 Rates III

Lesson 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:

$SPEED=\frac{DISTANCE}{TIME}$SPEED=DISTANCETIME

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

$S=\frac{D}{T}$S=DT

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?

Do:

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

Do:

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?

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?