Senior Phase Grade 8 • Study Guide 2/2
Mathematics
PL E SA M
Study Guide 2/2
2608-E-MAM-SG02
CAPS
Mathematics
exercises to refresh prior knowledge. • Revision explanations of concepts and techniques. • Detailed examples help learners to better understand new concepts. • Worked Varied exercises to entrench theory and practise mathematical skills. • Test papers and memorandums for exam preparation. • Formula sheets and accepted geometrical reasons for • quick reference. of mathematical terms. • Index facilitator’s guide contains step-by-step calculations • The and answers. • Use in school or at home.
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Reg. No.: 2006/002404/07 ISBN: 9781779841834
2608-E-MAM-SG02_Cover.indd 16
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Mathematics
SA M
PL
E
Study Guide 2/2 − Grade 8
2608-E-MAM-SG02
CAPS-aligned
Prof. C Vermeulen, Lead author H Otto M Sherman
TABLE OF CONTENTS
2.4
THEME 13: COMMON FRACTIONS................................................................. 1
3.
Percentages..................................................................................................16
3.1
Percentages of whole numbers.....................................................................17
Solving problems by dividing or multiplying fractions ...............................15 Exercise 76: Solving problems involving multiplication or division of fractions .................................................................................16
Introduction.............................................................................................................1 Calculations with common fractions...........................................................1
1.1
Add and subtract common fractions and mixed numbers..............................1
3.2
Exercise 68: Add and subtract common fractions and mixed numbers..........4
E
Exercise 77: Percentages of whole numbers.................................................17
1.
Determine fractions of whole numbers..........................................................4
1.3
Multiply common fractions and mixed numbers by each other.....................5
Exercise 78: Part of a whole as a percentage of the whole...........................18 Exercise 79: Percentage increase and decrease............................................19
3.4
Exercise 80: Calculate quantities if the percentage increase or decrease is given.......................................................................................19
Exercise 69: Multiply whole numbers by fractions and fractions by fractions................................................................................6
1.5
Divide whole numbers by fractions and fractions by ...................................... fractions..........................................................................................................7
3.5
Exercise 70: Divide whole numbers by fractions and fractions by fractions......................................................................................8
4.
Equivalence.................................................................................................21
4.1
Equivalent forms of common fractions........................................................21
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1.4
Squares, cubes, square roots and cube roots of common fractions.................9 Exercise 71: Squares and cubes of common fractions..................................10 Exercise 72: Square roots and cube roots of common fractions...................12
2.
Problem solving involving fractions .........................................................12
2.1
Problems involving fractions of several objects or fractions of a whole......12 Exercise 73: Find a fraction of a number......................................................13
2.2
Exercise 81: Problem solving that involve percentages................................20
Exercise 82: Equivalent forms of common fractions....................................22 4.2
Equivalent forms of common fractions and decimal fractions.....................22 Exercise 83: Equivalent forms of common fractions and decimal fractions....................................................................................23
4.3
Equivalent forms of common fractions, decimal fractions and percentages..............................................................................23
Self-evaluation.......................................................................................................25
Problems involving adding and subtracting fractions ..................................15
Summary of theme................................................................................................26
Exercise 75: Solve problems involving addition and subtraction of fractions.................................................................................15
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Solving problems involving percentages......................................................20
Exercise 84: Equivalent forms of common fractions, decimal fractions and percentages................................................................24
Problems involving division of objects in a certain ratio.............................14 Exercise 74: Divide a number of objects in a certain ratio...........................14
2.3
Calculate quantities if the percentage increase or decrease is given............19
PL
1.2
Part of a whole as a percentage of the whole...............................................17
End of theme exercise...........................................................................................27
iii
G08– Mathematics
G08 – Mathematics – Study Guide 2/2 THEME 14: DECIMAL FRACTIONS................................................................29
Self-evaluation.......................................................................................................41
Introduction...........................................................................................................29
Summary of theme................................................................................................42
1.
Revision........................................................................................................29
End of theme exercise...........................................................................................43
1.1
Place values..................................................................................................29
1.2
Compare and order decimal numbers...........................................................29
1.3
Round decimal fractions to at least two places.............................................30
THEME 15: THEOREM OF PYTHAGORAS.................................................... 45 Introduction...........................................................................................................45 1.
