A rate is a ratio between two measurements with different units. We've already looked at how to convert and compare rates.

A common example of a rate is speed (which is written in kilometres per hour or km/h). You can see that this describes a relationship between two measurements- kilometres and hours.

Think about what the expression km/h means. It is a rate that expresses a relationship between distance (kilometres) and time (hours). Speed is one of the most common rates that we see everyday. We can write this relationship as:


In maths, we always like to write things in short hand, so we write this relationship as:


We can also rearrange this formula in a couple of ways:

$T=\frac{D}{S}$T=DS or $D=ST$D=ST


We can substitute values into these kind of rate formulae to find unknown values. Let's look how with some examples.


Worked Examples

question 1

What is the time in minutes required for a car to travel $35$35 kilometres at a speed of $105$105 kilometres per hour?

Think: What is the relationship between $D$D, $S$S, and $T$T in a normal speed equation? Which of the three are we looking for here?


We are looking for time so $T$T. Expressing $T$T in terms of $D$D and $S$S we get:

$T$T $=$= $\frac{D}{S}$DS
  $=$= $\frac{35}{105}$35105
  $=$= $\frac{1}{3}$13 hours
  $=$= $\frac{1}{3}\times60$13×60 minutes
  $=$= $20$20 minutes


Question 2

At a busy subway station, $13440$13440 people went through the entrance gates in $4$4 hours. What's the station's commuter rate in people per minute?

Think: People per minute means we are dividing the number of people by the number of minutes


We don't currently have the time in minutes so we can convert that first.

$4$4 hours $=$= $4\times60$4×60 minutes
  $=$= $240$240 minutes

So then our rate is:

$\frac{13440}{240}$13440240 people per minute = $56$56 people per minute


question 3

In $2012$2012, the population growth of Oman was $91$91 per $1000$1000. This means that for every $1000$1000 people, the number of people at the end of the year would be $1091$1091.

For a town with a population of $810$810, what would you expect the population to be at the end of the year?

  1. Give your answer to the nearest integer.


Question 4

What is the time required for a car to travel $170$170 kilometres at a speed of $10$10 kilometres per hour?

Question 5

If $9600$9600 litres of water flow through a tap in $8$8 hours, what is the tap's flow rate per minute?




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