Ratio to fraction

Here 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.

What is converting ratio to fraction?

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.

What is converting ratio to fraction?

Common Core State Standards

How does this relate to 6 th and 7 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.

How to convert a part to whole ratio to a fraction

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

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

Ratio to fraction examples

Example 1: picture

The 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?

1. 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.

2State 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.

Example 2: standard question

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

Example 3: word problem

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

How to convert a part to part ratio to a fraction

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

1. Decide the order of the ratio.
2. Fill in the numerator and denominator.

Example 4: picture

There are 6 bananas and 2 oranges.

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

Example 5: word problem – improper fraction with decimals

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

Example 6: ratios with complex fractions

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.

Teaching tips for ratio to fraction

• 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.

Easy mistakes to make

• 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.

Related ratio lessons

• Ratio
• Unit rate math
• How to write a ratio
• Simplifying ratios
• Ratio to percent
• How to calculate exchange rates
• Ratio problem solving
• Dividing ratios
• How to find the unit rate
• Ratio scale
• Constant of proportionality

The next lessons are

• Proportion
• Converting fractions, decimals and percentages
• Percent

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