

\begin{align*}\frac{scale}{actual}\\
Unit \ scale &= \frac{1 \ cm}{28 \ km}\\
Distance &= \frac{2}{d}\end{align*}

Then, set the ratios equal to one another to form a proportion.

\begin{align*}\frac{1 \ cm}{28 \ km}=\frac{2}{d}\end{align*}

Next, cross multiply and solve.

The answer is that the actual distance from Baltimore to Washington is 56 km.

#### Example 4

A baker uses 22 cups of flour to make 4 loaves of bread. How many cups of flour will he need to use to make 31 loaves of bread?

First, write the ratios.

\begin{align*}\frac{flour}{loaves} \ \ \ \ \ \frac{22}{4} \ \ \ \ \ \ \frac{x}{31}\end{align*}

Next, write a proportion.

\begin{align*}\frac{22}{4}=\frac{x}{31}\end{align*}

Then, determine a method. Since the relationship between the terms in the denominators, 4 and 31, is not immediately obvious, cross multiplication is easier.

Next, cross multiply and solve.

\begin{align*}4x&=22\times 31\\
x&=170.5\end{align*}

The answer is 170.5. The baker will need 170.5 cups of flour to make 31 loaves of bread.

#### Example 5

If the baker cut the original recipe in half, how many cups of flour would he have needed?

First, write the ratios.

\begin{align*}\frac{flour}{loaves} \ \ \ \ \ \frac{22}{4} \ \ \ \ \ \ \frac{x}{2}\end{align*}

Next, write a proportion.

Then, determine a method. It is easy to see that 4 2 = 2. Use proportional reasoning.

\begin{align*}\frac{22 \div 2}{4 \div 2}=\frac{11}{2}\end{align*}

### Review

Use what you have learned to solve each problem. Consider more than one strategy for solving each problem. Then choose the strategy you think will work best and use it to solve the problem.

- A jar contains only pennies and nickels. The ratio of pennies to nickels in the jar is 2 to 7. If there are 14 nickels in the jar, how many pennies are in the jar?
- Anya charges $40 for 5 hours of babysitting. Lionel charges $14 for 2 hours of babysitting. Which babysitter charges the cheapest rate?
- On a map, Derek measured the straight-line distance between Toronto, Canada and Niagara Falls, New York to be 2 inches. The scale on the map shows that \begin{align*}\frac{1}{2} \ inch = 11 \ miles\end{align*}
12 inch=11 miles . What is the actual straight-line distance between Toronto and Niagara Falls? - A desk is 120 centimeters long. What is the length of the desk in meters? Use this unit conversion: 1 meter = 100 centimeters.
- On a field trip, the ratio of teachers to students is 1 : 25. If there are 5 teachers on the field trip, how many students are on the trip?
- Kara bought 5 pounds of Brand X roast beef for $43. Cameron bought 3 pounds of Brand Y roast beef for $27. Which brand of roast beef is the better buy?
- If two inches on a map are equal to three miles, how many miles are represented by four inches?
- If eight inches on a map are equal to ten miles, how many miles are 16 inches equal to?
- Casey drew a design for bedroom. On the picture, she used one inch to represent five feet. If her bedroom wall is ten feet long, how many inches will Casey draw on her diagram to represent this measurement?
- If two inches are equal to twelve feet, how many inches would be equal to 36 feet?
- If four inches are equal to sixteen feet, how many feet are two inches equal to?
- The carpenter chose a scale of 6” for every twelve feet. Given this measurement, how many feet would be represented by 3”?
- If 9 inches are equal to 27 feet, how many feet are equal to three inches?
- If four inches are equal to 8 feet, how many feet are equal to two inches?
- If six inches are equal to ten feet, how many inches are five feet equal to?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.14.