chapter 3 units of measurement and application

7/28/2019 Chapter 3 Units of Measurement and Application

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73

3.1 Units of measurement

Measurement is used to determine

the size o a quantity. It usually

involves using a measuringinstrument. For example, to measure

length, instruments that can be

used include the rule, tape measure,

caliper, micrometer, odometer and

GPS. There are a number o systems

o measurement that dene their

units o measurement. We use the SI

metric system.

3.1

C H A P T E R

3Units of measurement and

applications

Syllabus topic MM1 Units of measurementand applications

Determine and convert appropriate units of measurement

Convert units of area and volume

Calculate the percentage error in a measurement

Use numbers in scientific notation

Express numbers to a certain number of significant figures

Calculate and convert ratesFind ratios of two quantities and use the unitary method

Calculate repeated percentage changes

ISBN: 9781107627291

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74 Preliminary Mat hematics General

SI unitsThe SI is an international system o units o measurement based on multiples o ten. It is

a version o the metric system which allows easy multiplication when converting between

units. Units shown in red (below) are non-SI units approved or everyday or specialised usealongside SI units.

Quantity Name of unit Symbol Value

Length Metre

Millimetre

Centimetre

Kilometre

Nautical mile

m

mm

cm

km

nm

Base unit

1000 mm = 1 m

100 cm = 1 m

1 km = 1000 m

1 nm = 1852 m

Area Square metreSquare centimetre

Hectare

m2

cm2

ha

Base unit10 000 cm2 = 1 m2

1 ha = 10 000 m2

Volume Cubic metre

Cubic centimetre

Litre

Millilitre

Kilolitre

m3

cm3

L

mL

kL

Base unit

1 000 000 cm3 = 1 m3

1L = 1000 cm3

1000 mL = 1 L

1 kL = 1000 L

Mass Kilogram

Gram

Tonne

kg

g

t

Base unit

1000 g = 1 kg

1 t = 1000 kg

Time Second

Minute

Hour

Day

s

min

h

d

Base unit

1 min = 60 s

1 h = 60 min

1 d= 24 h

Converting between units

A prex is a simple way to convert between units. It indicates a multiple o 10. Some common

prexes are mega (1 000 000), kilo (1000), centi 1( ) and milli 1000( ).

Length, mass and volume Time

unit

kilo 1000

100

10

1000

100

10centi

milli

1000 1000mega

24

60

60

24

60

60

hours

days

minutes

seconds

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75Chapter 3 Units of measurement and applications

Example 1 Converting units of length

Complete the ollowing.

a =35 cm mm b 4500 m km

Solution

1 To change cm to mm multiply by 10.

2 To change m to km divide by 1000.

a 35 cm = 35 10 mm

= 350 mm

b 4500 = 4500 1000 km

= 4.5km

Example 2 Converting units of time

Complete the ollowing.

a =3 h and 15 min min b 10 080 min d

Solution

1 To change hours to minutes, multiply by 60.

2 To change minutes to hours, divide by 60.

3 To change hours to days, divide by 24.

a 3 h 15 min = 3 60 + 15 min

= 195 min

b 10 080 min = 10 080 60 h

= 168 h

= 168 24 d

= 7 d

Converting area and volume unitsTo convert area units, change the side length units and compare the values or area.

1 m 100 cm

cm

1 m2= 100 100 = 10 000 cm2

1 m2= 10 000 cm2

or1 cm 02 2

10 00

To convert volume units, change the side length units and compare the values or volume.

1 m

1 m 100 cm100 cm

100 cm

=1 m

1 m3= 100 100 100 = 1 000 000 cm3

1 m3= 1000 000 cm3

or1 cm01 0 00 00

=

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5 What unit o length is most appropriate to measure each o the ollowing?

a Length o a pen b Height o a building

c Thickness o a credit card d Distance rom Sydney to Newcastle

e Height o a person f Length o a ootball ield

6 What unit o mass is most appropriate to measure each o the ollowing?

a Weight o an elephant b Mass o a mug

c Bag o onions d Weight o a baby

e Mass o a truck f Mass o a teaspoon o sugar

7 What unit o time is most appropriate to measure each o the ollowing?

a Lesson at school b Reheating a meal in a microwave

c Age o a person d School holidays

e Accessing the internet f Movie

8 There are 20 litres o a chemical stored in a container.

a What amount o chemical remains i 750 mL is

removed rom the container? Answer in litres.

b How many containers are required to make a

kilolitre o the chemical?

9 Christopher bought 3 kg o sultanas. What mass osultanas remains i he ate 800 grams? Answer in

kilograms.

10 The length o the Murray River is 2575 km. The length

o the Hawkesbury River is 80 000 m. What is the

dierence in their lengths? Answer in metres.

11 There are three tonnes o grain in a truck. What is the mass i another 68 kg o grain isadded to the truck? Answer in kilograms.

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Development

12 A cyclist travels to and rom work over a 1200-metre long bridge. Calculate the distance

travelled in a week i the cyclist works or 5 days. Answer in kilometres.