Exercise 85: Values of digits in decimal fractions, ordering decimal fractions and rounding decimal fractions.....................................................31
Exercise 94: Identifying the parts of a right-angled triangle........................46 2.
Add and subtract decimal fractions..............................................................32
Exercise 95: Investigation into the sides of right-angled triangles ..............47
Exercise 86: Add and subtract decimal fractions..........................................32
3.
Multiply decimal fractions by whole numbers.............................................33
Exercise 96: Calculations with the theorem of Pythagoras..........................49
Exercise 87: Multiply decimal fractions by whole numbers........................33 1.6
4.
Divide decimal fractions by whole numbers................................................34 Extend multiplication and division...........................................................36
2.1
Multiply decimal fractions by decimal fractions..........................................36
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2.
Solving geometry problems using pythagoras..........................................50 Exercise 97: Solving geometry problems using Pythagoras ........................51
Exercise 88: Divide decimal fractions by whole numbers...........................35
Exercise 89: Multiply decimal fractions by decimal fractions.....................36 2.2
Calculations with the theorem of pythagoras..........................................47
PL
1.5
Investigation into the sides of a right-angled triangle.............................46
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1.4
Identifying the parts of a right-angled triangle.......................................45
Divide decimal fractions by decimal fractions.............................................37 Exercise 90: Divide decimal fractions by decimal fractions........................38
Exercise 98: Identifying right-angled triangles............................................53
5.
Solving problems in context using pythagoras.........................................54 Exercise 99: Solving problems in context using Pythagoras........................54
Self-evaluation.......................................................................................................55 Summary of theme................................................................................................56
3.
Squares, cubes, square roots and cube roots............................................38
End of theme exercise...........................................................................................56
3.1
Squares and cubes.........................................................................................38
THEME 16: PERIMETER AND AREA...............................................................58
Exercise 91: Squares and cubes ...................................................................39 3.2
Square roots and cube roots..........................................................................39
1.
Exercise 92: Square roots and cube roots ....................................................40 4.
Introduction...........................................................................................................58 Perimeter of polygons.................................................................................58 Exercise 100: Perimeter of simple shapes ...................................................60
Solving problems involving decimal fractions..........................................40
2.
Exercise 93: Solving problems involving decimal fractions.........................41
Perimeter of complex shapes.....................................................................60 Exercise 101: Perimeter of complex shapes.................................................61
iv
3. 4.
Circumference of a circle...........................................................................62
THEME 18: DATA HANDLING........................................................................92
Exercise 102: Circumference of a circle.......................................................63
Introduction...........................................................................................................92
Solving problems involving perimeter......................................................64
1.
Exercise 103: Solving problems involving perimeter ..................................65 5.
Exercise 111: Collect data ...........................................................................93
Area and units.............................................................................................66
2.
Exercise 104: Area of simple shapes............................................................67 6.
Organise and summarise data...................................................................93 Exercise 112: Organise data.........................................................................95
Area of a circle............................................................................................67
3.
Exercise 105: Area of a circle ......................................................................69
Summarise data .........................................................................................96 Exercise 113: Summarise data......................................................................97
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7.
Collect data..................................................................................................92
Breaking down shapes to find area...........................................................69
4.
Exercise 106: Areas of complex shapes.......................................................71
Represent data ...........................................................................................97
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Exercise 114: Represent data......................................................................102
Self-evaluation.......................................................................................................71
5.
Summary of theme................................................................................................72
Exercise 115: Interpret data represented in graphs.....................................107
End of theme exercise...........................................................................................73
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6.
THEME 17: SURFACE AREA AND VOLUME................................................... 76 Introduction...........................................................................................................76 1.
Volume and capacity..................................................................................76
Exercise 107: Volume and capacity of a rectangular prism ........................79 2. 3.
Interpret data............................................................................................102 Analyse and report data...........................................................................109 Exercise 116: Analyse and report data ......................................................110
Self-evaluation.....................................................................................................111 Summary of theme..............................................................................................112 End of theme exercise.........................................................................................113
Volume of triangular prisms......................................................................79
THEME 19: FUNCTIONS AND RELATIONSHIPS...........................................116
Exercise 108: Volume of a triangular prism.................................................82
Introduction.........................................................................................................116
Surface area of prisms................................................................................82
1.
