5. Ratios & Proportional Relationships

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

- 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.

Write the ratio of butterflies to ladybugs in three ways.

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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.

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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}.

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:

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:

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

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:

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

Consider the following diagram of a forest:

a

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

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b

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

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c

Consider this diagram of a second forest:

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

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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.

a

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

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b

What is Emma's weekly mileage?

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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.

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b

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

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c

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

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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