13 Madison travels 32 km to work each day. Her car uses 1 litre o petrol to travel 8 km.

a How many litres o petrol will she use to get to work?

b How many litres o petrol will she use or 5 days o work, including return travel?

14 Arrange 500 m, 0.005 km, 5000 cm and 5 000 000 mm in:

a Ascending order (smallest to largest)

b Descending order (largest to smallest)

15 Complete the ollowing.

a 1 2 2km

b

12

m

c 1 2 2cm d 1000 cm

2

e 2000 mm2 2 f 5000 m 2

g 3 2m h 310 2 2km

i 4 m = j 74300 m

k 6500 mm l 4000 cm =

16 The area o a ield is 80 000 square metres. Convert the area units to the ollowing.a Square kilometres b Hectares

17 Jackson swims 30 lengths o a

50-metre pool.

a How many kilometres does he

cover?

b I his goal is 4 kilometres, how

many more lengths must he swim?

18 Eliza worked rom 10.30 a.m. until 4.00 p.m. on Friday, rom 7.30 a.m. until 2.00 p.m.

on Saturday, and rom 12 noon until 5.00 p.m. on Sunday.

a How many hours did Eliza work during the week?

b Express the time worked on Friday as a percentage o the total time worked during

the week. Answer correct to the nearest whole number.

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3.2 Measurement errors

There are varying degrees o instrument error and measurement uncertainty when measuring.

Every time a measurement is repeated, with a sensitive instrument, a slightly dierent result

will be obtained. The possible sources o errors include mistakes in reading the scale, parallax

error and calibration error. The accuracy o a measurement is improved by making multiple

measurements o the same quantity with the same instrument.

Accuracy in measurementsThe smallest unit on the measuring instrument is

called the limit o reading. For example, a 30 cm

rule with a scale or millimetres has a limit oreading o 1 mm. The accuracy o a measurement is

restricted to 12

o the limit o reading. For example,

i the measurement on the ruler is 10 mm then the

range o errors is 10 0.5 mm. Here the upper limit

is 10 + 0.5 mm or 10.5 mm and the lower limit is

10 0.5 mm or 9.5 mm.

Every measurement is an approximation and has an error. The absolute error is the dierence

between the actual value and the measured value indicated by the instrument. The maximum

value or an absolute error is 12

o the limit o reading.

Limit of reading Absolute error

Smallest unit on measuring instrument Measured value Actual value

Maximum value is2

limit o reading

Relative error gives an indication o how good a measurement is relative to the size o the

quantity being measured. The relative error o a measurement is calculated by dividing the

limit o reading by the actual measurement. For example, the relative error or the above

measurement is10

0 05( ) = . The relative error is oten expressed as a percentage andcalled the percentage error. For example, the percentage error or the above measurement

is 510

5%( ) = .

1 cm 2 3 4 5

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Relative error Percentage error

Absolute error

Measurement

Absolute error

Measurement100%

Example 3 Finding the measurement errors

a What is the length indicated by the arrow on the above

ruler?

b What is the limit o reading?

c What is the upper and lower limit or each measurement?

d Find the relative error. Answer correct to three decimal places.

e Find the percentage error. Answer correct to one decimal place.

Solution

1 The arrow is pointing to

38 mm.

2 Limit o reading is the

smallest unit on the ruler

(millimetre).

3 Calculate hal the limit o

reading.

4Lower limit is the measuredvalue minus 1 the limit o

reading.

5 Upper limit is the measured

value plus 1 the limit o

reading.

6 Write the ormula or

relative error.

7 Substitute the values or

absolute error and the

measurement.

8 Evaluate correct to three

decimal places.

9 Write the ormula or

percentage error.

10 Substitute the values or

absolute error and the

measurement.

11 Evaluate.

a Length is 38 mm.

b Limit o reading is 1 mm.

c2

limit o reading =2

1

= 0.5 mm

Lower limit = 38 0.5 = 37.5 mm

Upper limit = 38 + 0.5 = 38.5 mm

d Relative error=Absolu e error

Measurement

= 0 5

38

= 0.013

e Percentage error=Absolute error

Measurement

100%

=0.5

38 100%

= 1.3%

0 10 20 30 40 50 60 70 80

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Exercise 3B

1 Four measurements o length are shown on the ruler below.

0 10 20 30 40 50 60 70 80 90 100

A B C D

a What length is indicated by each letter? Answer to the nearest millimetre.


c What the largest possible absolute error?

d What is the upper and lower limit or each measurement?e Calculate the relative error, correct to three decimal places, or each measurement.

f Calculate the percentage error, correct to one decimal place, or each measurement.

2 Two measurements o mass are shown on the scales below.

a What mass is indicated by each letter? Use the outer scale.


c What the largest possible absolute error?

d What is the upper and lower limit or each measurement?

e Calculate the relative error, correct to three decimal places, or each measurement.

f Calculate the percentage error, correct to one decimal place, or each measurement.

