NZ Level 5

Working Mathematically

Lesson

A rate is a ratio between two measurements with different units. 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.

**Evaluate:** If I drive at $55$55km/hour, how far will I travel in $4$4 hours?

**Think:** $55$55km represents the distance traveled in one hour, so to work out how far I went in $4$4 hours, I would need to multiply this distance by $4$4.

**Do:** $55\times4=220$55×4=220km

These questions don't always use multiplication. Sometimes we need to use division.

**Evaluate:** If Lee paid $\$11.85$$11.85 for $3$3kg of apples, how much did he pay per kilogram?

**Think: **To find out what $1$1kg costs, we need to divide the weight by $3$3. So we also need to divide the cost by $3$3.

**Do:**$11.85\div3=\$3.95$11.85÷3=$3.95 $/kg$/`k``g`

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 $300$300 mL/hr to mL/min.

Ratios tell us about the relative sizes of two or more values. They are often used in everyday life, whether it's for dividing up money, betting odds, cooking or mixing cement! So knowing how to apply your knowledge about ratios is really important.

You need to know!

- How to calculate the **total number of parts** (by adding all the numbers in the ratio).

- How to calculate what **one part is worth** (by dividing a value by the total number of parts) and

- How to calculate what **each share of the ratio is worth** (by multiplying what one part is worth with each number in the ratio)

**Evaluate: **Amir and Keira shared $\$720$$720 in the ratio $5:4$5:4. How much did each person get?

**Think: **There are $9$9 parts in total $\left(5+4\right)$(5+4) and $720\div9$720÷9 is $80$80.

That means that one part is worth $\$80$$80.

Amir gets $5$5 parts and Keira gets $4$4 parts.

**Do: **

Amir | gets | $5$5 parts |

$=$= | $5\times\$80$5×$80 | |

$=$= | $\$400$$400 | |

Keira | gets | $4$4 parts |

$=$= | $4\times\$80$4×$80 | |

$=$= | $\$320$$320 |

**Check:** The total of Amir and Keira's amounts should sum to the total amount ( $\$720$$720 )

$\$400+\$320=\$720$$400+$320=$720

**Evaluate: **Divide $60$60cm in the ratio $1:2:3$1:2:3

**Think:** There are $6$6 parts in total $\left(1+2+3\right)$(1+2+3) and $60\div6$60÷6 is $10$10. This means that one part is worth $10$10cm.

**Do:** $60$60cm divided in the ratio $1:2:3$1:2:3 would be $10$10cm:$20$20cm:$30$30cm.

Divide $24$24 kilograms into the ratio of $4:8$4:8.

What is the larger value?

What is the smaller value?

Reason with linear proportions.