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Simplifying fractions Equivalent fractions How to write a ratioHere you will learn about converting ratios to fractions, including using ratios to find fractions and using fractions to find ratios.

Students will first learn about ratio to fraction as part of ratio and proportions in 6 th grade and 7 th grade.

Converting **ratio to fraction** is a way of writing a ratio as a fraction. A **ratio** compares how much of one thing there is compared to another. It can be written using a ‘ \, \text{:} \, ’, the word ‘to’ or as a **fraction.**

There are two types of ratios that can be written as fractions: **part to whole** and **part to part.**

If a and b are parts of the same whole in the ratio a \, \text{:} \, b (read a to b ), you can write a part to whole fraction:

\cfrac{a}{a+b} \, and \, \cfrac{b}{a+b}.

For example,

The bar model below shows the ratio of blue \, \text{:} \, red as 3 \, \text{:} \, 2 ( 3 to 2 ). There are 3 blue blocks, 2 red blocks and 5 blocks in total.

The fraction of blue is \cfrac{3}{2+3}=\cfrac{3}{5} \, .

The fraction of red is \cfrac{2}{2+3}=\cfrac{2}{5} \, .

Here, the **total number** **of parts** in the ratio is equal to a+b (the denominator of each fraction) and the numerator of the fraction is the part of the ratio you are interested in.

This part to whole relationship allows us to make statements like…

- \cfrac{3}{5} \, of the blocks are blue.
- \cfrac{2}{5} \, of the blocks are red.
- \cfrac{5}{5} \, of the blocks are blue or red.

The ratio of blue \, \text{:} \, red as 3 \, \text{:} \, 2 can also be shown as a part to part fraction…

For example,

- Blue to red: \cfrac{3}{2} (read as 3 to 2 )
- Red to blue: \cfrac{2}{3} (read as 2 to 3 )

It is important to note that when a relationship is part to part, fraction language is used differently.

The model above does NOT show “ \cfrac{3}{2} of the blocks as blue” or “ \cfrac{2}{3} of the blocks as red.”

What the fractions do show is the ratio relationship BETWEEN the blue and red blocks. This allows us to make statements like…

- The number of blue blocks is \cfrac{3}{2} larger than red.
- The number of red blocks is \cfrac{2}{3} the amount of blue.

If a and b are NOT both parts of the same whole in the ratio a \, \text{:} \, b, you can ONLY write a part to part fraction:

\cfrac{a}{b} \, and \, \cfrac{b}{a}.

For example,

At a store, 4 apples costs \$ 2. This means the ratio of apples to dollars is 4 \, \text{:} \, \$ 2.

The ratio as a fraction is \cfrac{4}{\$ 2} \, .

The fraction in lowest terms is \cfrac{2}{\$ 1} and can be read as 2 apples per dollar.

The ratio as a fraction is \cfrac{\$ 2}{4} \, .

The fraction can be simplified to be \cfrac{\$ 1}{2}, then simplified even further to be \cfrac{\$ 0.5}{1} and can be read as \$ .50 per apple.

The simplified fractions with 1 as the denominator are unit rates. Notice that one fraction has a decimal as the top number (numerator).

When working with rates, particularly unit rate, it will often be necessary to divide by a number that is not a common factor. This will result in a rational number in one or both parts of the fraction.

How does this relate to 6 th grade math?

**Grade 6 – Ratios and Proportional Relationships (6.RP.A.1)**Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2 \, \text{:} \, 1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

**Grade 6 – Ratios and Proportional Relationships (6.RP.A.2)**

Understand the concept of a unit rate \cfrac{a}{b} associated with a ratio a \, \text{:} \, b with b≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is \cfrac{3}{4} cup of flour for each cup of sugar.” “We paid \$ 75 for 15 hamburgers, which is a rate of \$ 5 per hamburger.”

**Grade 7 – Ratios and Proportional Relationships (7.RP.A.1)**

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks \cfrac{1}{2} mile in each \cfrac{1}{4} hour, compute the unit rate as the complex fraction \cfrac{\cfrac{1}{2}}{\cfrac{1}{4}} miles per hour, equivalently 2 miles per hour.

**Grade 7 – Ratios and Proportional Relationships (7.RP.A.2b)**

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

In order to convert a part to whole ratio to a fraction:

**Add the parts of the ratio to create the denominator (the whole).****State the required part of the ratio as the numerator.**

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 FREEThe diagram below shows part of the repeating pattern of red and yellow beads on a bracelet.

What fraction of the beads in the bracelet are red?