0

0 88

8

8

88

8

8

8

88

0.5

1kg

1lb

2lb

3lb

4lb

5lb6lb

7lb

8lb

9lb

10lb

1.5

2kg

2.5

3kg

3.5

4kg

4.5

A

0

0 88

8

8

88

8

8

8

88

0.5

1kg

1lb

2lb

3lb

4lb

5lb6lb

7lb

8lb

9lb

10lb

1.5

2kg

2.5

3kg

3.5

4kg

4.5

B

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Development

3 A dishwasher has a mass o exactly 49.6 kg. Abbey

measured the mass o the dishwasher as 50 kg to the nearest

kilogram.a Find the absolute error.

b Find the relative error. Answer correct to three decimal

places.

c Find the percentage error to the nearest whole number.

4 An iPod has a mass o exactly 251 g. Jake measured the

mass o the iPod as 235 g to the nearest gram.

a Find the absolute error.

b Find the relative error. Answer correct to three decimalplaces.

c Find the percentage error correct to two decimal places.

5 An LCD screen has a mass o exactly 2.71 kg. Saliha

measured the mass o the screen as 3 kg to the nearest

kilogram.

a Find the absolute error.

b Find the relative error. Answer correct to three decimal

places.

c Find the percentage error correct to three decimal places.

6 A measurement was taken o a skid mark at the scene o a car accident. The actual length

o the skid mark was 25.15 metres, however it was measured as 25 metres.

a What is the absolute error?

b Find the relative error. Answer correct to three decimal places.

c Find the percentage error. Answer correct to one decimal place.

7 The length o a building at school is exactly 56 m. Cooper measured the length o the

building to be 56.3 m and Filip measured the building at 55.8 m.

a What is the absolute error or Coopers measurement?

b What is the absolute error or Filips measurement?

c Compare the relative error or both measurements. Answer correct to our decimal

places.

d Compare the percentage error or both measurements. Answer correct to three

decimal places.

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3.3 Scientific notation and significant figures

Scientific notation

Scientic notation is used to write very large or very small numbers more conveniently. Itconsists o a number between 1 and 10 multiplied by a power o ten. For example, the number

4 100 000 is expressed in scientic notation as 4.1 106. The power o ten indicates the

number o tens multiplied together. For example:

4.1 106= 4.1 (10 10 10 10 10 10)

= 4 100 000

When writing numbers in scientic notation, it is useul to remember that large numbers have

a positive power o ten and small numbers have a negative the power o ten.

Writing numbers in scientific notation

1 Find the rst two non-zero digits.

2 Place a decimal point between these two digits. This is the number between 1 and 10.

3 Count the digits between the new and old decimal point. This is the power o ten.

4 Power o ten is positive or larger numbers and negative or small numbers.

Example 4 Expressing a number in scientific notation

The land surace o the earth is

approximately 153 400 000 square

kilometres. Express this area more

conveniently by using scientic

notation.

Solution

1 The irst two non-zero digits are 1 and 5.

2 Place the decimal point between these numbers.

3 Count the digits rom the old decimal point (end o

the number) to position o the new decimal point.

4 Large number indicates the power o 10 is positive.

5 Write in scientiic notation.

1.534

1.53 400 000 eight digits

Power o 10 is +8 or 8

153 400 000 = 1.534 108

3.3

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Significant figuresSignicant gures are used to speciy the accuracy o a number. It is oten used to round a

number. Signicant gures are the digits that carry meaning and contribute to the accuracy o

the number. This includes all the digits except the zeros at the start o a number and zeros atthe nish o a number without a decimal point. These zeros are regarded as placeholders and

only indicate the size o the number. Consider the ollowing examples.

51.340 has our signicant gures: 5, 1, 3 and 4.

0.00871 has three signicant gures: 8, 7 and 1.

56091 has ve signicant gures: 5, 6, 0, 9 and 1.

The signicant gures in a number not containing a decimal point can sometimes be unclear.

For example, the number 8000 may be correct to 1 or 2 or 3 or 4 signicant gures. To prevent

this problem, the last signicant gure o a number is underlined. For example, the number

8000 has two signicant gures. I the digit is not underlined the context o the problem is aguide to the accuracy o the number.

Writing numbers to significant figures

1 Write the number in scientic notation.

2 Count the digits in the number to determine its accuracy (ignore zeros at the end).

3 Round the number to the required signicant gures.

Example 5 Writing numbers to significant figures

Write these numbers correct to the signicant gures indicated.

a 153 400 000 (3 signicant gures)

b 0.000 657 (2 signicant gures)

Solution


2 Count the digits in the number.

3 Round the number to 3 signiicant igures.

4 Write answer in scientiic notation correct

to 3 signiicant igures.


6 Count the digits in the number.

7 Round the number to 2 signiicant igures.

8 Write answer in scientiic notation correct

to 2 signiicant igures.

a 153 400 000 = 1.534 108

1.534 has 4 digits

1.53 rounded to 3 sig. ig.

153 400 000 = 1.53 108

b 0.000 657 = 6.57 104

6.57 has 3 digits

6.6 rounded to 2 sig. ig.

