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Factors and multiples Greatest common factor (GCF) Simplifying fractions Prime factors How to write a ratioHere you will learn about simplifying ratios, including how to simplify a ratio and how to write equivalent ratios.

Students will first learn about simplifying ratios as part of ratios and proportions in 6 th grade.

**Simplifying ratios** is a way of using **common factors** to divide all the numbers in a ratio until they cannot be divided further.

A simplified ratio is **equivalent** to a non simplified ratio.

You can use bar modeling to simplify ratios.

For example,

Here are some circles.

Rearrange them into colored bars to show the number of yellow circles and the number of blue circles.

The number of yellow circles is 4, and the number of blue circles is 6. This can be written as the ratio 4:6.

Rearrange these circles into another pattern, where yellow and blue have the same number of complete rows:

The overall ratio is still 4:6, but now you have two rows that show the ratio 2:3. This means that you can split the ratio into 2 groups, and still obtain an equivalent ratio of yellow to blue.

Using bar modeling simplifies the ratio 4:6 to get 2:3.

You can also divide by the **greatest common factor** or \bf{GCF} to simplify a ratio.

For example,

The ratio of red counters to blue counters is 12:16.

You can simplify the ratio by finding the **greatest common factor** or \bf{GCF} of each of the numbers in the ratio.

Factors of 12: 1, \, 2, \, 3, \, 4, \, 6, \, 12

Factor of 16: 1, \, 2, \, 4, \, 8, \, 16

The greatest common factor is 4. To simplify the ratio, you divide both sides by 4.

How does this relate to 6 th grade math?

**Grade 6 – Ratios and Proportional Relationships (6.RP.A.3)**Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

In order to simplify a ratio:

**Calculate the greatest common factor of the parts of the ratio.****Divide each part of the ratio by the greatest common factor.**

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEAssess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEUse a bar model to simplify the ratio 8:10.

**Calculate the greatest common factor of the parts of the ratio.**

Representing the ratio 8:10 as a bar model, you have…

Where red represents 8 shares and purple represents 10 shares. Thinking about how both 8 and 10 can be grouped equally, they are both divisible by 2 \, - which is also the GCF.

2**Divide each part of the ratio by the greatest common factor.**

Divide the bar above into two groups with the same amount of red shares and purple shares.

Each bar is the ratio 4:5, which is 8:10 in lowest terms.

Write the ratio 20:32 in lowest terms.

**Calculate the greatest common factor of the parts of the ratio.**

Factors of 20 = 1, \, 2, \, 4, \, 5, \, 10, \, 20

Factors of 32 = 1, \, 2, \, 4, \, 8, \, 16, \, 32

GCF (20,32) = 4

**Divide each part of the ratio by the greatest common factor.**

5:8 is in lowest terms.

Write the ratio 21:14 in lowest terms.

**Calculate the greatest common factor of the parts of the ratio.**

Factors of 21 = 1, \, 3, \, 7, \, 21

Factors of 14 = 1, \, 2, \, 7, \, 14

GCF (21,14) = 7

**Divide each part of the ratio by the greatest common factor**

3:2 is in lowest terms.

Write the ratio 16:24 in lowest terms.

**Calculate the greatest common factor of the parts of the ratio.**

Factors of 16 = 1, \, 2, \, 4, \, 8, \, 16

Factors of 24 = 1, \, 2, \, 3, \, 4, \, 6, \, 8, \, 12, \, 24

GCF (16, 24) = 8

**Divide each part of the ratio by the greatest common factor.**

2:3 is in lowest terms.

Write the ratio 55:95 in lowest terms.

**Calculate the greatest common factor of the parts of the ratio.**

Factors of 55 = 1, \, 5, \, 11, \, 55

Factors of 95 = 1, \, 5, \, 19, \, 95

GCF (55,95) = 5

**Divide each part of the ratio by the greatest common factor.**

11:19 is in lowest terms.

Write the ratio 145:212 in lowest terms.

**Calculate the greatest common factor of the parts of the ratio.**

Factors of 145 = 1, \, 5, \, 29, \, 145

Factors of 212 = 1, \, 2, \, 4, \, 53, \, 106, \, 212

GCF (145,212) = 1

**Divide each part of the ratio by the greatest common factor.**

Dividing each side by 1 will not change the ratio. The ratio is already in lowest terms.

