8.01 Working with 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.

 

Expressing ratios

A ratio can express a "part to part" relationship like we saw in the dot example above. We can also describe a "part to whole relationship". For example, if we wanted to describe the ratio of green dots to all the dots, we could write it as $3:4$3:4, because there are $3$3 green dots and $4$4 dots in total.

The order that the words are written correspond to the order of the values in the ratio, so it is important that we don't jumble them around.

We can also express fractions, decimals and percentages as ratios.

Worked examples

Example 1

During one day at the airport, $5$5 flights were delayed and $21$21 flights were on-time. Write a ratio comparing the number of delayed flights to the number of on-time flights.

Think: As both values are in the same units, the number of flights, we can compare the two quantities as a ratio $a:b$a:b. The order is important, so the first number in the ratio will represent the number of delayed flights.

Do: $5:21$5:21

Example 2

There are $53$53 grams of sugar in $100$100 grams of chocolate spread. Express the amount of sugar to the total amount of spread as a ratio.

Think: Both values are in grams, so we can express the two values directly as a ratio. We want to express $53$53 parts to $100$100 parts as a ratio.

Do: $53:100$53:100

 

Converting between units

Often we might want to compare two quantities that have different units, such as a number of minutes compared to a number of hours, a distance in kilometres to a distance in metres, a duration in days to a duration in weeks, and so on.

To compare these types of quantities, we will need to convert one of the quantities to use the same units as the other. It does not matter which one we convert, we will end up with exactly the same ratio in the end.

Worked example

example 3

Fernando takes $23$23 minutes to travel to work every day, and he spends $4$4 hours at work. Write a ratio comparing the time spent travelling to work, to the time spent at work.

Think: We are comparing two quantities with different units, one is in minutes and the other is in hours, so we need to convert one to the other before we compare them as a ratio.

There are $60$60 minutes in an hour, so we can convert $4$4 hours to minutes by multiplying $4$4 by $60$60.

Do: First, we will convert $4$4 hours into minutes, $4\times6=240$4×6=240, so now we can compare $23$23 minutes to $240$240 minutes.

Now that both values are in the same units, we can write the ratio of $23$23 minutes to $240$240 minutes as $23:240$23:240.

Reflect: We could have compared both quantities in units of hours instead of minutes. First we would convert $23$23 minutes to an amount of hours by dividing by $60$60, since there are $60$60 minutes in $1$1 hour. So Fernando spends $\frac{23}{60}$2360​ hours travelling to work each day.

Next we compare the two durations by writing the ratio $\frac{23}{60}:4$2360​:4. Notice that we can multiply both sides of this ratio by $60$60 to end up with the same ratio as above. That is, $\left(\frac{23}{60}\times60\right):\left(4\times60\right)$(2360​×60):(4×60) is equivalent to $23:240$23:240.

As you can see, sometimes converting one number will lead to an easier calculation than if we converted the other.

 

Practice questions

Question 1

Question 2

Question 3