0.000 657 = 6.6 104

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Exercise 3C

1 Write these numbers in scientiic notation.

a 7600 b 1 700 000 000

c 590 000 d 6 800 000

e 35 000 f 310 000 000

g 77 100 000 h 523 000 000 000

i 95 400 000 000

2 Write these numbers in scientiic notation.

a 0.000 56 b 0.000 068 7

c 0.000 000 812 d 0.0043

e 0.000 058 f 0.000 003 12g 0.26 h 0.092

i 0.000 000 000 167

3 A microsecond is one millionth o a second. Write 5 microseconds in scientiic notation.

4 Sharks existed 410 million years ago.

a Write this number in scientiic notation.

b Express this number correct to one

signiicant igure.

5 Write each o the ollowing as a basic numeral.

a 1.12 105 b 5.34 108

c 5.2 103 d 8.678 107

e 2.4 102 f 7.8 109g 3.9 106 h 2.8 101

i 6.4 104


a 3.5 104 b 7.9 106

c 1.63 107 d 5.81 103

e 4.9 102 f 9.8 101

g 4.12 108 h 6.33 105

i 3.0 109

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7 Convert a measurement o 5.81 103 grams into kilograms. Express your answer in

scientiic notation.

8 Evaluate the ollowing and express your answer in scientiic notation.

a (2.5 103) (5.9 106) b (4.7 105) (6.3 102)

c (7.1 105) (4.2 102) d (3.0 104) (6.2 105)

9 Evaluate the ollowing and express your answer in scientiic notation.

a 9.1 10

2 8 10

b 7.2 10

8 103

c .8 10

3 2 105

10 Write these numbers correct to signiicant igures indicated.

a 1 561 231 (2 sig. g.) b 3 677 720 (4 sig. g.) c 789 001 (5 sig. g.)d 3 300 000 (1 sig. g.) e 777 777 (3 sig. g.) f 3 194 729 (5 sig. g.)

g 821 076 (4 sig. g.) h 7091 (1 sig. g.) i 49 172 (2 sig. g.)

11 Write these numbers correct to signiicant igures indicated.

a 0.0035 (1 sig. g.) b 0.191 785 (4 sig. g.) c 0.001 592 (3 sig. g.)

d 0.111 222 33 (6 sig. g.) e 0.000 0271 (1 sig. g.) f 0.019 832 6 (5 sig. g.)

g 0.008 12 (2 sig. g.) h 0.092 71 (3 sig. g.) i 0.000 419 (2 sig. g.)

12 A bacterium has a radius o 0.000 015 765 m.Express this length correct to two

signiicant igures.

13 Convert a measurement o 2654 kilograms into centigrams. Express your answer correct

to two signiicant igures.

14 Convert a measurement o 4 239 810 milligrams into grams. Express your answer correct

to our signiicant igures.

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Development

15 I y x1 2 , ind the value oy when:

a x= 2.4 103

b x= 9.8 103

16 The arc length o a circle is

l r360

where is the angle at the centre andris the

radius o the circle. Use this ormula to calculate the arc length o a circle when = 30

andr= 7.4 108. Answer in scientiic notation correct to one signiicant igure.

17 Given that V = ind the value orin scientiic notation when:

a V= 5 104 andh= 9 106 b V= 6 107 andh= 4 102

18 Use the ormulaE=mc2 to indc correct to three signiicant igures given that:

a m =0.08 andE= 5.5 109 b m= 2.7 103 andE= 1.6 104

19 Findx givenx3= 2.7 1012. Answer correct to our signiicant igures.

20 Light travels at 300 000 kilometres per second. Convert this measure to metres per

second and express this speed in scientiic notation.

21 Use the ormulaE= 3pq to evaluateEgiven thatp= 7.5 105 andq= 2.5 104.

Answer in scientiic notation correct to one signiicant igure.

22 The volume o a cylinder is V=r2h where ris the radius o the cylinder andh is the

height o the cylinder. Use this ormula to calculate the volume o the cylinder i

r= 5.6 104 andh= 2.8 103. Answer in scientiic notation correct to three signiicant

igures.

23 The Earth is 1.496 108km rom the Sun. Calculate the distance travelled by the Earth

in a year using the ormula c= 2r. Answer in scientiic notation correct to two

signiicant igures.13.3

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3.4 Calculations with ratios

A ratio is used to compare amounts o the

same units in a denite order. For example,

the ratio 3:4 represents 3 parts to 4 parts or3 or 0.75 or 75%.

A ratio is a raction and can be simplied

in the same way as a raction. For example,

the ratio 15:20 can be simplied to 3:4

by dividing each number by 5. Equivalent

ratios are obtained by multiplying or

dividing each amount in the ratio by the

same number.

15 : 12 = 5 : 4

3 3

5 : 4 = 15 : 12

3 3

15:12 and 5:4 are equivalent ratios.

When simpliying a ratio with ractions, multiply each o the amounts by the lowest common

denominator. For example, to simpliy8

:4

multiply both sides by 8. This results in the

equivalent ratio o 1:6.

Ratio

A ratio is used to compare amounts o the same units in a denite order.

