4.02 Simplify ratios

Expressing ratios as fractions

Last lesson we saw that we could express a ratio as a part-to-part relationship or a part-to-whole relationship. For example, if we wanted to describe the ratio of green dots to blue dots in the image above, we would write it as ratio $2:3$2:3. And alternatively, if we wanted to express the ratio of green dots to the total amount of dots we could use the ratio $2:5$2:5.

We can also express ratios as fraction in these two ways. The ratio of green to blue dots is $\frac{2}{3}$23​, which we can interpret as there are two-thirds as many green dots as blue dots. We can also express the green dots as a ratio out of the total amount of dots with the ratio $\frac{2}{5}$25​, that is two-fifths of the total number of dots are green. When expressing ratios take care if you want a direct comparison between parts or the proportion those parts make up of a whole.

 

Worked example

Example 1

A cordial mix requires the ratio of $30$30 ml of cordial to $100$100 ml of water.

(a) Express the ratio of cordial to water as a fraction in simplest form.

Think: As we want the ratio of cordial to water, the amount of cordial will be in the numerator and water will be in the denominator.

Do:

Ratio cordial to water	$=$=	$\frac{30}{100}$30100​
	$=$=	$\frac{30\div10}{100\div10}$30÷10100÷10​	Divide top and bottom by a common factor of $10$10 to simplify the fraction.
	$=$=	$\frac{3}{10}$310​

This means there is $\frac{3}{10}$310​ the amount of cordial compared to water. 

(b) Write the ratio of cordial in a mixed drink as a fraction in simplest form.

Think: This time we want the ratio of cordial to the total liquid in a mixed drink. We have $30$30 ml of cordial and $100$100 ml of water, so the total will be $130$130 ml.

Do:

Ratio cordial to total	$=$=	$\frac{30}{130}$30130​
	$=$=	$\frac{30\div10}{130\div10}$30÷10130÷10​	Divide top and bottom by a common factor of $10$10 to simplify the fraction.
	$=$=	$\frac{3}{13}$313​

Thus, in a drink made to the given ratio cordial will be $\frac{3}{13}$313​ of the total drink.

 

Practice questions

Question 1

Question 2

Question 3

 

Equivalent and simplified ratios

Just as we can make equivalent fractions and can simplify fractions we can also do this with ratios written in the form $a:b$a:b. To make an equivalent fraction, recall we can multiply or divide the numerator and denominator by the same value. Having seen above how we can express a ratio as a fraction, this would be the same as multiplying or dividing both sides of a ratio by the same number. Multiplication and division by the same number preserves the proportion of the values in the ratio.

Consider the following example: a cake recipe that uses $1$1 cup of milk and $4$4 cups of flour, that is, the ratio of milk to flour is $1:4$1:4. What if we want to make two cakes? We would need to double the amount of milk and flour we use. This means we will need $2$2 cups of milk and $8$8 cups of flour. Now the ratio of milk to flour is $2:8$2:8.

But do we get two different ratios from the same recipe? No, the two ratios actually represent the same proportion of milk to flour. We say that $1:4$1:4 and $2:8$2:8 are equivalent ratios.

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$1. Just as we simplify fractions by dividing by the numerator and denominator by a common factor we can divide 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 example

Example 2

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

 

Hence, 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 highest 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 if rather than measuring in millilitres we put $1$1 tablespoon on vinegar we know we need to put $3$3 tablespoons of olive oil.

 

Practice questions

Question 4

Question 5

Question 6

Question 7