11.01 Identify ratios

A ratio is an ordered comparison of two or more quantities. It tells us 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.

 

Writing 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.

 

Worked examples

Example 1

During one day at the airport, $5$5 flights were delayed and $21$21 flights were on-time.

(a) Write a ratio comparing the number of delayed flights to the number of on-time flights.

Think: 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 and the second will represent the number of on-time flights.

Do: The ratio is $5:21$5:21.

(b) Write a ratio comparing the number of delayed flights to all flights that day.

Think: The total number of flights is $5+21=26$5+21=26. The first number in our ratio will be the number of delayed flights and the second number will be the total number of flights.

Do: The ratio is $5:26$5:26.

 

Converting units in a ratio

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. If the quantities in the ratio represent the same type of measurement (for example both are lengths) it is important that both quantities are in the same units for comparison.

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 an equivalent ratio in the end.

 

Worked example

Example 2

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. We will look more closely at equivalent and simplified ratios in our next lesson.

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