Equivalent ratios are obtained by multiplying or dividing by the same number.

Dividing a quantity in a given ratioRatio problems may be solved by dividing a quantity in a given ratio. This method divides

each amount in the ratio by the total number o parts.

Dividing a quantity in a given ratio

1 Calculate the total number o parts by adding each amount in the ratio.

2 Divide the quantity by the total number o parts to determine the value o one part.

3 Multiply each amount o the ratio by the result in step 2.

4 Check by adding the answers or each part. The result should be the original quantity.

3.4

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Example 6 Dividing a quantity in a given ratio

Mikhail and Ilya were given $450 by their grandparents to share in the ratio 4:5. How much

did each person receive?

Solution

1 Calculate the total number o parts by adding each

amount in the ratio (4 parts to 5 parts).

2 Divide the quantity ($450) by the total number o

parts (9 parts) to determine the value o one part.

3 Multiply each amount o the ratio by the result in

step 2 or $50.

4 Check by adding the answers or each part. The

result should be the original quantity or $450.

5 Write the answer in words.

To al par s

9 par s $450

1 part$450

=

=

5+ 9

9$ 05

200

250

4 par s

5 par s

$50 =

$50 =

($200 + $250 = $450)

Mikhail receives $200 and

Ilya receives $250.

The unitary methodThe unitary method involves nding one unit o an amount by division. This result is then

multiplied to solve the problem.

Using the unitary method

1 Find one unit o an amount by dividing by the amount.

2 Multiply the result in step 1 by a number to solve the problem.

Example 7 Using the unitary method

A car travels 360 km on 30 L o petrol. How ar does it travel on 7 L?

Solution1 Write a statement using inormation rom the

question.

2 Find 1 L o petrol by dividing 360 km by the amount

or 30.

3 Multiply the 36030

by a 7 to solve the problem.

4 Evaluate.

5 Write answer to an appropriate degree o accuracy.


30 L km

1 L km

7 L km

84 km

360

30

360

307

The car travels 84 km.

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Exercise 3D

1 Express each ratio in simplest orm.

a 15:3 b 10:40 c 24:16

d 14:30 e 8:12 f 49:14

g 9:18:9 h 5:10:20 i 11

3:

j1

2

1

5: k

2

3

3

7: l

31:

2 A delivery driver delivers 1 parcel on average every 20 minutes. How many hours does it

take to drop 18 parcels?

3 Divide 240 into the ollowing ratios.a 2:1 b 3:2

c 1:5 d 7:5

4 A bag o 500 grams o chocolates is divided into the ratio 7:3. What is the mass o the

smaller amount?

5 At a concert there were 7 girls or every 5 boys. How many girls were in the audience

o 8616?

6 Molly, Patrick and Andrew invest in a business in the ratio 6:5:1. The total amount

invested is $240 000. How much was invested by the ollowing people?

a Molly b Patrick c Andrew

7 In a boiled ruit cake recipe the ratio o mixed ruit to

lour to sugar is 5:3:2. A 250 g packet o mixed ruit is

used to make the cake. How much sugar and lour is

required?

8 A 5 kg bag o potatoes costs $12.80. Find the cost o:

a 1 kg b 10 kg

c 14 kg d 6 kg

9 The cost o 3 pens is $42.60. Find the cost o:

a 1 pen b 4 pens

c 6 pens d 10 pens

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Development

10 A punch is made rom pineapple juice, lemonade and passionruit in the ratio 3:5:2.

a How much lemonade is needed i one litre o pineapple juice is used?

b How much pineapple juice is required to make 10 litres o punch?

11 Angus, Ruby and Lily share an inheritance o $500 000 in the ratio o 7:5:4. How much

will be received by the ollowing people?

a Angus b Ruby c Lily

12 Samantha and Mathilde own a restaurant. Samantha gets 3 o the proits and Mathilde

receives the remainder.

a What is the ratio o proits?

b Last week the proit was $2250. How much does Mathilde receive?

c This week the proit is $2900. How much does Samantha receive?

13 A jam is made by adding 5 parts ruit to 4 parts o sugar. How much ruit should be

added to 22

kilograms o sugar in making the jam?

14 A local council promises to spend $4 or every $3 raised in public subscriptions or a

community hall. The cost o the hall is estimated at $1.75 million. How much does thecommunity need to raise?

15 The ratio o $5 to $10 notes in Stephanies purse is 3:5. There are 24 notes altogether.

What is the total value o Stephanies $5 notes?

16 Nathan makes a blend o mixed lollies using 5 kg jelly babies, 4 kg licorice and 1 kg

skittles. What is the cost o the blend per kilogram to the nearest cent?

Mixed lollies

Jelly babies $5.95 per kg

Licorice $6.95 per kg

Skittles $11.90 per kg

17 The three sides o a triangle are in the ratio o 1:3:5. The longest side o the triangle is

16.2 mm. What is the perimeter o the triangle?

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3.5 Rates and concentrations

Rates

A rate is a comparison o amounts withdierent units. For example, we may compare

the distance travelled with the time taken.