**Add the parts of the ratio to create the denominator (the whole).**

Since the 5 yellow beads and 3 red beads repeat, the ratio of yellow to red beads is 5 \, \text{:} \, 3.

The total of the parts is 5+3=8.

2**State the required part of the ratio as the numerator.**

3 out of the 8 beads are red, so \cfrac{3}{8} of the beads in the bracelet are red.

Ann and Bob share a box of cookies in the ratio of 3 \, \text{:} \, 4. What fraction of the cookies does Bob receive?

**Add the parts of the ratio to create the denominator (the whole).**

Ann gets 3 parts and Bob gets 4 parts, so there are 7 parts in the whole: 3+4=7.

**State the required part of the ratio as the numerator.**

Bob receives 4 parts, so he receives \cfrac{4}{7} of the box of cookies.

\cfrac{3}{4} of a school of fish are male. The rest are female. Write the ratio of females to males in the school.

**Add the parts of the ratio to create the denominator (the whole).**

In this case, you need to work backwards. The parts have already been added. The numerator is 3, so that is one part – the males. The denominator is 4, so that is the total.

3 \; + \; ?=4

**State the required part of the ratio as the numerator.**

Since 3+1=4, the missing numerator must be 1 – the females.

The ratio of females to males is 1 \, \text{:} \, 3.

In order to convert a part to part ratio to a fraction:

**Decide the order of the ratio.****Fill in the numerator and denominator.**

There are 6 bananas and 2 oranges.

Complete the sentence: The amount of oranges is *__* the amount of bananas.

**Decide the order of the ratio.**

The sentence is comparing oranges to bananas, which is the ratio 2 \, \text{:} \, 6.

**Fill in the numerator and denominator.**

The ratio 2 \, \text{:} \, 6, can be written as \cfrac{2}{6} or \cfrac{1}{3}.

The amount of oranges is \cfrac{1}{3} the amount of bananas.

A biker travels 34 miles in 5 hours. Write the ratio as a fraction comparing miles per hour.

**Decide the order of the ratio.**

You are asked to compare miles to hours, which is the ratio 34 \, \text{:} \, 5.

**Fill in the numerator and denominator.**

The ratio 34 \, \text{:} \, 5, can be written as \cfrac{34}{5}, which is read 34 to 5.

Dividing each part of the ratio by 5 :

\cfrac{34 \div 5}{5 \div 5}=\cfrac{6.8}{1}

This is the rate of 6.8 miles per hour (since the simplified denominator represents 1 hour).

A recipe calls for \cfrac{1}{2} cup of chopped tomatoes and \cfrac{3}{4} cup of cheese.

Write the ratio of cups of chopped tomatoes to cups of cheese as a fraction.

**Decide the order of the ratio.**

You are asked to compare tomatoes to cheese, which is the ratio \cfrac{1}{2} \, \text{:} \, \cfrac{3}{4}.

**Fill in the numerator and denominator.**

The ratio \cfrac{1}{2} \, \text{:} \, \cfrac{3}{4}, can be written as \cfrac{\cfrac{1}{2}}{\cfrac{3}{4}}, which is read \cfrac{1}{2} to \cfrac{3}{4}.

Multiplying each part of the ratio by \cfrac{4}{3}…

\cfrac{\cfrac{1}{2} \times \cfrac{4}{3}}{\cfrac{3}{4} \times \cfrac{4}{3}}=\cfrac{\cfrac{4}{6}}{1} or \cfrac{\cfrac{2}{3}}{1}

The fraction is read as the rate \cfrac{2}{3} cup of chopped tomatoes for every cup of cheese (since the simplified denominator represents 1 cup of cheese).

- Be mindful of the ratios being used when choosing worksheets for your students. Students should begin working only with whole number ratios and then progress to use rational numbers. Complex fractions should be introduced later.

- Be very clear about the difference between part to whole ratios and part to part ratios. Use precise language for all ratios written in fraction form and correct students when they read a part to part ratio in incorrect fraction language.

**Ratios and fractions confusion**

For example, the ratio 2 \, \text{:} \, 3 can be expressed as the fraction \cfrac{2}{3} or \cfrac{2}{5}, depending on the context. It depends on whether you are asked for a ratio of a part to whole or part to part.

**Incorrect value for the numerator**

Always pay attention to the order of the ratio and what part is being asked for in the question.

For example, the number of mugs to glasses in a kitchen is 4 \, \text{:} \, 3. What fraction of cups in the kitchen are mugs?

The correct answer is \cfrac{4}{7}. The “mugs” value is the first number in the ratio, m \, \text{:} \, g=4 \, \text{:} \, 3 as you use the same order as the written sentence. Writing the ratio as \cfrac{3}{7} is incorrect.

