Lesson Objectives

- Demonstrate an understanding of how to simplify fractions
- Learn how to compare two quantities using a ratio
- Learn how to write a ratio using different formats
- Learn how to compare two quantities using a unit rate

## What is a Ratio?

A ratio is a comparison of two quantities. Let's start by looking at an example.
In our above image, we can see that there are a total of 10 apples and 6 bananas. We can use a ratio to describe the relationship between apples and bananas.
The ratio of apples to bananas is:

10 to 6

This can also be written with a ":" colon, but still read as 10 to 6.

10:6

Lastly, we can write this ratio in fractional form. Again, this is still read as 10 to 6. $$\frac{10}{6}$$ Ratios can be simplified using the same strategies we used with fractions. With 10/6, we think about the GCF of 10 and 6, which is 2. If we divide each part of the ratio by 2, we get our simplified ratio as:

5 to 3

This ratio is comparing the number of apples to the number of bananas. In other words, the ratio is stating that for every 5 apples, there are 3 bananas. We can rearrange our picture to show this concept: If we change the order of the wording, the ratio changes. If we want the ratio of bananas to apples we get:

3 to 5

3 bananas for every 5 apples

Let's think about another example. In our above image, we have a group of students. There are 4 girls and 2 boys in this group. What can we say about the ratio of girls to boys?

girls to boys » 4:2 or 2:1

There are 4 girls for every 2 boys. This can be simplified to 2 girls for each boy. We can rearrange our picture to make this clear: When we work with fractions, we can write: $$\frac{2}{1} = 2$$ We don't do that with ratios, although 2/1 is 2, with a ratio we are comparing two quantities. We need the 1 here to show the second part of the relationship. Let's take a look at a few examples.

Example 1:

On a field trip, there are 20 girls and 4 boys. What is the ratio of girls to boys?

The ratio of girls to boys is:

20:4

Which can be simplified to:

5:1

This means there are 5 girls for each boy on the field trip.

Example 2:

At a local orchard, there are 850 apple trees and 700 orange trees. What is the ratio of orange trees to apple trees?

The ratio of orange trees to apple trees is:

700:850

Which can be simplified to:

14:17

This means there are 14 orange trees for every 17 apple trees.

Ratios can be very helpful when solving everyday problems. Suppose there is a social function at school and we are told the ratio of girls to boys will be 3 to 1. Gift bags need to be arranged based on gender. Girls will get pink bags and boys blue bags. If we plan on having a total of 40 students at the event, how many of each bag should be prepared?

Let's break this down using a table:

We can see from our table that 40 students gives us 30 girls and 10 boys. How could we determine this without a table? When we look at a ratio we can think about
grouping. A ratio of 3 girls to 1 boy tells us there are 3 girls and 1 boy for each group of 4 students. If we wanted to know how many girls
are in a group of 80 students, we can first divide by 4 and determine the number of groups of 4 that can be made from 80.

80 ÷ 4 = 20

20 is not the number of girls in a group of 80 students. 20 would be the number of groups of 4 that can be made from 80. Each group has 4 students: 3 girls and 1 boy. So 20 groups with 3 girls in each group would give us 60 girls: 20 x 3 = 60. In a class of 80 students, 60 would be girls and 20 would be boys. Now a faster method just uses multiplication with fractions. If 3 out of 4 students are girls, we can multiply 3/4 by the number of students and get our answer: $$80 \cdot \frac{3}{4} = 60$$ What if we had 120 students? How many are girls? $$120 \cdot \frac{3}{4} = 90$$ With 120 students, 90 would be girls. Let's take a look at an example.

Example 3:

A mixture contains 30 gallons of alcohol for every 90 gallons of water. If the mixture is 3600 gallons, how much water is present?

To solve this problem, we think about the ratio of alcohol to water:

30:90

which simplifies to:

1:3

Since there is 1 gallon of alcohol for every 3 gallons of water, this means in every 4 gallons of the mixture we have 1 gallon of alcohol and 3 gallons of water. How many groups of 4 can be made out of 3600?

3600 ÷ 4 = 900

900 is not our answer, in each group of 4 gallons there are 3 gallons of water and 1 gallon of alcohol. We can multiply 900 x 3 = 2700. This means 2700 gallons are water and 900 gallons are alcohol.

We can also use the quicker method here and multiply 3600 by the fraction 3/4 $$3600 \cdot \frac{3}{4} = 2700$$

9 ÷ 3 = 3

10 ÷ 5 = 2

This tells us that 3 gallons for $9 will be purchased at a cost of $3 per gallon, and 5 gallons for $10 will be purchased at a cost of $2 per gallon. Purchasing 5 gallons will give us the better per gallon cost. $$\frac{\$9}{3 \hspace{.25em} gallons} = \frac{\$3}{1 \hspace{.25em} gallon}$$ $$\frac{\$10}{5 \hspace{.25em} gallons} = \frac{\$2}{1 \hspace{.25em} gallon}$$ With a unit rate, we read the division symbol as "per" so $3/1 gallon can be read as $3 per gallon. Let's take a look at an example.

Example 4:

Jason earned $72 dollars for 8 hours of work. How much was he paid per hour?

Setup the rate as: $$\frac{\$72}{8 \hspace{.25em} hours}$$ Now we can divide the numerator by the denominator:

72 ÷ 8 = 9 $$\frac{\$72}{8 \hspace{.25em} hours} = \frac{$9}{1 \hspace{.25em} hour}$$ We can say that Jason earned $9 per hour.

10 to 6

This can also be written with a ":" colon, but still read as 10 to 6.