- Writing ratios is a prerequisite skill for simplifying ratios. Give students plenty of practice writing ratios from the real world, before asking them to simplify them.

- Simplifying ratios worksheets can be useful, but be sure they involve a mixture of question types, including word problems that encourage ratio problem solving.

- Let struggling students use a ratio calculator. Ask them to draw bar models of the equivalent ratios and explain how the drawings prove equivalency. Once they have enough practice doing this, they can begin solving problems on their own.

**Not fully simplifying the ratio**Ratios can be simplified by other common factors, but will only be in simplest form when divided by the GCF.

For example,

Simplify the ratio 12:18.

You can simplify by dividing parts of the ratio by dividing both numbers by 2 :

While this is an equivalent ratio, this is not the lowest terms, since it can be simplified further by dividing by 3 to get the ratio 2:3.

**Thinking a ratio with smaller numbers is not equivalent**

Simplifying the ratio does not change the relationship – it just shows it in a different way.

For example,

The three models of 1:3 have the same number of blue and yellow in total as the 3:9 model. That is because these two ratios are equivalent.

**Simplifying just one side of the ratio**

A ratio shows the relationship between two or more numbers, so when simplifying, you must simplify all parts of the ratio.

**Using subtraction to simplify a ratio**

A ratio represents a multiplicative relationship. To make a ratio smaller, divide both sides by the same number. Subtracting the same number from both sides will not keep the relationship equivalent.

**Simplifying a ratio to include decimal numbers**

A simplified ratio should not include decimals.

For example,

While 0.75:1 is an equivalent ratio, when simplifying a ratio, your solution should have only whole numbers (integers). The ratio in its lowest terms is 3:4.

1) Which bar model shows the ratio 6:10 in lowest terms?

The original ratio 6:10 is shown with 6 blue squares and 10 orange squares.

The greatest common factor is 2, so divide the bar above into two groups with the same amount of blue shares and orange shares.

Each bar is the ratio 3:5, which is 6:10 in lowest terms.

2) Write the ratio 72:24 in lowest terms.

17:4

14:11

36:12

3:1

Factors of 24 = 1, \, 2, \, 3, \, 4, \, 6, \, 8, \, 12, \, 24

Factors of 72 = 1, \, 2, \, 3, \, 4, \, 6, \, 8, \, 9, \, 12, \, 18, \, 24, \, 36, \, 72

GCF (24,72) = 24

3) Write the ratio 36:30 in lowest terms.

6:5

5:6

12:10

13:15

Factors of 36 = 1, \, 2, \, 3, \, 4, \, 6, \, 9, \, 12, \, 18, \, 36

Factors of 30 = 1, \, 2, \, 3, \, 5, \, 6, \, 10, \, 15, \, 30

GCF (36,30) = 6

4) Write the ratio 28:42 in lowest terms.

4:6

1:14

14:21

2:3

Factors of 28 = 1, \, 2, \, 4, \, 7, \, 14, \, 28

Factors of 42 = 1, \, 2, \, 3, \, 6, \, 7, \, 14, \, 21, \, 42

GCF (28,42) = 14

5) Write the ratio 78:91 in lowest terms.

78:91

6:7

13:14

1:3

Factors of 78 = 1, \, 2, \, 3, \, 6, \, 13, \, 26, \, 39, \, 78

Factors of 91 = 1, \, 7, \, 13, \, 91

GCF (78,91) = 13

6) Write the ratio 102:81 in lowest terms.

102:81

51:41

34:27

4:5

Factors of 102 = 1, \, 2, \, 3, \, 6, \, 17, \, 34, \, 51, \, 102

Factors of 81 = 1, \, 3, \, 9, \, 27, \, 81

GCF(102,81) = 3

Yes, you can write one part of the ratio as the numerator and the other part as the denominator.

This is another name for the greatest common factor (GCF).

Yes, part of a ratio can be a fraction, decimal, mixed number, percentage, or other types of numbers. It can also be algebraic and show a relationship represented by polynomials.

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