Have you ever tried to solve a dilemma in a grocery store? Take a look at this situation.

Jeffrey went to the grocery store to purchase a chicken for dinner. He was amazed at the prices and had to do some quick math while he was there. Here is what he found out. At the store, 3 pounds of chicken was selling for $13.50. If Jeffrey has $30.00, how many pounds of chicken can he purchase with that amount of money?

The best way to solve this problem is to use a proportion. Pay attention to this Concept and you will see how to connect proportions to dilemmas.

### Guidance

A proportion is created when two ratios are equal. Sometimes, you will see real-world situations that describe ratios and proportions. When these situations occur, you can solve them by using equal ratios or cross - products.

Let's take a look at a situation where a proportion can be used to solve a real-world dilemma.

**Amanda read 18 pages in 23 minutes. At this rate, how many pages will she read in 45 minutes?**

**First, set up a proportion.**

\begin{align*}\frac{18}{23} = \frac{x}{45}\end{align*}

**You cannot easily use equivalent fractions to solve this proportion. Cross-multiply to solve the proportion.**

\begin{align*}23x = 18(45)\end{align*}

**Now simplify the equation and solve for \begin{align*}x\end{align*}.**

\begin{align*}23x &= 810\\ x & \approx 35.2\end{align*}

**Amanda will read about 35.2 pages in 45 minutes.**

What about a situation using an equal ratio?

**John ate three hot dogs in six minutes. At this rate, how many will he eat in twelve minutes?**

First, set up a proportion.

\begin{align*}\frac{3}{6} = \frac{x}{12}\end{align*}

Now look at the relationship between the denominators.

\begin{align*}6 \times 2 = 12\end{align*}

We can use equal ratios and multiply the numerator by two also.

\begin{align*}3 \times 2 = 6\end{align*}

**At this rate, John will eat 6 hot dogs in 12 minutes.**

#### Example A

Carmen ran one mile in 7 minutes. At this rate, how long will it take her to run 5 miles?

**Solution: 35 minutes**

#### Example B

Jack bought 5 oranges for $3.99. What was the cost for one orange?

**Solution: \begin{align*}.80\end{align*}**

#### Example C

Jessie read three books in one week. At this rate, how many books will she read in 3 weeks?

**Solution: 9 books**

Now let's go back to the dilemma from the beginning of the Concept.

**Set up a proportion.**

\begin{align*}\frac{3}{13.50} = \frac{x}{30}\end{align*}

**Cross-multiply and use algebra to solve for \begin{align*}x\end{align*}.**

\begin{align*}13.5x &= 3(30)\\ 13.5x &= 90\\ x & \approx 6.67\end{align*}

**You could buy about 6.67 pounds of chicken for $30.**

### Vocabulary

- Ratio
- a comparison between two quantities. Ratios can be written in fraction form, with a colon or by using the word “to”.

- Equivalent
- means equal.

- Proportion
- formed when two ratios are equivalent. We compare two ratios, they are equal and so they form a proportion.

### Guided Practice

Kelvin measured the distance from his door to the park. It is 1.5 miles. The distance from Kelvin's house to the library is twice the distance from Kelvin's door to the park. How far is it from Kelvin's to the library?

**Solution**

We know that the distance from Kelvin's door to the park is 1.5 miles.

The distance from his house to the library is double that distance.

\begin{align*}\frac{1}{1.5} = \frac{2}{x}\end{align*}

Now cross - multiply and solve.

**It is 3 miles from Kelvin's house to the library.**

### Video Review

Khan Academy Ratios and Proportions

### Practice

Directions: Write a proportion and use equivalent ratios to solve the following problems.

- Marco makes $25 for every 2 hours he works. If he works for 12 hours, how much will he make?
- If Marco works for 6 hours, how much will he make?
- If Marco works for 4 hours, how much will he make?
- If Marco made $50 for every 2 hours he works, how much will he make in 10 hours?
- Corinne runs 2.8 miles in 30 minutes. If she runs for 150 minutes this week, how many miles will she have run?
- If she runs 300 minutes, how many miles will she run?
- Adam drives 45 miles per hour. If he drives for 3.5 hours, how many miles will he have driven?
- If he drives 7 hours, how many miles will he have driven?

Directions: Write a proportion and use cross-multiplying to solve the following problems. You may round when necessary.

- Marni buys 2.5 pounds of grapefruit for $4.48. To the nearest cent, how much would 6 pounds of grapefruit cost?
- Sarah buys 3 pounds of bananas for $2.50. What is the cost for bananas per pound?
- Glenn can make 8 flyers in 35 minutes. How long will it take him to make 50 flyers?
- At this rate, how many flyers could Glenn make in 70 minutes?
- How many in two hours?
- A store sells 21 pieces of clothing every 45 minutes. How long will it take the store to sell 100 pieces of clothing?
- The basketball team scored 85 points in the last 2 games. How many points can they expect to score after 5 games?