Suppose that in 5 minutes, you can type 150 words. If you continued at this rate, do you know how many words you could type in 12 minutes? In order to answer this question, you could use ratios or a proportion.

### Equations with Ratios and Proportions

**Ratios** and **proportions** have a fundamental place in mathematics. They are used in geometry, size changes, and trigonometry.

A **ratio** is a fraction comparing two things with the same units.

A **rate** is a fraction comparing two things with different units.

You have experienced rates many times: 65 mi/hour, $1.99/pound, and $3.79/\begin{align*}yd^2\end{align*}. You have also experienced ratios. A “student to teacher” ratio shows approximately how many students one teacher is responsible for in a school.

#### Let's solve the following problem:

The State Dining Room in the White House measures approximately 48 feet long by 36 feet wide. Compare the length of the room to the width, and express your answer as a ratio.

\begin{align*}\frac{48\ feet}{36\ feet} = \frac{4}{3}\end{align*}

The length of the State Dining Room is \begin{align*}\frac{4}{3}\end{align*} the width.

#### Proportions

A **proportion** is a statement in which two fractions or ratios are equal: \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}.

For example, \begin{align*}\frac{2}{3} = \frac{6}{12}\end{align*} is not a proportion because if they are not equal.

\begin{align*}\frac{2}{3} = \frac{8}{12} \neq \frac{6}{12}\end{align*}

A ratio can also be written using a colon instead of the fraction bar.

\begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*} can also be read, “\begin{align*}a\end{align*} is to \begin{align*}b\end{align*} as \begin{align*}c\end{align*} is to \begin{align*}d\end{align*}” or \begin{align*}a:b = c:d.\end{align*}

The values of \begin{align*}a\end{align*} and \begin{align*}d\end{align*} are called the **extremes** of the proportion and the values of \begin{align*}b\end{align*} and \begin{align*}c\end{align*} are called the **means**. To solve a proportion, you can use the **cross products:**

For a proportion, if \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}ad=bc\end{align*}.

#### Let's solve the following proportion using cross products:

\begin{align*}\frac{a}{9} = \frac{7}{6}\end{align*}.

Apply the Cross Products of a Proportion.

\begin{align*}6a & =7(9) \\ 6a & =63\end{align*}

Solve for \begin{align*}a\end{align*}.

\begin{align*}a=10.5\end{align*}

### Examples

#### Example 1

Earlier, you were told that you can type 150 words in 5 minutes. If you continued at this rate, how many words could you type in 12 minutes?

This is an example of a problem that can be solved using several methods, including proportions. To solve using a proportion, you need to translate the statement into an algebraic sentence. The key to writing correct proportions is to keep the units the same in each fraction.

\begin{align*}\frac{words}{minutes} = \frac{words}{minutes} && \frac{words}{minutes} \neq \frac{minutes}{words}\end{align*}

Now, we can substitute the information given in the problem.

\begin{align*}\frac{words}{minutes}=\frac{words}{minutes}\\ \frac{150}{5} = \frac{words}{12}\end{align*}

To make the equation more clear, let \begin{align*}w\end{align*} represent the words that we are solving for and use the Cross Product of Proportions to solve:

\begin{align*}\frac {150}{5} = \frac{w}{12}\\ 150 \times 12 = 5 \times w\\ 1800=5w 1800 \div 5 = 5w \div 5\\ 360 = w\end{align*}

You can type 360 words in 12 minutes.

#### Example 2

A train travels at a steady speed. It covers 15 miles in 20 minutes. How far will it travel in 7 hours, assuming it continues at the same rate?

As in Example 1, make sure that you have the units the same for the ratios.

\begin{align*}\frac{miles}{time} = \frac{miles}{time} && \frac{miles}{time} \neq \frac{time}{miles}\end{align*}

Before we substitute in miles and time into our proportion, we need to have our units consistent. We are given the train's rate in minutes, and asked how far the train will go in 7 hours, so we need to figure out how many minutes are an equivalent amount of time to 7 hours. Since there are 60 minutes in each hour:

\begin{align*}7 \times 60=420\end{align*}

So 7 hours is equivalent to 420 minutes.

\begin{align*}&\frac{miles}{time} = \frac{miles}{time}\\ &\frac{15}{20} = \frac{miles}{420}\\ \end{align*}

Now we solve for the number of miles using the Cross Product of Proportions:

\begin{align*}&15\times 420 =miles \times 20\\ &15\times 420\times \frac{1}{20} =miles \times 20 \times \frac{1}{20}\\ &15\times \frac{420}{20} =miles\\ &15\times 21 =miles\\ &315 =miles \end{align*}

The train traveled 315 miles in 7 hours.