In a rate the units are dierent and must be

specied.

The order o a rate is important. A rate is

written as the rst amount per one o the

second amount. For example, $2.99/kg

represents $2.99 per one kilogram or 80 km/hrepresents 80 kilometres per one hour.

Converting a rate

1 Write the rate as a raction. First quantity is the numerator and 1 is the denominator.

2 Convert the rst amount to the required unit.

3 Convert the second amount to the required unit.

4 Simpliy the raction.

Example 8 Converting a rate

Convert each rate to the units shown.

a 55 200 m/h to m/min

b $6.50/kg to c/g

Solution

1 Write the rate as a raction.

2 The numerator is 55 200 m and the denominator

is 1 h.3 No conversion required or the numerator.

4 Convert the 1 hour to minutes by multiplying by 60.



7 The numerator is $6.50 and the denominator is 1 kg.

8 Convert the $6.50 to cents by multiplying by 100.

9 Convert the 1 kg to g by multiplying by 1000.


a 55 20055 200 m

h

55 200 mmin

m/min

=

=

=

1

1 6 0

920

b 6.50$6.50

kg

6.50 c

g

c/g

=

=

=

1

100

1 1000

0 65

3.5

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ConcentrationsA concentration is a measure o how much o a given substance is mixed with another

substance. Concentrations are a rate that has particular applications in nursing and agriculture.

It oten involves mixing chemicals. Concentrations may be expressed as: weight per weight such as 10 g/100 g

weight per volume such as 5 g/10 mL

volume per volume such as 20 mL/10 L.

Finding a percentage concentration

1 Write the two quantities as a raction.

2 Multiply the raction by 100 to convert it to a percentage.

Example 9 Converting a concentration

A medicine is given as a concentration o 2.5 mL per 10 kg. What is the dosage rate or this

medicine in mL/kg?

Solution


2 The numerator is 2.5 mL and the denominator is

10 kg.

3 Divide the numerator by the denominator.

4 Evaluate.

5 Write answer to an appropriate degree o

accuracy.


2.5 mL/10 kg2.5 mL

kg

2.5 mL

kg

=

=

=

10

10

0 m25 /L kg

The dosage rate is 0.25 mL/kg.

Example 10 Expressing as a percentage concentration

Express 6.2 g o sugar per 50 g as a percentage concentration.Solution

1 Write as a raction. The irst amount is divided by

the second amount.

2 Multiply the raction by 100 to convert it to a

percentage.

3 Evaluate.


6.2 g/50 g6.2 g

50 g

6.2

=

=

=

50100

12

%

. %4

Percentage composition

is 12.4%.

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Exercise 3E

1 Use the rate provided to answer the ollowing questions.

a Cost o apples is $2.50/kg. What is the cost o 5 kg?

b Tax charge is $28/m. What is the tax or 7 m2?

c Cost savings are $35/day. How much is saved in 5 days?

d Cost o a chemical is $65/100 mL. What is the cost o 300 mL?

e Cost o mushrooms is $5.80/kg. What is the cost o2

kg?

f Distance travelled is 1.2 km/min. What is the distance travelled in 30 minutes?

g Concentration o a chemical is 3 mL/L. How many mL o the chemical is needed

or 4 L?

h Concentration o a drug is 2 mL/g. How many mL is needed or 10 g?

2 Express each rate in simplest orm using the rates shown.

a 300 km on 60 L [km per L] b 15 m in 10 s [m per s]

c $640 or 5 m [$ per m] d 56 L in 0.5 min [L per min]

e 78 mg or 13 g [mg per g] f 196 g or 14 L [g per L]

3 Convert each rate to the units shown.

a 39 240 m/min [m/s] b 2 m/s [cm/s]

c 88 cm/h [mm/h] d 55 200 m/h [m/min]

e 0.4 km/s [m/s] f 57.5 m/s [km/s]

g 6.09 g/mL [mg/mL] h 4800 L/kL [mL/kL]

i 12 600 mg/g [mg/kg]

4 Mia earns $37.50 per hour working in a cae.

a How much does Mia earn or working a 9-hour day?

b How many hours does Mia work to earn $1200?

c What is Mias annual income i she works 40 hours a week? Assume she works

52 weeks in the year.

5 Patrick mixes 35 mL o a pesticide per 20 L as a percentage concentration.

a Express this concentration in litres per litre.

b What is the percentage concentration?

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Development

6 A tap is dripping water at a rate o 70 drops per minute. Each drop is 0.2 mL.

a How many millilitres o water drip rom the tap in one minute?

b How many litres o water drip rom the tap in a day?

7 Natural gas is charged at a rate o 1.4570 cents per MJ.

a Find the charge or 12 560 MJ o natural gas. Answer to the nearest dollar.

b The charge or natural gas was $160.27. How many megajoules were used?

8 Olivias council rate is $2915 p.a. or land valued at $265 000. Lucy has a council rate o

$3186 on land worth $295 000 rom another council.

a What is Olivias council charge as a rate o $/$1000 valuation?b What is Lucys council charge as a rate o $/$1000 valuation?

