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Chapter 1

Ratios and Rates

1A Rates (pg. 2)

1B Heart Rate (pg. 4)

1C Energy (pg. 5)

1D Fuel Consumption (pg. 7)

1E Ratios (pg. 8)

1F Dividing a Quantity in a Given Ratio (pg. 9)

1G Scale Drawings (pg. 10)

1H Plans and Elevations (pg. 11)

1I Practical Applications of Perimeter, Area and Volume (pg. 13)

Written by

Benjamin Odgers

Maths Teacher

B Teaching / B Science

The following theory booklet lines up with the Cambridge Year 12 NSW Standard Mathematics 2

Textbook. This can be found using the following link: https://www.cambridge.edu.au/education/titles/CambridgeMATHS-Stage-6-Mathematics-Standard-2-Year-12-print-

and-interactive-textbook-powered-by-HOTmaths/#.XYgHTUszaUk

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1A Rates

Rates are used to compare two amounts. When we talk about rates we use the word “per.” For example we

travel at 50 kilometres per hour (50km/h) or we purchase apples at $3 per kilogram ($3/kg). We use rates all

the time in real life. Notice that rates commonly use the forward slash “/” symbol which represents the word

“per.” We can usually solve rate problems using the 4 boxes, 3 numbers and 2 arrows technique.

Example 1 https://youtu.be/xDt9bWBR7Cw

Convert the rates below into the units shown in brackets.

(a) $4.50/kg ($/g) (b) 12L/min (mL/min) (c) 85km/h (m/s)

The Unitary Method

The unitary method involves turning a quantity into a unit of one before turning it into another quantity.

Example 2 https://youtu.be/IRWW9rR9W_8

a) Water is gushing out of a tap at a rate of 15L/min. How much water would come out in 3.5 minutes?

b) Peter paid $12.50 for 5kg of tomatoes. How much would it cost for 3.5kg of tomatoes?

c) Jared used 45L of petrol and travelled 607.5km. How far can he travel on 13 litres of petrol?

d) Joe’s Health Shop sells protein powder at $6 per 100 grams or you can buy a 1.25kg container of

protein for $50. Which option is the better buy? Why?

(this chapter continues on the next page)

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Speed

𝑆 =𝐷

𝑇 𝑇 =

𝐷

𝑆 𝐷 = 𝑆 × 𝑇

S = Speed T = Time S = Speed

Example 3 https://youtu.be/8REpDs-P2g0

a) Calculate the average speed of a car that travels 278km in 3 hours.

b) If a car is travelling at an average speed of 62km/h, how far will it travel in 3.5 hours?

c) Harry needs to travel 189km in order to get to his destination. How long will it take him to get to his

destination if he travels at an average speed of 75km/h?

D

S T

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1B Heart Rate https://youtu.be/xZ_IHPSKwQk

Your heart rate is measured in beats per minute (bpm). Notice that once again we are using the word “per.”

We usually measure a person’s heart rate by checking their pulse and counting the number of beats in 15

seconds. We then multiply this by 4 to calculate the beats per minute. A person’s resting heart rate can be

anywhere from 60 to 100 beats per minute. When we exercise or do strenuous activity our heart rate will

increase.

Maximum Heart Rate (MHR)

As people get older they must be careful not to raise their heart rate too high. The following formula gives

us a rough guide on what our maximum heart rate should be. Obviously the older you are the lower your

MHR becomes.

MHR = 220 − age (in years)

Target Heart Rate (THR)

When we exercise, the target heart rate is the desired heart rate that is most beneficial for the lungs and

heart. The THR ranges between 65% and 85% of the MHR.

Example 1 https://youtu.be/jrnOJPtZyPc

Frank is 37 years old.

a) Calculate his maximum heart rate (MHR).

b) What is Frank’s target heart rate?

Example 2 https://youtu.be/dMHrLOcA0w4

The table below will tell a woman the condition of her health based on her resting heartbeat.

