chapter 3 units of measurement and application
TRANSCRIPT
-
7/28/2019 Chapter 3 Units of Measurement and Application
1/30
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
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
2/30
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
3/30
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
=
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
4/30
-
7/28/2019 Chapter 3 Units of Measurement and Application
5/30
77Chapter 3 Units of measurement and applications
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
6/30
78 Preliminary Mat hematics General
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
7/30
79Chapter 3 Units of measurement and applications
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
8/30
80 Preliminary Mat hematics General
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
9/30
81Chapter 3 Units of measurement and applications
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.
b What is the limit o reading?
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.
b What is the limit o reading?
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
10/30
82 Preliminary Mat hematics General
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
11/30
83Chapter 3 Units of measurement and applications
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
12/30
84 Preliminary Mat hematics General
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
1 Write in scientiic notation.
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.
5 Write in scientiic notation.
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
13/30
85Chapter 3 Units of measurement and applications
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
6 Write each o the ollowing as a basic numeral.
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
14/30
86 Preliminary Mat hematics General
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
15/30
87Chapter 3 Units of measurement and applications
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
16/30
88 Preliminary Mat hematics General
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
17/30
89Chapter 3 Units of measurement and applications
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.
6 Write the answer in words.
30 L km
1 L km
7 L km
84 km
360
30
360
307
The car travels 84 km.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
18/30
90 Preliminary Mat hematics General
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
19/30
91Chapter 3 Units of measurement and applications
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?
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
20/30
92 Preliminary Mat hematics General
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.
5 Simpliy the raction.
6 Write the rate as a raction.
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.
10 Simpliy the raction.
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
21/30
93Chapter 3 Units of measurement and applications
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
1 Write the rate as a raction.
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.
6 Write the answer in words.
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.
4 Write the answer in words.
6.2 g/50 g6.2 g
50 g
6.2
=
=
=
50100
12
%
. %4
Percentage composition
is 12.4%.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
22/30
94 Preliminary Mat hematics General
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?
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
23/30
95Chapter 3 Units of measurement and applications
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
24/30
96 Preliminary Mat hematics General
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.
5 Write the answer in words.
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%.
2 Write the quantity (new price) to be ound.
3 Multiply the above percentage (120%) by the
amount.4 Evaluate and write using correct units.
5 Subtract the 20% decrease rom 100%.
6 Write the quantity (new price) to be ound.
7 Multiply the above percentage (80%) by the
amount.
8 Evaluate and write using correct units.
9 Write the answer in words.
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
25/30
97Chapter 3 Units of measurement and applications
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
26/30
98 Preliminary Mat hematics General
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
40% and then a decrease
o 50%
c Store C: Increase o
25% and then a decrease
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?
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
27/30
99Chapter 3 Units of measurement and applications
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.
4 Simpliy the raction.
Percentage change 1 Add the % increase or subtract the % decrease rom 100%.
2 Multiply the above percentage by the amount.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
28/30
100 Preliminary Mat hematics General
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
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
29/30
101Chapter 3 Units of measurement and applications
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.
4 Write each o the ollowing as a basic numeral.
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.
ISBN: 9781107627291
Photocopying is restricted under law and this material must not be transferred to another party
The Powers Family Trust 2013 Cambridge University Press
-
7/28/2019 Chapter 3 Units of Measurement and Application
30/30
102 Preliminary Mat hematics General
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