What if you were told that every two inches on a map represented 500 miles? How could you write this comparison mathematically? After completing this Concept, you'll be able to write and understand ratios like this one.

### Watch This

CK-12 Foundation: 0309S Ratios (H264)

### Guidance

A **ratio** is a way to compare two numbers, measurements or quantities. When we write a ratio, we divide one number by another and express the answer as a fraction. There are two distinct ratios in the example below.

#### Example A

Nadia is counting out money with her little brother. She gives her brother all the nickels and pennies. She keeps the quarters and dimes for herself. Nadia has four quarters and six dimes. Her brother has fifteen nickels and five pennies and is happy because he has more coins than his big sister. How would you explain to him that he is actually getting a bad deal?

**Solution:**

The ratio of the **number** of Nadia’s coins to her brother’s is \begin{align*}\frac{4 + 6}{15 + 5}\end{align*}, or \begin{align*}\frac{10}{20} = \frac{1}{2}\end{align*}. (Ratios should always be simplified.) In other words, Nadia has half as many coins as her brother.

Another ratio we could look at is the **value** of the coins. The value of Nadia’s coins is \begin{align*}(4 \times 25) + (6 \times 10) = 160 \ cents\end{align*}. The value of her brother’s coins is \begin{align*}(15 \times 5) + (5 \times 1) = 80 \ cents\end{align*}. The ratio of the **value** of Nadia’s coins to her brother’s is \begin{align*}\frac{160}{80} = \frac{2}{1}\end{align*}. So the value of Nadia’s money is twice the value of her brother’s.

Notice that even though the denominator is one, we still write it out and leave the ratio as a fraction instead of a whole number. A ratio with a denominator of one is called a **unit rate**.

#### Example B

*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.*

**Solution**

We could compare the numbers by expressing the difference between them: \begin{align*}\$10.00 - \$6.50 = \$3.50.\end{align*}

We can also use a ratio to compare them: \begin{align*}\frac{10.00}{6.50} = \frac{100}{65} = \frac{20}{13}\end{align*} (after multiplying by 10 to remove the decimals, and then simplifying).

So we can say that **the new book is $3.50 more than the used book**, or we can say that **the new book costs** \begin{align*}\frac{20}{13}\end{align*} **times as much as the used book**.

#### Example C

*A tournament size shuffleboard table measures 30 inches wide by 14 feet long. Compare the length of the table to its width and express the answer as a ratio.*

**Solution**

We could just write the ratio as \begin{align*}\frac{14 \ feet}{30 \ inches}\end{align*}. But since we’re comparing two lengths, it makes more sense to convert all the measurements to the same units. 14 feet is \begin{align*}14 \times 12 = 168 \ inches\end{align*}, so our new ratio is \begin{align*}\frac{168}{30} = \frac{28}{5}\end{align*}.

Watch this video for help with the Examples above.

### Vocabulary

- A
**ratio**is a way to compare two numbers, measurements or quantities by dividing one number by the other and expressing the answer as a fraction. -
**A proportion**is formed when two ratios are set equal to each other. -
**Cross multiplication**is useful for solving equations in the form of proportions. To cross multiply, multiply the bottom of each ratio by the top of the other ratio and set them equal. For instance, cross multiplying results in \begin{align*}11 \times 3 = 5x\end{align*}. -
**Scale**is a proportion that relates map distance to real life distance.

### Guided Practice

*A family car is being tested for fuel efficiency. It drives non-stop for 100 miles and uses 3.2 gallons of gasoline. Write the ratio of distance traveled to fuel used as a* **unit rate**.

**Solution**

The ratio of distance to fuel is \begin{align*}\frac{100 \ miles}{3.2 \ gallons}\end{align*}. But a unit rate has to have a denominator of one, so to make this ratio a unit rate we need to divide both numerator and denominator by 3.2. \begin{align*}\frac{\frac{100}{3.2} \ miles}{\frac{3.2}{3.2} \ gallons} = \frac{31.25 \ miles}{1 \ gallon}\end{align*} or **31.25 miles per gallon.**

### Practice

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.
- Write the following ratios as a unit rate.
- 54 hotdogs to 12 minutes.
- 5000 lbs to 250 square inches.
- 20 computers to 80 students.
- 180 students to 6 teachers .
- 12 meters to 4 floors .
- 18 minutes to 15 appointments.