Age 18 – 25 26 – 35 36 – 45 46 – 55 56 – 65 65 +

Athlete 54 – 60 54 – 59 54 – 59 54 - 60 54 – 59 54 – 59

Excellent 61 – 65 60 – 64 60 – 64 61 – 65 60 – 64 60 – 64

Great 66 – 69 65 – 68 65 – 69 66 – 69 65 – 68 65 – 68

Good 70 – 73 69 – 72 70 – 73 70 – 73 69 – 73 69 – 72

Average 74 – 78 73 – 76 74 – 78 74 – 77 74 – 77 73 – 76

Below Average 79 – 84 77 – 82 79 – 84 78 – 83 78 – 83 77 – 84

Poor 85 + 83 + 85 + 84 + 84 + 85 +

a) What is the average resting heart rate of a 27-year-old woman whose health is in excellent condition?

b) What is the health of a woman who is 47 years old and has a resting heart rate of 67bpm?

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1C Energy https://youtu.be/cCbciCTElxA

Power is the rate at which energy is consumed, we measure it in watts (W). For example, a heater has a

power rating of 2000W while an LED light bulb might only use 10W.

Kilowatt-hours (kWh)

When we pay for electricity it is not enough to know how many watts each

appliance uses, we need to know how long they have been running for. For

example, a 2kW heater will use 2kWh in 1 hour or 4kWh in 2 hours. The number

of kilowatt-hours we use will determine the cost of our electricity bill.

The Australian Government requires appliances to have the following energy rating

sticker so that consumers can compare energy costs. When reading the sticker, you

will notice the following:

• The more stars the more energy efficient

• The lower the score they lower the energy consumption

Example 1 https://youtu.be/EDV0TbCCzYE

Calculate the cost of running the following appliances if electricity costs 47c/kWh.

a) A washing machine that uses 66kWh per year.

b) A 1.2kW oil heater that has been running for 9 hours.

c) A phone charger can use 0.5W of electricity without a phone plugged in when turned on at the wall.

How much would this cost if it was left on every day in the month of November (30 days)?

d) An 85kW Tesla takes 75 minutes to fully charge its battery. How much does it cost to charge a

Tesla?

(this chapter continues on the next page)

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BASIX https://youtu.be/0ce_QU9QxFw

Basix (Building Sustainability Index) is an Australian government scheme that makes sure that people build

more sustainable and environmentally friendly homes. In order for your home to pass the BASIX

requirements you can do some or all of the following:

• Have a North facing home so that it is heated naturally by the sun.

• Have water tanks for water usage.

• Insulate the walls so that you don’t have to heat/cool the home as much

• Water efficient taps and shower heads

• Solar panels for electricity usage

• And many more…

Example 2 https://youtu.be/iZNZ8MewdbY

Grant has a 10kW solar system on his roof (this can produce about 29 to 46 kWh of energy per day). His

electricity provider charges him 26 cent per kWh of energy used and pays him 11.1 cents per kWh of solar

energy that goes back to the grid.

a) How much will he have to pay the energy provider on a day where he uses 41 kWh of energy

(assuming the solar system produced 32 kWh of energy that day)?

b) How much money did he save in part (a) due to the solar system?

c) How much money will he get from the energy provider if he uses only 21 kWh of energy in a day

(assuming the solar system produced 32 kWh of energy that day)?

d) How much money did he save in part (c) due to the solar system?

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1D Fuel Consumption

Example 1 https://youtu.be/93A5LeVhNwQ

a) A car consumed 47.04 litres of fuel after travelling 560 km. What was its fuel consumption for this

journey?

b) My car has a fuel consumption of 7.6 L/100 km. How much fuel will I use after travelling 840 km?

Fuel Consumption =Amount of fuel consumed (Litres) × 100

Distance travelled (kilometres)

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1E Ratios https://youtu.be/ekcGHvMP_Tc

Ratios are used to compare amounts. Ratios are usually used in cases where the amounts change depending

on the circumstance. For example, in childcare 1 adult is required to look after 4 babies (aged 0 to 24

months). We give this the ratio 1:4 since this can change depending on how many babies we have. It could

be 2 adults to 8 babies or 3 adults to 12 babies, either way it is still a ratio of 1:4.

Ratios work in the same way as fractions. You can multiply the denominator and numerator of a fraction by

the same number and the fraction remains the same. Similarly, you can multiply and divide both sides of a

ratio by the same number and it remains the same ratio.

Example 1 https://youtu.be/NkAcfvDAYmg

Express the following ratios in simplest form.