**Ratio written in the wrong order**

The parts of the ratio are written in the wrong order. For example the number of trees to flowers is given as the ratio 11 \, \text{:} \, 13 but the solution is written as 11 \, \text{:} \, 12.

1. The ratio of tyrannosaurus rex to velociraptor fossils is 3 \, \text{:} \, 8.

What fraction of the fossils are tyrannosaurus rex? Give your answer as a fraction in its simplest form.

\cfrac{3}{8}

\cfrac{8}{11}

\cfrac{3}{11}

\cfrac{8}{3}

Total number of fossils is 3+8=11.

3 out of 11 are T-Rex fossils which is written as the fraction \cfrac{3}{11}.

2. A dealership sells new cars to used cars in a ratio of 8 \, \text{:} \, 7.

Calculate the fraction of cars sold that are new.

\cfrac{8}{7}

\cfrac{7}{8}

\cfrac{7}{15}

\cfrac{8}{15}

Calculate the total number of cars sold: 8+7=15,

8 out of 15 cars sold are new which is written as the fraction \cfrac{8}{15}.

3. A football team won \cfrac{3}{5} of their games in the league. They did not tie any games. Write the ratio of wins : losses.

3 \, \text{:} \, 2

3 \, \text{:} \, 5

3 \, \text{:} \, 0 \, \text{:} \, 2

2 \, \text{:} \, 3

In this case, you need to work backwards. The parts have already been added. The numerator is 3, so that is one part – the wins. The denominator is 5, so that is the total number of games.

3 \; + \; ? = 5

Since 3+2=4, the missing numerator must be 2 – the losses.

The ratio of wins to losses is 3 \, \text{:} \, 2.

4. Which statement correctly compares the bananas and oranges?

The number of oranges is \cfrac{1}{4} the number of bananas.

The number of oranges is \cfrac{1}{5} the number of bananas.

The number of bananas is \cfrac{1}{4} the number of oranges.

The number of bananas is \cfrac{1}{5} the number of oranges.

Comparing bananas to oranges the ratio is 1 \, \text{:} \, 4.

The ratio 1 \, \text{:} \, 4, can be written as \cfrac{1}{4}.

The amount of bananas is \cfrac{1}{4} the amount of oranges.

5. 45 baseball bats weigh 135 pounds. Write the ratio as a fraction in simplest form, comparing the number of bats to weight in pounds.

\cfrac{1}{3}

\cfrac{1}{4}

3

\cfrac{135}{45}

You are asked to compare the number of bats to weight in pounds, which is the ratio 45 \, \text{:} \, 135.

The ratio 45 \, \text{:} \, 135, can be written as \cfrac{45}{135}, which reads 45 to 135.

Divide each part of the ratio by 45…

\cfrac{45 \div 45}{135 \div 45}=\cfrac{1}{3}

The fraction becomes \cfrac{1}{3} and is read as the rate 1 bat to 3 pounds.

6. A paint mixture is \cfrac{1}{3} tablespoon red paint and \cfrac{2}{5} tablespoon blue paint. Which of the following ratios is blue paint to red paint written as a non-complex fraction?

\cfrac{3}{8}

\cfrac{2}{15}

\cfrac{6}{5}

\cfrac{5}{6}

You are asked to compare blue paint to red paint, which is the ratio \cfrac{2}{5} \, \text{:} \, \cfrac{1}{3}.

The ratio \cfrac{2}{5} \, \text{:} \, \cfrac{1}{3}, can be written as \cfrac{\cfrac{2}{5}}{\cfrac{1}{3}}, which is read \cfrac{2}{5} to \cfrac{1}{3}.

Multiply each part of the ratio by \cfrac{3}{1}….

\cfrac{\cfrac{2}{5} \times \cfrac{3}{1}}{\cfrac{1}{3} \times \cfrac{3}{1}}=\cfrac{\cfrac{6}{5}}{1}

The fraction is the rate \cfrac{6}{5} tablespoons of blue paint for every tablespoon of red paint (since the simplified denominator represents 1 tablespoon of red paint).

*Note: This ratio can also be expressed with a mixed number \cfrac{1 \cfrac{1}{5}}{1} or rate 1 \cfrac{1}{5} tablespoons of blue paint for every tablespoon of red paint.

When using fraction notation to represent ratios, the numerator (top number) and denominator (bottom number) can be whole numbers, decimals, fractions, integers, mixed fraction or any rational number.

When a fraction ratio is in lowest terms, both the numerator and denominator are prime numbers.

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