What if you were told that there were 15 birds on a pond and that the ratio of ducks to geese is 2:3? How could you determine how many ducks and how many geese are on the pond? After completing this Concept, you'll be able to solve problems like this one.

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### Guidance

A **ratio** is a way to compare two numbers. Ratios can be written in three ways: \begin{align*}\frac{a}{b}\end{align*}, \begin{align*}a:b\end{align*}, and \begin{align*}a\end{align*} to \begin{align*}b\end{align*}.

We always reduce ratios just like fractions. When two or more ratios reduce to the same ratio they are called **equivalent ratios**. For example, 50:250 and 2:10 are **equivalent ratios** because they both reduce to 1:5.

One common use of ratios is as a way to convert measurements.

#### Example A

There are 14 girls and 18 boys in your math class. What is the ratio of girls to boys?

Remember that order matters. The question asked for the ratio of *girls* to *boys*. The ratio would be 14:18. This can be simplified to 7:9.

#### Example B

Simplify the following ratios.

a) \begin{align*}\frac{7 \ ft}{14 \ in}\end{align*}

b) 9m:900cm

c) \begin{align*}\frac{4 \ gal}{16 \ gal}\end{align*}

First, change each ratio so that each part is in the same units. Remember that there are 12 inches in a foot.

a) \begin{align*}\frac{7 \ \bcancel{ft}}{14 \ \cancel{in}} \cdot \frac{12 \ \cancel{in}}{1 \ \bcancel{ft}} = \frac{84}{14} = \frac{6}{1}\end{align*}

The inches and feet cancel each other out. *Simplified ratios do not have units.*

b) It is easier to simplify a ratio when written as a fraction.

\begin{align*}\frac{9 \ \bcancel{m}}{900 \ \cancel{cm}} \cdot \frac{100 \ \cancel{cm}}{1 \ \bcancel{m}} = \frac{900}{900} = \frac{1}{1}\end{align*}

c) \begin{align*}\frac{4 \ \bcancel{gal}}{16 \ \bcancel{gal}} = \frac{1}{4}\end{align*}

#### Example C

A talent show has dancers and singers. The ratio of dancers to singers is 3:2. There are 30 performers total, how many of each are there?

To solve, notice that 3:2 is a reduced ratio, so there is a number, \begin{align*}n\end{align*}, that we can multiply both by to find the total number in each group. Represent dancers and singers as expressions in terms of \begin{align*}n\end{align*}. Then set up and solve an equation.

\begin{align*}\text{dancers} = 3n, \ \text{singers} = 2n \ \longrightarrow \ 3n+2n &= 30\\ 5n &= 30\\ n &= 6\end{align*}

There are \begin{align*}3 \cdot 6 = 18\end{align*} dancers and \begin{align*}2 \cdot 6 = 12\end{align*} singers.

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### Guided Practice

The total bagel sales at a bagel shop for Monday is in the table below.

Type of Bagel |
Number Sold |
---|---|

Plain | 80 |

Cinnamon Raisin | 30 |

Sesame | 25 |

Jalapeno Cheddar | 20 |

Everything | 45 |

Honey Wheat | 50 |

1. What is the ratio of cinnamon raisin bagels to plain bagels?

2. What is the ratio of honey wheat bagels to total bagels sold?

3. What is the ratio of cinnamon raisin bagels to sesame bagels to jalapeno cheddar bagels?

**Answers:**

1. The ratio is 30:80. Reducing the ratio by 10, we get 3:8.

2. Order matters. Honey wheat is listed first, so that number comes first in the ratio (or on the top of the fraction). Find the total number of bagels sold, \begin{align*}80 + 30 + 25 + 20 + 45 + 50 = 250\end{align*}.

The ratio is \begin{align*}\frac{50}{250} = \frac{1}{5}\end{align*}.

3. You can have ratios that compare more than two numbers and they work just the same way. The ratio for this problem is 30:25:20, which reduces to 6:5:4.

### Explore More

- The votes for president in a club election were: \begin{align*}\text{Smith}:24 \text{ Munoz}:32 \text{ Park}:20\end{align*}Find the following ratios and write in simplest form.
- Votes for Munoz to Smith

- Votes for Park to Munoz

- Votes for Smith to total votes

- Votes for Smith to Munoz to Park

Use the picture to write the following ratios for questions 2-6.

\begin{align*}AEFD \ \text{is a square} \qquad ABCD \ \text{is a rectangle}\end{align*}

- \begin{align*}AE:EF\end{align*}
- \begin{align*}EB:AB\end{align*}
- \begin{align*}DF:FC\end{align*}
- \begin{align*}EF:BC\end{align*}
- Perimeter \begin{align*}ABCD\end{align*}:Perimeter \begin{align*}AEFD\end{align*}:Perimeter \begin{align*}EBCF\end{align*}

Simplify the following ratios. Remember that there are 12 inches in a foot, 3 feet in a yard, and 100 centimeters in a meter.

- \begin{align*}\frac{25 \ in}{5 \ ft}\end{align*}
- \begin{align*}\frac{9 \ ft}{3 \ yd}\end{align*}
- \begin{align*}\frac{95 \ cm}{1.5 \ m}\end{align*}

- The measures of the angles of a triangle are have the ratio 3:3:4. What are the measures of the angles?
- The length and width of a rectangle are in a 3:5 ratio. The perimeter of the rectangle is 64. What are the length and width?
- The length and width of a rectangle are in a 4:7 ratio. The perimeter of the rectangle is 352. What are the length and width?
- A math class has 36 students. The ratio of boys to girls is 4:5. How many girls are in the class?
- The senior class has 450 students in it. The ratio of boys to girls is 8:7. How many boys are in the senior class?
- The varsity football team has 50 players. The ratio of seniors to juniors is 3:2. How many seniors are on the team?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.1.