Ratios and Rates

[Pages:14]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:





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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

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

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?

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Speed

=

=

= ?

S = Speed

T = Time

S = Speed

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

D S T

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?



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1B Heart Rate 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 Frank is 37 years old.

a) Calculate his maximum heart rate (MHR).

b) What is Frank's target heart rate?

Example 2 The table below will tell a woman the condition of her health based on her resting heartbeat.

Age Athlete Excellent Great Good Average Below Average Poor

18 ? 25 54 ? 60 61 ? 65 66 ? 69 70 ? 73 74 ? 78 79 ? 84

85 +

26 ? 35 54 ? 59 60 ? 64 65 ? 68 69 ? 72 73 ? 76 77 ? 82

83 +

36 ? 45 54 ? 59 60 ? 64 65 ? 69 70 ? 73 74 ? 78 79 ? 84

85 +

46 ? 55 54 - 60 61 ? 65 66 ? 69 70 ? 73 74 ? 77 78 ? 83

84 +

56 ? 65 54 ? 59 60 ? 64 65 ? 68 69 ? 73 74 ? 77 78 ? 83

84 +

65 + 54 ? 59 60 ? 64 65 ? 68 69 ? 72 73 ? 76 77 ? 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 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 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?

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BASIX 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 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 Amount of fuel consumed (Litres) ? 100

Fuel Consumption = Distance travelled (kilometres) Example 1

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?



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1E Ratios 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

Express the following ratios in simplest form.

(a) 10: 15

(b) 3: 15: 9

(c)

1 2:4

(d) 5: 1.25

(e)

24 3:7

(f) $2.50: $3.25

(g) 500g: 1.25kg (h) 15: 10



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