2.01 Ratios

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

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$1 blue dot for every $3$3 green dots. We could also express this as a ratio, which we would write as $1:3$1:3.

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 $a:b$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 greatest common factor of $1$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.

Worked examples

Example 1

A recipe for a salad dressing includes $20$20 ml of vinegar and $60$60 ml of olive oil.

a) Fill in the table shown below to make equivalent ratios for a larger amount of dressing.

Vinegar	to	Olive Oil
$20$20	:	$60$60
$30$30	:	$90$90
$40$40	:	$\editable{}$
$\editable{}$	:	$300$300

Think: What do you need to multiply one side of the ratio by to get the known value? Multiply both sides by this value to get an equivalent ratio.

Do:

Vinegar	to	Olive oil		Vinegar	to	Olive oil
$20$20	:	$60$60		$20$20	:	$60$60
$\times2$×2		$\times2$×2		$\times5$×5		$\times5$×5
$40$40	:	$\editable{}$		$\editable{}$	:	$300$300

 

So the completed table is:

Vinegar	to	Olive Oil
$20$20	:	$60$60
$30$30	:	$90$90
$40$40	:	$120$120
$100$100	:	$300$300

b) What is the simplified ratio of vinegar to olive oil in the dressing?

Think: What is the greatest common factor of $20$20 and $60$60? Divide both sides of the ratio by this number.

Do:

Vinegar	to	Olive oil
$20$20	:	$60$60
$\div20$÷20		$\div20$÷20
$1$1	:	$3$3

 

The simplified ratio of vinegar to olive oil is $1:3$1:3. A simplified ratio is great for simple recipes, this one tells us we need three times as much olive oil than vinegar. So rather than measuring in millilitres, if we have $1$1 tablespoon on vinegar we know we need to put $3$3 tablespoons of olive oil.

 

Example 2

For every two metres Alex walks to the right, Hayley walks five metres to the left. 

a) Write this scenario as a ratio.

Think: Just the distances can be written in a ratio as $2:5$2:5. But we also need to take into account the different directions. So to show that the direction that Hayley walks is the opposite of the direction that Alex walks, we can make the $5$5 a negative. 

Do: So, the final ratio is $2:-5.$2:−5.

b) If Alex walks $240$240 metres to the right, how far has Hayley walked and in what direction?

Think: We need to find how many multiples of $2$2 there are in $240$240. Then we can multiply this number by $5$5. 

Do: $240\div2=120$240÷​2=120, so $-5\times120=-600$−5×120=−600. So Hayley walked $600$600 metres in the left direction.

 

Practice question

question 1

 

Triple ratio

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

Worked example

example 3

A certain paint colour has a ratio of blue, yellow and red given by $2:3:6$2:3:6. If $20$20 ml of blue paint is added how much red paint will be added?

Think: We can consider just the relevant parts of the ratio for this question, the blue to red ratio will be $2:6$2:6.

Do: We can multiply both sides by $20$20 to find an equivalent ratio:

$2:6$2:6	$=$=	$1:3$1:3	Simplifying the ratio
	$=$=	$20:60$20:60	Multiplying both sides by $20$20
Amount of red paint	$=$=	$60$60 mL

 

Reflect: Notice that the ratio of just two parts can be simplified, however the original ratio can not be simplified as not all three parts can simplify.

Practice question

question 2