9 Miras car uses 9 litres o petrol to travel 100 kilometres. Petrol costs $1.50 per litre.

a What is the cost o travelling 100 kilometres?

b How ar can she drive using $50 worth o petrol? Answer to the nearest kilometre.

10 A motor bike is moving at a steady

speed. When the speed is 90 km/hthe bike consumes 5 litres o

petrol or every 100 kilometres

travelled.

a The petrol tank holds 30 litres.

How many kilometres can the

bike travel on a ull tank o

petrol when its speed is

90 km/h?

b When the speed is 110 km/h

the bike consumes 30% more petrol per kilometre travelled. Calculate the number o

litres per 100 kilometres consumed when the bike travels at 110 km/h.

11 A plane travelled non-stop rom Los Angeles to Sydney, a distance o 12 027 kilometres

in 13 hours and 30 minutes. The plane started with 180 kilolitres o uel, and on landing

had enough uel to ly another 45 minutes.

a What was the planes average speed in kilometres per hour? Answer to the nearest

whole number.

b How much uel was used? Answer to the nearest kilolitre.

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3.6 Percentage change

Percentage change involves increasing or decreasing a quantity as a percentage o the original

amount o the quantity.

Percentage increase Percentage decrease

1 Add the % increase to 100%.

2 Multiply the above percentage by the

amount.

1 Subtract the % decrease rom 100%.

2 Multiply the above percentage by

the amount.

Example 11 Calculating the percentage change

The retail price o a toaster is $36 and is to be increased by 5%. What is the new price?

Solution

1 Add the 5% increase to 100%.

2 Write the quantity (new price) to be ound.

3 Multiply the above percentage (105%) by the

amount.

4 Evaluate and write using correct units.


100% + 5% = 105%

New price of $36

$37.80

=

=

=

105

3 6

%

New price is $37.80.

Example 12 Calculating repeated percentage changes

Increase $75 by 20% and then decrease the result by 20%.

Solution

1 Add the 20% increase to 100%.



amount.4 Evaluate and write using correct units.

5 Subtract the 20% decrease rom 100%.



amount.

8 Evaluate and write using correct units.


100% + 20% = 120%

New price of $75

$90

=

=

=

120

7 5

%

100% 20% = 80%

New price of $90

$ 2

=

=

=

0

9 0

%

New price is $72.

3.6

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Exercise 3F

1 What is the amount o the increase in each o the ollowing?

a Increase o 10% on $48 b Increase o 30% on $120

c Increase o 15% on $66 d Increase o 25% on $88

e Increase o 40% on $1340 f Increase o 36% on $196

g Increase o 4.5% on $150 h Increase o 1% on $24

2 What is the amount o the decrease in each o the ollowing?

a Decrease o 20% on $110 b Decrease o 60% on $260

c Decrease o 35% on $320 d Decrease o 75% on $1096

e Decrease o 6% on $50 f Decrease o 32% on $36

g Decrease o 12.5% on $640 h Decrease o1 4% on $56

3 David Jones clearance sale has a discount o

30% o the retail price o all clothing. Find

the amount saved on the ollowing items.

a Mens shirt with a retail price o $80

b Pair o jeans with a retail price o $66

c Ladies jacket with a retail price o $450

d Boys shorts with a retail price o $22

e Jumper with a retail price o $124f Girls skirt with a retail price o $50

4 A manager has decided to award a salary increase o 6% or all employees. Find the new

salary awarded on the ollowing amounts.

a Salary o $46 240

b Salary o $94 860

c Salary o $124 280

d Salary o $64 980

5 Molly has a card that entitles her to a 2.5% discount at the store where she works.

How much will she pay or the ollowing items?

a Vase marked at $190

b Cutlery marked at $240

c Painting marked at $560

d Pot marked at $70

30%OFForiginalprice

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Development

6 A used car is priced at $18 600 and oered or sale at a discount o 15%.

a What is the discounted price o the car?

b The car dealer decides to reduce the price o this car by another 15%. What is thenew price o the car?

7 Find the repeated percentage change on the ollowing.

a Increase $100 by 20% and then decrease the result by 20%.

b Increase $280 by 10% and then increase the result by 5%.

c Decrease $32 by 50% and then increase the result by 25%.

d Decrease $1400 by 5% and then decrease the result by 5%.

e Increase $960 by 15% and then decrease the result by 10%.

f Decrease $72 by 12.5% and then increase the result by 33 1% .

8 An electronic store oered a $30 discount on a piece o sotware marked at $120. What

percentage discount has been oered?

9 The cost price o a sound system is $480. Retail stores have oered a range o successive

discounts. Calculate the inal

price o the sound system at

the ollowing stores.

a Store A: Increase o

10% and then a decrease

o 5%

b Store B: Increase o


o 50%

c Store C: Increase o


o 15%

d Store D: Increase o

30% and then a decreaseo 60%

10 The price o a clock has been reduced rom $200 to $180.

a What percentage discount has been applied?

b Two months later the price o the clock was increased by the same percentage

discount. What is new price o the clock?