### Review

Write the following comparisons as ratios. Simplify fractions where possible.

- $150 to $3
- 150 boys to 175 girls
- 200 minutes to 1 hour
- 10 days to 2 weeks

In 5 – 10, write the ratios as a unit rate.

- 54 hot dogs to 12 minutes
- 5000 lbs to 250 \begin{align*}in^2\end{align*}
- 20 computers to 80 students
- 180 students to 6 teachers
- 12 meters to 4 floors
- 18 minutes to 15 appointments
- Give an example of a proportion that uses the numbers 5, 1, 6, and 30
- In the following proportion, identify the means and the extremes: \begin{align*}\frac{5}{12} = \frac{35}{84}\end{align*}

In 13 – 23, solve the proportion.

- \begin{align*}\frac{13}{6} = \frac{5}{x}\end{align*}
- \begin{align*}\frac{1.25}{7} = \frac{3.6}{x}\end{align*}
- \begin{align*}\frac{6}{19} = \frac{x}{11}\end{align*}
- \begin{align*}\frac{1}{x} = \frac{0.01}{5}\end{align*}
- \begin{align*}\frac{300}{4} = \frac{x}{99}\end{align*}
- \begin{align*}\frac{2.75}{9} = \frac{x}{\left (\frac{2}{9} \right )}\end{align*}
- \begin{align*}\frac{1.3}{4} = \frac{x}{1.3}\end{align*}
- \begin{align*}\frac{0.1}{1.01} = \frac{1.9}{x}\end{align*}
- \begin{align*}\frac{5p}{12} = \frac{3}{11}\end{align*}
- \begin{align*}- \frac{9}{x} = \frac{4}{11}\end{align*}
- \begin{align*}\frac{n+1}{11} = -2\end{align*}
- A restaurant serves 100 people per day and takes in $908. If the restaurant were to serve 250 people per day, what might the cash collected be?
- The highest mountain in Canada is Mount Yukon. It is \begin{align*}\frac{298}{67}\end{align*} the size of Ben Nevis, the highest peak in Scotland. Mount Elbert in Colorado is the highest peak in the Rocky Mountains. Mount Elbert is \begin{align*}\frac{220}{67}\end{align*} the height of Ben Nevis and \begin{align*}\frac{44}{48}\end{align*} the size of Mont Blanc in France. Mont Blanc is 4800 meters high. How high is Mount Yukon?
- At a large high school, it is estimated that two out of every three students have a cell phone, and one in five of all students have a cell phone that is one year old or less. Out of the students who own a cell phone, what proportion own a phone that is more than one year old?
- The price of a Harry Potter Book on Amazon.com is $10.00. The same book is also available used for $6.50. Find two ways to compare these prices.
- To prepare for school, you purchased 10 notebooks for $8.79. How many notebooks can you buy for $5.80?
- It takes 1 cup mix and \begin{align*}\frac{3}{4}\ \end{align*} cup water to make 6 pancakes. How much water and mix is needed to make 21 pancakes?
- Ammonia is a compound consisting of a 1:3 ratio of nitrogen and hydrogen atoms. If a sample contains 1,983 hydrogen atoms, how many nitrogen atoms are present?
- The Eagles have won 5 out of their last 9 games. If this trend continues, how many games will they have won in the 63-game season?

**Mixed Review**

- Solve \begin{align*}\frac{15}{16} \div \frac{5}{8}\end{align*}.
- Evaluate \begin{align*}|9-108|\end{align*}.
- Simplify: \begin{align*}8(8-3x)-2(1+8x)\end{align*}.
- Solve for \begin{align*}n: \ 7(n+7)=-7\end{align*}.
- Solve for \begin{align*}x: \ -22=-3+x\end{align*}.
- Solve for \begin{align*}u: \ 18=2u\end{align*}.
- Simplify: \begin{align*}- \frac{1}{7}- \left (-1 \frac{1}{3} \right )\end{align*}.
- Evaluate: \begin{align*}5\times \frac{p}{6} |n|\end{align*} when \begin{align*}n=10\end{align*}and\begin{align*}p=-6\end{align*}.
- Make a table when \begin{align*}-4 \le x \le 4\end{align*}for\begin{align*}f(x)= \frac{1}{8} x + 2\end{align*}.
- Write as an English phrase: \begin{align*}y + 11\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.7.