Exercise 109: Surface areas of prisms..........................................................85
Input and output values...........................................................................116 Exercise 117: Input and output values........................................................118
Exercise 110: Problems involving volume and surface area .......................87
2.
Input and output values using formulas.................................................120
Self-evaluation.......................................................................................................88
Exercise 118: Input and output values using formulas...............................121
Summary of theme................................................................................................88
Self-evaluation.....................................................................................................122
End of theme exercise...........................................................................................89
Summary of theme..............................................................................................122 End of theme exercise.........................................................................................123
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G08– Mathematics
G08 – Mathematics – Study Guide 2/2 THEME 20: ALGEBRAIC EQUATIONS......................................................... 124
Self-evaluation.....................................................................................................149
Introduction.........................................................................................................124
Summary of theme..............................................................................................150
1.
Solve equations..........................................................................................124
End of theme exercise.........................................................................................151
Exercise 119: Solve equations using additive and multiplicative inverses and exponential laws.....................................................................126
THEME 22: TRANSFORMATION GEOMETRY............................................. 153
2.
Introduction.........................................................................................................153
Use equations to solve problems..............................................................127 Exercise 120: Use equations to solve problems..........................................128
3.
Use equations to complete tables and find ordered number pairs.......129
Translation of a point...............................................................................154
1.1
Translation of a point within a quadrant.....................................................154
1.2
Translation of a point across quadrants......................................................155
1.3
Combining translations of a point..............................................................156
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Exercise 121: Complete a table and find ordered number pairs.................130
1.
Self-evaluation.....................................................................................................131
Exercise 126: Translation of a point ..........................................................157
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Summary of theme..............................................................................................131 End of theme exercise.........................................................................................132
2.
THEME 21: GRAPHS..................................................................................... 133
3.
Exercise 127: Translation of triangles .......................................................160
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Introduction.........................................................................................................133 1.
Analyse and interpret graphs..................................................................133
1.1
Linear graphs..............................................................................................133
1.2
Non-linear graphs.......................................................................................135 Exercise 122: Analyse and interpret graphs ..............................................137
2.
Draw graphs from information...............................................................139 Exercise 123: Draw graphs from information............................................141
3.
Points on the cartesian plane...................................................................142
3.1
Ordered number pairs.................................................................................142
3.2
Ordered number pairs and graphs...............................................................143
Reflection of a point in an axis................................................................161 Exercise 128: Reflection of a point in the y-axis (Investigation 1)............162 Exercise 129: Reflection of a point in either axis.......................................163
4.
Reflection of a triangle in an axis............................................................164 Exercise 130: Reflection of a triangle in an axis........................................165
5.
Rotation of a triangle................................................................................165 Exercise 131: Rotation of a triangle ..........................................................168
6.
Enlargements.............................................................................................169 Exercise 132: Enlargements........................................................................173
7.
Reductions.................................................................................................174 Exercise 133: Reductions...........................................................................176
Exercise 124: Points on the Cartesian plane...............................................145 4.
Translation of a triangle ..........................................................................158
Self-evaluation.....................................................................................................177
Plot points from tables.............................................................................146
Summary of theme..............................................................................................177
Exercise 125: Plot points from tables.........................................................148
End of theme exercise.........................................................................................178 vi
THEME 23: GEOMETRY OF 3D OBJECTS................................................... 182
2.
Probability of single events......................................................................197
Introduction.........................................................................................................182
2.1
What is probability?....................................................................................198
1.
Platonic solids............................................................................................182
2.2
Determine probability.................................................................................198
1.1
2D Shapes and 3D objects..........................................................................182
Exercise 145: Determine probability..........................................................199 3.
Exercise 134: Word quiz ............................................................................182
Relative frequency....................................................................................201 Exercise 146: Relative frequency...............................................................203
Exercise 135: Platonic solids .....................................................................183
Self-evaluation.....................................................................................................204
Exercise 136: Compare Platonic solids .....................................................184
Summary of theme..............................................................................................204
2.
