Topics
5. Ratios & Proportional Relationships
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United States of AmericaVirginia
Grade 6

5.01 Introduction to ratios

Lesson
Practice

The language of ratios

Exploration

Consider the following image that shows ingredients for a batch of smoothies:

An image with 4 bananas and 6 strawberries.
  1. What kind of comparisons can you make between the number of strawberries and the number of bananas in the image?

A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another. In other words, a ratio is an association between two or more quantities.

The ratio of strawberries to bananas is 6 to 4. The ratio of bananas to strawberries is 4 to 6.

We can also associate the two quantities by saying for every 6 strawberries, there are 4 bananas. For every 4 bananas, there are 6 strawberries.

We can also write ratios in the form a:b which is read as "a to b".

If we want to describe the relationship between the number strawberries to the number of bananas, we could write the ratio as 6:4.

Finally, ratios can be represented as a fraction.

We can describe the ratio of bananas to strawberries as the fraction \dfrac{4}{6}.

The order that the words are written correspond to the order of the values in the ratio, so it is important that pay attention to the order.

Examples

Example 1

Write the ratio of butterflies to ladybugs in three ways.

An image with 3 butterflies and 8 ladybugs.
Worked Solution
Create a strategy

Consider all the ways we talked about how to compare quantities using both words and symbols.

Apply the idea

The ratio of butterflies to ladybugs is 3 to 8.

Using ratio notation, we can also express the ratio as either 3:8 or \dfrac{3}{8}.

Reflect and check

Consider whether 8:3 would describe the desired ratio.

This would mean that there are 8 butterflies for every 3 ladybugs. This does not match the image since there should be more ladybugs than butterflies. The order of the numbers in a ratio is important.

Example 2

Lily describes the ratio between two quantities at the beach as \dfrac{5}{4}. Use words to describe the relationship that the ratio may represent and draw a picture to represent the description of your ratio.

Worked Solution
Create a strategy

Remember that ratios describe the relationships between two quantities. In a fraction, the numerator describes one quantitiy and the denominator describes the other.

Apply the idea

We have 5 of one quantity and 4 of another.

At the beach, she could be noticing the number of sea shells and the number of sand dollars.

An image of 5 seashells and 4 sand dollars.

We could say there are 5 seashells for every 4 sand dollars; or the ratio of seashells to sand dollars is \dfrac{5}{4} or 5:4.

Idea summary

A ratio compares the relationship between two quantities. It tells us how much there is of one thing compared to another.

We can also write ratios in the form a:b which is read as "a to b" or write them as the fraction \dfrac{a}{b}.

Ratios as comparisons

We can also use compare quantities by using part to part ratios, part to whole ratios, and whole to whole ratios. These ratios can describe different relationships in the same scenario.

Recall this recipe for smoothies:

An image with 4 bananas and 6 strawberries.

A part to part ratio describes the ratio between two parts of a whole. In this recipe, the number of strawberries and the number of bananas are both parts of the whole group of fruit. The ratio of strawberries to bananas is 6:4.

A part to whole ratio describes the relationship between one quantitiy and the total group of quantities. For example, if we wanted to describe the ratio of bananas to the total amount of fruit in this recipe, we could say "for every 4 bananas, we need 10 total fruit for this batch of smoothies". We could write it as 4 : 10.

We now want to decide between two recipes for a batch of smoothies. Consider these two recipes:

The image shows Recipe 1 and Recipe 2. Recipe 1 have 4 bananas and 6 strawberries. Recipe 2 have 2 bananas and 4 strawberries.

A whole to whole ratio is a ratio that compares the total of one quantity to the total of another. In this example, Recipe A has 10 total fruit, and Recipe B has 6 total fruit. So the ratio of the total fruit in Recipe A to the total fruit in Recipe B is 10:6.

A part to part ratio can also compare between parts of one group to parts of another group. For example, for every 6 strawberries needed in Recipe A, we need 4 strawberries for Recipe B. This ratio can be written as 6:4 or \dfrac{6}{4}.

Tape diagrams can be used to visually represent ratios without drawing the items to compare. This diagram shows represents the relationships in Recipe A

The image titled Recipe 1. There are 2 tape diagrams: 1st tape diagram divided into 6 parts and labeled as Strawberries, and 2nd tape diagram divided into 4 parts and labeled as Bananas. There are 2 arrows that says '10 total fruit.'

Each box of the same size will represent the same amount of an item. Tape diagrams may be simplified and each box may represent more than a single item. This tape diagram also represents Recipe A:

The image titled Recipe 1. There are 2 tape diagrams: 1st tape diagram divided into 3 parts with label 2 each and labeled as Strawberries, and 2nd tape diagram divided into 2 parts with label 2 each and labeled as Bananas. There are 2 arrows that says '10 total fruit.'

Tape diagrams are a usefool tool to model and solve problems involving ratios.