(a)

10: 15 (b) 3: 15: 9 (c) 1

2: 4

(d) 5: 1.25

(e)

2

3:4

7

(f) $2.50: $3.25 (g) 500g: 1.25kg (h) 15𝑎𝑏: 10𝑎

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1F Dividing a Quantity in a Given Ratio

Example 1 https://youtu.be/hVAvT7zOdwU

Grant and Jenny have two children, Abby and Ben aged 7 and 9 respectively. They decide to share a total of

$24 amongst the children each fortnight for pocket money. They share it in the ratio 7:9 so that the older

child gets a larger portion of the money. How much does each child get for pocket money?

Example 2 https://youtu.be/GoLP7_PatJY

Fred and Sarah have three children, Angela, Bryce and Cher aged 8, 11 and 15 respectively. They decide to

share a total of $40 amongst the children each fortnight for pocket money. They share it in the ratio 8:11:15

so that the older child gets a larger portion of the money. How much does each child get for pocket money?

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1G Scale Drawings https://youtu.be/adNdezNQO8c

Scale drawings represent a scaled down (and sometimes scaled up) drawing of an actual object. For

example, a map is a scaled down drawing of an actual location. If the scale for the map was 1:500 then

everything you see on the map is actually 500 times bigger than that in real life.

The following YouTube video illustrates a scaled down model from the Zoolander movie.

https://www.youtube.com/watch?v=0KC_rd7-bf0.

A scale can be given in one of two ways:

• Units included – if 1 centimetre on the map represented 5 metres in real life then the scale would be

1 cm = 5 m.

• Without Units – we could write the above scale as 1:500 since 5 metres is 500 times bigger than 1

centimetre.

Example 1 https://youtu.be/Slqskv8K6eg

A map has a scale of 1:15000.

a) Jane measured a straight road to be 4.5 cm long (this is called the drawing length). What is the actual

length of the road (in metres)?

b) A straight road has an actual length of 3.6 kilometres. What is the drawing length of the road?

What is the difference between drawing length and actual length?

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1H Plans and Elevations

A plan is a view from above an

object. This is commonly known as a birds-eye view. On the

right we can see a plan of a house, this is commonly known as a

floor plan. Floor plans will commonly have a scale such as

1:200 which would imply that the actual distances are 200

times larger than what you can see in the drawing.

Below are some common symbols used for floor plans.

Kitchen sink

Toilet

Door – showing

the direction the

door will open

Stove top

Window

Shower – shows

location of tap

and drain

Example 1 https://youtu.be/dFtbiy4mq2M

The floor plan above does not come with a scale, however we do know that the bedroom has dimensions

3000 mm × 3300 mm.

a) What is the scale for the above floor diagram?

b) What are the actual dimensions of the bathroom?

c) What is the floor area of the storage room?

(this chapter continues on the next page)

Floor plan taken from https://www.roomsketcher.com/features/2d-floor-plans/

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Elevation

An elevation is a view of an object from one of many sides.

This could be from the front, rear, top or the side of an object.

Elevations are usually given so that people can see different

views of an object they are interested in purchasing.

For the pictures of cars at right and below we can see

3 different elevations.

Example 2 https://youtu.be/GhDCh6CM5TA

For the object below, draw the following:

a) Plan view

b) Side elevation

c) Front elevation

Plan view

Side elevation

Front elevation

Front elevation

Side elevation

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1I Practical Applications of Perimeter, Area and Volume

Example 1 https://youtu.be/J2JoEh3dUvQ

The following image is an aerial photograph

taken using Google Earth. It is an image of the

Murwillumbah Mustangs football field. In

rugby league, the distance between the football

posts is 100 metres (as shown).

a) Calculate the scale for the following

image

b) The Murwillumbah Mustangs need to

install barrier fencing for the field (as

shown with a white border). What is the

perimeter of the barrier fencing?

c) The Mustangs are thinking of installing a water tank next to the change rooms. They must first

calculate the area of the roof to see how much water will be collected when it rains. What is the area

of the roof? Note: the roof is made from a rectangle and a trapezium shape.

(this chapter continues on the next page)

Change

Rooms

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Example 2 https://youtu.be/nauVyqIMYkc

The diagram at right is a plan view of a

swimming pool. The diagram below that is a

3-d image showing the height of the pool as

1.5 metres.

a) Estimate the area of the base of the

pool using the trapezoidal rule.

𝐴 ≈ℎ

2(𝑑𝑓 + 𝑑𝑙)

b) What volume of water is required to fill the pool

right to the top?

3.2 m

2.2 m

4.6 m

1.5 m