Prisms and pyramids................................................................................184
End of theme exercise.........................................................................................205
2.1
Prisms.........................................................................................................184
2.2
Pyramids.....................................................................................................185
PL
NOVEMBER: PAPER 1 (A)............................................................................ 206
Exercise 137: Investigate vertices and edges of a pyramid........................186
NOVEMBER: PAPER 1 (B)............................................................................ 209
Working with nets.....................................................................................186
NOVEMBER: PAPER 2 (A).............................................................................211
Exercise 138: Understand nets....................................................................188
NOVEMBER: PAPER 2 (B)............................................................................. 215
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3.
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The platonic solids......................................................................................182
1.2
Exercise 139: Sketch a net..........................................................................189 Exercise 140: Construct a net and build a model.......................................190
INDEX........................................................................................................... 219
Exercise 141: Construct a net and build a model.......................................190 Exercise 142: Build a model of a triangular prism.....................................191 Exercise 143: Relationship between vertices, edges and faces...................191 Self-evaluation.....................................................................................................192 Summary of theme..............................................................................................192 End of theme exercise.........................................................................................193 THEME 24: PROBABILITY............................................................................ 196 Introduction.........................................................................................................196 1.
Possible outcomes......................................................................................196 Exercise 144: List the possible outcomes...................................................197
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vii
G08– Mathematics
Numbers, operations and relationships (1)
1.
THEME 13
A common fraction is part of a whole and consists of a numerator and a denominator.
COMMON FRACTIONS
Denominator – The number of equal parts into which the whole is divided.
E
Introduction In this theme, you will:
•
3 8
Numerator – it tells us how many of the equal parts, into which the whole is divided, are taken.
If the numerator is smaller than the denominator, the fraction is called a proper fraction. If the numerator is bigger than the denominator, the fraction is called an improper fraction.
PL
•
cover the following operations involving common fractions and mixed numbers: ◦ addition and subtraction ◦ finding fractions of whole numbers ◦ multiplication ◦ dividing whole numbers by fractions using the reciprocal of a number ◦ dividing a fraction by a fraction using the reciprocal ◦ finding squares and cubes ◦ finding square roots and cube roots
•
SA M
•
CALCULATIONS WITH COMMON FRACTIONS
solve problems where you have to: ◦ find a fraction of a number of people or objects ◦ divide a number of objects among people according to their contributions ◦ add or subtract fractions ◦ multiply or divide by fractions
•
work with percentages: ◦ percentages of whole numbers ◦ part of a whole as a percentage of the whole ◦ percentage increase or decrease ◦ calculate quantities if the percentage increase or decrease is given ◦ problem solving involving percentages
•
find and work with equivalent forms of common fractions, decimal fractions and percentages.
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1
In the sketch, the circle is divided into eight equal parts of which three are shaded. So, the fraction represented by the shaded part is __38 . •
1.1
A mixed number, that is a number that consists of a whole number and a fraction, can be written as an improper fraction. For example, 1 __14 can be written as __ 54 . Improper fractions simplify calculations with fractions.
Add and subtract common fractions and mixed numbers.
If we want to add fractions, they must be of the same kind. That means that their denominators must be equal.
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2
Solutions
Worked example 1: Add and subtract fractions with the same denominators
a)
Calculate the following: a) b)
__ 35 + __15 __ 78 − __58
__ 78 − __58 5 = ____ 7 − 8 = __ 82 = __ 41
b)
a) b)
__ 34 + __ 23 __ 56 − __ 14
LCM (lowest common multiple) is the smallest non-zero number into which different whole numbers can divide without leaving a remainder.