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Rev

iew

Chapter summary Units of measurement and applications Study guide 3

Units of measurement

unit

kilo 1000

100

10

1000

1000

100

10centi

milli

mega 24

60

60

24

60

60

hours

days

minutes

seconds

10 000 cm2 = 1 m2

1 ha = 10 000 m2

1 000 000 cm3 = 1 m3

Writing numbers in

scientific notation

1 Find the rst two non-zero digits.

2 Place a decimal point between these two digits.

3 Power o ten is number o the digits between the new and the

old decimal point. (Small number negative value, Large

number positive value)

Writing numbers in

significant figures

1 Write the number in scientic notation.

2 Count the digits in the number to determine its accuracy.

3 Round the number to the required signicant gures.

Ratios A ratio is used to compare amounts o the same units in a denite

order. Equivalent ratios are obtained by multiplying or dividing by

the same number.

Unitary method 1 Find one unit o an amount by dividing by the amount.

2 Multiply the result in step 1 by the number.

Converting a rate 1 Write the rate as a raction. First quantity is the numeratorand 1 is the denominator.

2 Convert the rst amount to the required unit.

3 Convert the second amount to the required unit.


Percentage change 1 Add the % increase or subtract the % decrease rom 100%.

2 Multiply the above percentage by the amount.

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ReviewSample HSC Objective-response questions

1 Convert 7.5 metres to millimetres.

A 0.0075 mm B 75 mm C 750 mm D 7500 mm

2 How many square millimetres are in a square centimetre?

A 10 B 100 C 1000 D 10 000

3 Write 4 500 000 in scientic notation.

A 4.5 106 B 4.5 105 C 4.5 105 D 4.5 106

4 Express 0.0655 correct to two signicant gures.

A 0.06 B 0.07 C 0.065 D 0.066

5 The ratio o adults to children in a park is 5:9. How many adults are in the park i there are

630 children?

A 70 B 126 C 280 D 350

6 A 360 gram lolly bag is divided in the ratio 7:5. What is the mass o the smaller amount?

A 150 g B 168 g C 192 g D 210 g

7 A hose lls a 10 L bucket in 20 seconds. What is the rate o fow in litres per hour?

A 0.0001 B 30 C 1800 D 7200

8 Which o the ollowing is the slowest speed?

A 60 km/h B 100 m/s C 10 000 m/min D 6000 m/h

9 The concentration o a drug is 3 mL/g. How many mL are required or 30 g?

A 0.1 mL B 10 mL C 27 mL D 90 mL

10 What is the new price when $80 is increased by 20% then decreased by 20%?

A $51.20 B $76.80 C $80.00 D $115.20

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Rev

iew

Sample HSC Short-answer questions

1 There are six tonnes o iron ore in a train. What is the mass (in tonnes) i another 246 kg o

iron ore is added to the train?

2 Complete the ollowing.

a 500 2 2cm b 40002

cm c 32 2

km

3 A eld has a perimeter o exactly 400 m. Lily measured the eld to be 401.2 m using a

long tape marked in 0.1 m intervals.

a Calculate the limit o reading.

b What is the absolute error or Lilys measurement?

c What is the percentage error or Lilys measurement? Answer correct to three decimal

places.


a 4.8 106 b 6.25 104 c 1.9 102

5 Write these numbers in scientic notation.a 50 800 b 0.0036 c 381 000 000

6 Evaluate the ollowing and express your answer in scientic notation.

a (7.2 105) (2.1 104) b4 6

2 3

1

1 2

7 Convert a measurement o 3580 tonnes into milligrams. Express your answer in scientic

notation correct to two signicant gures.

8 Find the value o 45 154 and express your answer in scientic notation correct to two

signicant gures.

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Review 9 Simpliy the ollowing ratios.

a 500:100 b 20:30 c 28:7

d 10:15:30 e 12:9 f 56:88

g 4.8:1.6 h 34

12

:

10 A 5 kg bag o rice costs $9.20. What is the cost o the ollowing amounts?

a 10 kg b 40 kg c 3 kg

d 7 kg e 500 g f 250 g

11 Convert each rate to the units shown.

a $15/kg to $/g b 14 400 m/h to m/minc 120 cm/h to mm/min d 4800 kg/g to kg/mg

e 14 L/g to mL/kg f $3600/g to c/mg

12 A car travels 960 km on 75 litres o petrol. How ar does it travel on 50 litres?

13 Daniel and Ethan own a business and share the prots in the ratio 3:4.

a The prot last week was $3437. How much does Daniel receive?

b The prot this week is $2464. How much does Ethan receive?

14Jill has a shareholder card that entitles her to a 5% discount at a supermarket. How muchwill she pay or the ollowing items? Answer to the nearest cent.

a Breakast cereal at $7.60 b Milk at $4.90

c Coee at $14.20 d Cheese at $8.40

15 An electrician is buying a light tting or $144 at a hardware store. He receives a clearance

discount o 15% then a trade discount o 10%. How much does the electrician pay or the

light tting?

Challenge questions 3

chapter 3 units of measurement and application

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