10:6

Lastly, we can write this ratio in fractional form. Again, this is still read as 10 to 6. $$\frac{10}{6}$$ Ratios can be simplified using the same strategies we used with fractions. With 10/6, we think about the GCF of 10 and 6, which is 2. If we divide each part of the ratio by 2, we get our simplified ratio as:

5 to 3

This ratio is comparing the number of apples to the number of bananas. In other words, the ratio is stating that for every 5 apples, there are 3 bananas. We can rearrange our picture to show this concept: If we change the order of the wording, the ratio changes. If we want the ratio of bananas to apples we get:

3 to 5

3 bananas for every 5 apples

Let's think about another example. In our above image, we have a group of students. There are 4 girls and 2 boys in this group. What can we say about the ratio of girls to boys?

girls to boys » 4:2 or 2:1

There are 4 girls for every 2 boys. This can be simplified to 2 girls for each boy. We can rearrange our picture to make this clear: When we work with fractions, we can write: $$\frac{2}{1} = 2$$ We don't do that with ratios, although 2/1 is 2, with a ratio we are comparing two quantities. We need the 1 here to show the second part of the relationship. Let's take a look at a few examples.

Example 1:

On a field trip, there are 20 girls and 4 boys. What is the ratio of girls to boys?

The ratio of girls to boys is:

20:4

Which can be simplified to:

5:1

This means there are 5 girls for each boy on the field trip.

Example 2:

At a local orchard, there are 850 apple trees and 700 orange trees. What is the ratio of orange trees to apple trees?

The ratio of orange trees to apple trees is:

700:850

Which can be simplified to:

14:17

This means there are 14 orange trees for every 17 apple trees.

Ratios can be very helpful when solving everyday problems. Suppose there is a social function at school and we are told the ratio of girls to boys will be 3 to 1. Gift bags need to be arranged based on gender. Girls will get pink bags and boys blue bags. If we plan on having a total of 40 students at the event, how many of each bag should be prepared?

Let's break this down using a table:

Students Attending | Girls | Boys |
---|---|---|

4 | 3 | 1 |

8 | 6 | 2 |

12 | 9 | 3 |

16 | 12 | 4 |

20 | 15 | 5 |

24 | 18 | 6 |

28 | 21 | 7 |

32 | 24 | 8 |

36 | 27 | 9 |

40 | 30 | 10 |

80 ÷ 4 = 20

20 is not the number of girls in a group of 80 students. 20 would be the number of groups of 4 that can be made from 80. Each group has 4 students: 3 girls and 1 boy. So 20 groups with 3 girls in each group would give us 60 girls: 20 x 3 = 60. In a class of 80 students, 60 would be girls and 20 would be boys. Now a faster method just uses multiplication with fractions. If 3 out of 4 students are girls, we can multiply 3/4 by the number of students and get our answer: $$80 \cdot \frac{3}{4} = 60$$ What if we had 120 students? How many are girls? $$120 \cdot \frac{3}{4} = 90$$ With 120 students, 90 would be girls. Let's take a look at an example.

Example 3:

A mixture contains 30 gallons of alcohol for every 90 gallons of water. If the mixture is 3600 gallons, how much water is present?

To solve this problem, we think about the ratio of alcohol to water:

30:90

which simplifies to:

1:3

Since there is 1 gallon of alcohol for every 3 gallons of water, this means in every 4 gallons of the mixture we have 1 gallon of alcohol and 3 gallons of water. How many groups of 4 can be made out of 3600?

3600 ÷ 4 = 900

900 is not our answer, in each group of 4 gallons there are 3 gallons of water and 1 gallon of alcohol. We can multiply 900 x 3 = 2700. This means 2700 gallons are water and 900 gallons are alcohol.

We can also use the quicker method here and multiply 3600 by the fraction 3/4 $$3600 \cdot \frac{3}{4} = 2700$$

### How to Find the Unit Rate

A rate is a special type of ratio where the units are different. When we work with rates, we usually divide the numerator by the denominator to obtain a unit rate (denominator of 1). The unit rate gives us the amount of one thing per one unit of another. We see unit rates used everywhere. Some examples would be: miles per gallon, dollars per hour, and cost per ounce. Let's suppose we visit the local grocery and there are two scenarios: We can purchase 3 gallons of milk for $9 or 5 gallons of milk for $10. Which is the better buy? Assuming you could use all of the milk before expiration, you would set up a unit rate for each to determine the cost per 1 gallon of milk: $$\frac{\$9}{3 \hspace{.25em} gallons}$$ $$\frac{\$10}{5 \hspace{.25em} gallons}$$ To obtain the unit rate, we simply divide the numerator by the denominator:9 ÷ 3 = 3

10 ÷ 5 = 2

This tells us that 3 gallons for $9 will be purchased at a cost of $3 per gallon, and 5 gallons for $10 will be purchased at a cost of $2 per gallon. Purchasing 5 gallons will give us the better per gallon cost. $$\frac{\$9}{3 \hspace{.25em} gallons} = \frac{\$3}{1 \hspace{.25em} gallon}$$ $$\frac{\$10}{5 \hspace{.25em} gallons} = \frac{\$2}{1 \hspace{.25em} gallon}$$ With a unit rate, we read the division symbol as "per" so $3/1 gallon can be read as $3 per gallon. Let's take a look at an example.

Example 4:

Jason earned $72 dollars for 8 hours of work. How much was he paid per hour?

Setup the rate as: $$\frac{\$72}{8 \hspace{.25em} hours}$$ Now we can divide the numerator by the denominator:

72 ÷ 8 = 9 $$\frac{\$72}{8 \hspace{.25em} hours} = \frac{$9}{1 \hspace{.25em} hour}$$ We can say that Jason earned $9 per hour.

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