Ratios and Rates

NZ Level 6 (NZC) Level 1 (NCEA)

Rates II

Lesson

When we talk about speed limits, we often talk about them in terms of kilometres per hour. 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`

Let's look at how to use this formula in an example.

**Question**: What is the speed of a car that travels $360$360km in $6$6 hours?

**Do**:

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

$=$= | $\frac{360}{6}$3606 | |

$=$= | $60$60 km/h |

The speed of the car is $60$60km/h

This same process applies for all rates, not just speed.

**Question**: If Charlie earns $14 per hour and he works for $17$17 hours, how much money does he earn?

**Think:** Charlie's rate of pay (R) could be expressed as dollars/hour. We could write this as:

$\text{Rate of Pay}=\frac{\text{Total Amount Earned}}{\text{Total Hours Worked}}$Rate of Pay=Total Amount EarnedTotal Hours Worked

So to calculate the total amount Charlie earned, we would calculate *Rate of pay* x *Total Hours worked*.

**Do:** $14\times17$14×17 = $\$238$$238

Charlie earned $\$238$$238.

What is the speed of a car that travels $56$56 kilometres in $7$7 hours?

We can also change the units of measurement that our rates are expressed in. For example kilometres/hour can be changed to metres/second. We just need to be aware of how the values are changing when we are converting the quantities as we need to keep our rate in the same proportion.

**Convert:** $13$13 L/hr to L/day.

**Think:** How are our units of measurement are changing. We only need to change one of our units of measurement- that is, we need to change hours to days. Since there are $24$24 hours in a day, we need to multiply both sides of our rate by $24$24.

**Do:**

$13$13 L/hr | $=$= | $312$312 L/ 24 hrs | multiply by 24 to get 24hrs |

$=$= | $312$312 L/day |

We may need to change both units of measurement in some questions.

Convert $468$468 km/hr into m/s.

First convert $468$468 km/hr into m/hr.

Now convert $468000$468000 m/hr into m/s.

This type of conversion is also useful when we look at exchange rates.

Assuming that $1$1 AUD can buy $0.90$0.90 USD, how many USD's is equivalent to $88$88 AUD?

Assuming that $1$1 AUD can buy $0.90$0.90 USD, how many AUD's are equivalent to $24$24 USD?

Round your answer to the nearest cent.

Apply direct and inverse relationships with linear proportions

Apply numeric reasoning in solving problems