Ratios and Rates

UK Secondary (7-11)

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:

$SPEED=\frac{DISTANCE}{TIME}$`S``P``E``E``D`=`D``I``S``T``A``N``C``E``T``I``M``E`

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

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

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

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

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

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 |

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

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?

Give your answer to the nearest integer.

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

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