The LCM of 4 and 3 is 12
NOTE If you multiply above the line and below the line by the same number, you actually multiply by 1. Multiplying by 1 does not change a number. The LCM of 6 and 4 is 12
Worked example 3: Add and subtract mixed numbers
Divide both the numerator and the numerator by 2
Worked example 2: Add and subtract fractions which do not have the same denominators Calculate the following:
__ 56 − __14 2 ____ = ____ 56 × − 1 × 3 ×2 4×3 = __ 10 − __ 3 12 12 7 = __ 12
PL
b)
__35 + __ 15 1 = ____ 3 + 5 = __ 45
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a)
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Solutions
__ 34 + __ 23 3 ____ = ____ 34 × + 2 × 4 ×3 3×4 9 8 = __ 12 + __ 12 = __ 17 12 = __ 12 + __ 5 12 12 5 = 1 __ 12
Calculate the following and leave your answer in its simplest fraction form: a) b) c)
3 __14 + 2 __18
1 2 __35 − 4 __ 10
4 − (1 __13 + 1 __83 )
Solutions a)
3 __14 + 2 __18
= ( ____ 3 ×4 4 + __ 41 ) + ( ____ 2 ×8 8 + __ 18 )
2
Each whole has four quarters and each whole has eight eights
= __ 13 + __ 17 4 8
× 2 __ = _____ 13 + 17 4×2 8
c)
1 × 3 __ = 4 − [(____ + 13 ) + (____ 1 ×8 8 + __ 83 )] 3
The LCM of 4 and 8 is 8
= __ 26 + __ 17 8 8
= 4 − ( __ 43 + __ 11 8)
= __ 43 8
41 = __ 13 − __ 10 5
×2 = _____ 13 − __ 41 10 5×2
= __ 26 − __ 41 10 10 − 41 = ______ 26 10
15 = − __ 10 10 __ = ___ −10 + –5 10 5 = − 1 __ 10 = − 1 __21
The LCM of 3 and 8 is 24
E
= 4 − __ 65 24
24 __ = _____ 41 × − 65 × 24 24
PL
= (____ 2 ×5 5 + __ 35 ) − ( __ 40 + __ 1 10 10 )
Do the calculation inside the bracket
= 4 − (__ 32 + __ 33 24 24 )
Write the whole number as an improper fraction
__ = 96 − __ 65 24 24
Write mixed numbers as improper fractions
= __ 31 24
= __ 24 + __ 7 24 24
The LCM of 5 and 10 is 10
SA M
1 2 __35 − 4 __ 10
Write mixed numbers as improper fractions
8 _____ ×3 = 4 − ( ____ 43 × + 11 ×8 8×3)
= __ 40 + __ 83 8 = 5 __38 b)
4 − (1 __13 + 1 __83 )
5 goes into 5 once and into 10 twice
7 = 1 __ 24
NOTE A short method to write a mixed number as an improper fraction is: • • • •
Multiply the whole number by the denominator. Then add the numerator of the fraction to your answer. Write the answer above the line. Write the denominator below the line. Example: 3 _34 = _ 3 × 44 + 3 = _ 12 4+ 3 = _ 15 4
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3
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 9 2.7 __45 − __ 34 − __ 10
Exercise 68: Add and subtract common fractions and mixed numbers 1.
8 2.8 __35 + __ 35 − __ 67
Calculate the following and leave the answers in their simplest form:
3.
1.1 __52 + __ 35
3.1 4 __12 + 2 __23
1.2 1 __52 + 2 __35
3.2 3 __34 − 2 __13
9 1.3 __ − __ 7 20 20
7 41 − 4 ___ 1.7 3 ___ 100 100
1.8 1 __31 − 2 __23 2.
E
3 3.4 3 __ − 4 __35 10
PL
7 41 + 4 ___ 1.6 3 ___ 100 100
3.3 − 1 __25 + 4 __23
5 3.5 3 __58 + 1 __13 − 2 __ 12
3.6 3 __14 − ( __ 23 + __ 34 )
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1.4 3 __51 − 2 __45 1 − __ 1.5 __ 3 20 20
Calculate the following and leave the answers in their simplest form:
2.1 __43 + __ 56 2.2 __83 − __ 56 2.3 __53 − __ 18
Calculate the following and leave the answers in their simplest form:
7 2.4 __ − __ 7 15 30
3.7 6 − ( 3 __13 + 2 __81 )
3.8 7 − ( 1 __56 − 2 __32 ) 7 3.9 3 __ + 1 __3 − 2) 12 ( 8
3.10 2 __18 − (7 __12 − 2 __31 )
1.2
Determine fractions of whole numbers
We usually say, ‘a fraction of a whole number’. If we do such a calculation, ‘of’ means that we multiply the fraction by the whole number.