Examples

Example 3

Consider the following diagram of a forest:

A set of 15 trees with 2 trees inside the shaded region.
a

Write the ratio of the shaded region of trees to the total number of trees.

Worked Solution
Create a strategy

This is a part to whole ratio. Count the number of shaded trees and the total number of trees including the shaded ones.

Write the ratio in the form a:b

Apply the idea

The number of shaded trees =2.

The total number of trees =15.

The ratio is 2:15

b

Write the ratio of the shaded trees to the unshaded trees.

Worked Solution
Create a strategy

This is a part to part ratio. Count the number of shaded trees and the number of unshaded trees.

Write the ratio in the form a:b

Apply the idea

The number of shaded trees =2.

The total number of unshaded trees =13.

The ratio of the shaded trees to the unshaded trees is 2:13

c

Consider this diagram of a second forest:

A set of 10 trees with 3 trees inside the shaded region.

Write the ratio of the total number of trees in the first forest to the total number of trees in the second forest.

Worked Solution
Create a strategy

This is a whole to whole ratio. Count the total number of trees in the first forest diagram and the total number of trees in the second forest diagram.

Write the ratio in the form a:b

Apply the idea

The total number of trees in the first forest is =15.

The total number of trees in the second forest is =10.

The ratio of the shaded trees to the unshaded trees is 15:10

Example 4

Emma and Riley are both on their school's track team. The ratio of Emma's weekly mileage to Riley's weekly mileage is 5:3. Their combined weekly mileage is 24 miles.

The ratio 5:3 is diplayed using a tape diagram.

The image shows 2 tape diagrams: 1st tape diagram for Emma is divided into 5 parts, and 2nd tape diagram for Riley is divided into 3 parts. There are 2 arrows that says '24 total miles.'
a

How many miles does each part represent? Justify your reasoning.

Worked Solution
Create a strategy

First count up the total number of parts in the diagram.

We know the values of each part must add up to 24.

Apply the idea

There are 8 total parts.

8 \text{ parts} = 24 \text{ miles}

\dfrac{8 \text{ parts}}{8} = \dfrac{24 \text{ miles}}{8}

1 \text{ part} = 3 \text{ miles}

Since 8 of the parts is equal to 24 total miles, each part must represent 3 miles.

b

What is Emma's weekly mileage?

Worked Solution
Create a strategy

From part (a), we discovered that each part is worth 3 miles. If we think of each part in Emma's tape diagram as representing 3 miles, we can calculate her total weekly mileage.

Apply the idea

In the tape diagram, Emma had 5 parts. We established that each part was worth 3 miles.

A tape diagram for Emma is divided into 5 parts with label of 3 each.

In total, Emma's weekly mileage is 5 \cdot 3 = 15 miles.

Example 5

The ratio 3:4 represents a comparison between items at a grocery store.

a

Create a context using this ratio in a part to part comparison.

Worked Solution
Create a strategy

A part to part ratio describes the ratio between two parts of a whole.

Consider any two quantities that are "part" of the grocery store.

Apply the idea

The ratio of oranges to apples in the grocery store is 3:4. This means for every 3 oranges, there are 4 apples in the grocery store.

Reflect and check

Remember that the ratio 3:4 does not mean there are only 3 oranges and 4 apples in the grocery store.

An image with 3 groups of 3 oranges and 4 apples.

This group would also represent the ratio 3:4 because for every 3 oranges, there are 4 apples.

b

Create a context using this ratio in a part to whole comparison.

Worked Solution
Create a strategy

A part to whole ratio describes the relationship between one quantitiy and the total group of quantities.

Apply the idea

We could compare the amount of white bread to the total amount of bread sold at the grocery store.

The ratio 3:4 would mean that for every 3 white breads sold, there are 4 total breads sold.

c

Create a context using this ratio in a whole to whole comparison.

Worked Solution
Create a strategy

A whole to whole ratio is a ratio that compares the total of one quantity to the total of another.

Apply the idea

We could compare the total number of frozen vegetables to the total number of fresh vegetables.

The ratio of 3:4 would mean that for every 3 frozen vegetables, there are 4 fresh vegetables.

Idea summary

A ratio can represent different comparisons within the same quantity or between different quantities.

  • part to whole - compares part of a whole to the entire whole
  • part to part - compares part of a whole to another part of the same whole
  • whole to whole - compares all of one whole to all of another whole

Outcomes

6.PFA.1

The student will use ratios to represent relationships between quantities, including those in context.

6.PFA.1a

Represent a relationship between two quantities using ratios.

6.PFA.1b

Represent a relationship in context that makes a comparison by using the notations a/b, a:b, and a to b.

6.PFA.1c

Represent different comparisons within the same quantity or between different quantities (e.g., part to part, part to whole, whole to whole).

6.PFA.1d

Create a relationship in words for a given ratio expressed symbolically.

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