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3 Ratios, rates, and time
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AustraliaVIC
Level 8

3.01 Ratios

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Introduction

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

One blue dot and 3 green dots

If we want to describe the relationship between the number of blue dots and the number of green dots, we could say that there is 1 blue dot for every 3 green dots. We could also express this as a ratio, which we would write as 1\text{:}3.

It is important that both quantities are in the same units, otherwise we cannot compare them with a ratio. In the example above, we are comparing one blue dot to three green dots, so a single dot can be thought of as our unit.

Equivalent and simplified ratios

We can make equivalent ratios written in the form a:b by multiplying or dividing each value in the ratio by the same number to preserve the proportions.

A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values. This is the same as saying that the two integers in the ratio have a highest common factor of 1. We simplify ratios by dividing both sides of a ratio expression by a common factor. A simplified ratio uses only integers. A ratio that includes fractions or decimals is not yet fully simplified and can be increased or decreased by an appropriate multiple to simplify it.

Examples

Example 1

Write 30 to 70 as a fully simplified ratio.

Worked Solution
Create a strategy

Divide the ratio by the highest common factor.

Apply the idea
\displaystyle 30:70\displaystyle =\displaystyle \dfrac{30}{10}: \dfrac{70}{10}Divide by 10
\displaystyle =\displaystyle 3:7Evaluate
Reflect and check

So we see that 30:70 and 3:7 are equivalent ratios, since we can divide the first ratio by 10 to get the second.

Idea summary

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.

A ratio is a simplified ratio if the values are integers and have no common factors other than 1.

Divide a quantity by a ratio

We may also want to divide a quantity into a given ratio. This can be done using two main methods. The unitary method and the fraction method.

The unitary method is named for the key step in which we find one part (one unit) of the whole amount. From there we can find the value of any number of parts. The fraction method finds what fraction of the whole each side of the ratio is.

Unitary method

  • Calculate the total number of parts (by adding all the numbers in the ratio)

  • Calculate what one part is worth (by dividing the given value by the total number of parts)

  • Calculate what each share of the ratio is worth (by multiplying what one part is worth with each number in the ratio)

Fraction method

  • Divide each side of the ratio by the total number of parts (by adding all the numbers in the ratio)

  • Each share is worth the corresponding fraction multiplied by the total amount

Examples

Example 2

A salad dressing is supposed to have a 5\text{:}16 ratio of vinegar to oil. If there are 13\text{ mL} of vinegar, how many \text{mL} of oil should be added? Round your answer to the nearest whole \text{mL}.

Worked Solution
Create a strategy

We can use the fraction method.

Apply the idea
\displaystyle \text{Oil}\displaystyle =\displaystyle \dfrac{16}{5} \times 13Multiply the fraction by 13
\displaystyle =\displaystyle 41\dfrac{3}{5}Evaluate
\displaystyle =\displaystyle 42\text{ mL}Round the answer
Idea summary

Unitary method

  • Calculate the total number of parts (by adding all the numbers in the ratio)

  • Calculate what one part is worth (by dividing the given value by the total number of parts)

  • Calculate what each share of the ratio is worth (by multiplying what one part is worth with each number in the ratio)

Fraction method

  • Divide each side of the ratio by the total number of parts (by adding all the numbers in the ratio)

  • Each share is worth the corresponding fraction multiplied by the total amount

Triple ratio

We can also use ratios to relate three quantities in the form a:b:c.

This works using the same rules as before, however all three quantities must be multiplied or divided by the same number when simplifying or finding equivalent ratios.

Examples

Example 3

A piece of rope is cut into three lengths in the ratio 3\text{:}4\text{:}8. The shortest length of rope is measured to be 18\text{ m} long.

a

Find the middle length of the rope.

Worked Solution
Create a strategy

Divide the length of the rope by the parts for the shortest length and multiply it by the parts for the middle length.

Apply the idea
\displaystyle \text{Length}\displaystyle =\displaystyle \dfrac{18}{3} \times 4Multiply one part by 4
\displaystyle =\displaystyle 6\times 4Evaluate the division
\displaystyle =\displaystyle 24\text{ m}Evaluate
b

Find the longest length of the rope.

Worked Solution
Create a strategy

Divide the length of the rope by the parts for the shortest length and multiply it by the parts for the longest length.

Apply the idea
\displaystyle \text{Length}\displaystyle =\displaystyle \dfrac{18}{3} \times 8Multiply one part by 8
\displaystyle =\displaystyle 6 \times 8Evaluate the division
\displaystyle =\displaystyle 48\text{ m}Evaluate
Idea summary

We can use ratios to relate three quantities in the form a:b:c.

Outcomes

VCMNA277

Solve a range of problems involving rates and ratios, including distancetime problems for travel at a constant speed, with and without digital technologies

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