4.05 Simplifying ratios

A ratio compares the relationship between two or more quantities of the same type. It shows how much there is of one thing compared to another.

Suppose a train carriage has $45$45 people in it, of which $25$25 are male and $20$20 are female. We can express the ratio of men to women on the train as $25:20$25:20, and we can say there are $25$25 men for every $20$20 women.

Expressing ratios

A ratio can express a part-to-part relationship, as in the example above. But it can also describe a part-to-whole relationship. For example, if we wanted to describe the ratio of males to all passengers on the train, we would write it as $25:45$25:45 because there are $25$25 males out of the $45$45 people in total. The order that the words are written in the question corresponds to the order of the values in the ratio, so we need to be careful not to jumble them around.

Simplifying ratios

Often we would like to express ratios as whole numbers, using the simplest numbers possible. We can simplify ratios in a similar way to how we simplify fractions. If we multiply or divide one side by a number, we do the same thing to the other side. A ratio is said to be in its simplest form when all terms have a highest common factor of $1$1.

Worked examples

Question 1

Express $25:20$25:20 as a ratio in its simplest form.

Think: $5$5 is a factor of both $25$25 and $20$20, so we can simplify by dividing both sides of the ratio by $5$5.

Do: $25\div5=5$25÷​5=5 and $20\div5=4$20÷​5=4, so the ratio $25:20$25:20 can be written as $5:4$5:4.

Ok, now let's look at an example of a ratio that is not made up entirely of whole numbers.

question 2

Simplify the ratio $1\frac{1}{2}:6$112​:6.

Think: Notice that $1\frac{1}{2}$112​ is the same as $\frac{3}{2}$32​, so we can write it as a whole number by multiplying by $2$2. Remember that we will then need to perform the same multiplication to the other part of the ratio. Once we have a ratio of two whole numbers, we can then look to simplify.

Do:

$1\frac{1}{2}:6$112​:6	$=$=	$\frac{3}{2}:6$32​:6	(Convert to an improper fraction)
	$=$=	$3:12$3:12	(Multiply both parts of the ratio by $2$2)
	$=$=	$1:4$1:4	(Simplify by dividing by the common factor $3$3)

Keeping things in the same units

Ratios are useful to describe the relationship between two values. However, to make sure that everyone can understand the ratio, it is only really useful if we compare things with the same unit of measurement.

Let's say a cookie recipe required $150$150 grams of butter for every kilogram of flour being used. We might be tempted to say that the ratio of butter to flour is $150:1$150:1. But does this seem correct? That is a lot of butter compared to the amount of flour.

We can find the right ratio by first making sure the measurements of each quantity are in the same units. We can change the $1$1 kg of flour to $1000$1000 g of flour, and then write the ratio as $150:1000$150:1000 which can then be simplified to $3:20$3:20.

Notice that the ratio itself has no units at all. This means we can use the ratio with any unit. For example, we could make a huge batch of cookies by using $3$3 kilograms of butter and $20$20 kilograms of flour. Or we could make only one cookie by using $15$15 grams of butter and $100$100 grams of flour. In both cases the ratio of butter to flour is the same.

Worked example

Write the ratio for $50c$50c to $\$2.10$$2.10 by first converting to the same unit, and then simplifying.

Think: One price is given in cents and the other in dollars. We can first write both in cents, then look to simplify the ratio.

Do: The price $\$2.10$$2.10 is the same as $210c$210c, so now our ratio becomes $50:210$50:210. The highest common factor of $50$50 and $210$210 is $10$10, so if we divide both sides of the ratio by $10$10 we have $5:21$5:21, which is in the most simplified form.

Equivalent ratios

Just like two expressions, say $2+3$2+3 and $9-4$9−4, are equivalent when they simplify to the same value, two ratios are equivalent when we can simplify them to give the same ratio.

Worked example

Is $10:20$10:20 equivalent to $4:8$4:8?

Think: If we simplify $10:20$10:20, we would get an answer of $1:2$1:2. If we simplify $4:8$4:8, we also get an answer of $1:2$1:2.

Do: Since both ratios simplify to $1:2$1:2, we can say that $10:20$10:20 is equivalent to $4:8$4:8.

Practice questions

Question 1

Question 2

Question 3