7 2.5 __ − __ 23 12 24 8 2.6 __76 + __ 21 − __ 11 42
4
1.3
Worked example 4: Determine a fraction of a whole number
When we multiply common fractions and mixed numbers by each other:
Calculate the following without using a calculator. Leave your answers in their simplest form. b)
• •
__ 34 of 20 5 __ 12 of 20
•
Solutions
b)
5 __ 12 of 20 5 = __ 12 × __ 20 1 5 5 = __ 12 × __ 20 1 3 5 × 5 = ____ 3 × 1 = __ 25 3 = __ 24 + __ 13 3 = 8 __13
©Optimi
PL
__ 34 of 20 = __ 34 × __ 20 1 5 = __ 34 × __ 20 1 1 3 × 5 = ____ 1 × 1 = 15
Worked example 5: Multiply common fractions by each other
Calculate the following without using a calculator. Leave answers in their simplest form.
Write the whole number as a fraction over 1 4 divides into 20 five times and into itself once
a)
SA M
a)
Change the mixed numbers to improper fractions. Write these fractions in their simplest form by dividing the numerators and denominators by the factors that they have in common. Multiply the numerators by each other and the denominators by each other.
E
a)
Multiply common fractions and mixed numbers by each other
Write the whole number as a fraction over 1
b)
__ 23 × __34 9 5 − __ × __ × __7 12 14 9
Solutions a)
4 divides into 12 three times and into 20 five times
b)
__ 23 × __ 34 1 = ____ 11 × ×2 = __ 12
3 divides into 3 once and 2 divides into 4 twice
9 5 __ − × __ × __7 12 14 9
5×1×1 = − _______ 12 × 2 × 1
9 divides into 9 once and 7 divides into 14 twice
5 = − __ 24
5
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 1.
Worked example 6: Multiply common fractions and mixed numbers by each other
1.1 __38 of 2 4
5 1.2 − __ of 2 8 12
Calculate the following without using a calculator. Leave answers in their simplest form. a) b)
1.3 __45 of 30
1 __37 × 5 __35
1.4 __56 of 27
__ 34 × 1 __59 × 2 __85
1.5 __35 of 100
E
Solutions 1 __37 × 5 __35
= __ 10 × __ 28 7 5
4 = ____ 21 × ×1
= 8 b)
__ 34 × 1 __59 × 2 __58 1
7
2
3
7
= __ 34 × __ 14 × __ 21 9 8 7×7 = _______ 12 × ×1×8
= __ 49 16
= __ 48 + __ 1 16 16 1 = 3 __ 16
PL
7 1.6 __ of 30 12
1.7 1 __12 of 12
7 divides into 28 four times and 5 divides into 10 twice
4 1.8 __37 of 3 __ 15
SA M
a)
Calculate:
1.9 − 1 __78 of (− 3 __59 ) 2.
Calculate:
2.1 __12 × __ 12
8 2.2 __34 × __ 15 9 2.3 − __59 × __ 20 8 2.4 __38 × __ 15
2.5 − __14 × __ 14 × ___ −14
Exercise 69: Multiply whole numbers by fractions and fractions by fractions
2.6 (− __41 ) × (− __13 ) × (− __52 ) 2.7 __23 × (___ −43 ) × (− __67 )
You are not allowed to use a calculator in this exercise. Leave your answers in their simplest form. 6
7 2.8 __ × __ 45 × __ 7 15 49 18
3.
The reciprocal of a whole number is simply ___________ whole 1number . The reciprocal of 3, for 1 1 __ __ instance, is 3 , because 3 × 3 = 1.
Calculate:
If we divide by mixed numbers, we do the following:
3.1 − 1 __45 × 1 __19
• • •
3.2 1 __13 × 1 __12 3.3 1 __17 × 6 __18
Worked example 7: Divide whole numbers by fractions and fractions by whole numbers
a)
b)
Divide whole numbers by fractions and fractions by fractions
SA M
1.4
Calculate the following without a calculator and leave your answers in their simplest form if necessary:
PL
5 3.6 5 __12 × 2 __ × 1 __37 22
E
3.4 − __25 × (− 4 __16 ) × (− 5 __17 ) 8 3.5 1 __79 × 4 __67 × 2 __ 17
When we divide a whole number by a fraction, we multiply the number by the reciprocal (the multiplicative inverse) of the fraction. The reciprocal of __ 32 , for 3 3 2 __ __ __ example, is 2 , because 3 × 2 = 1. So, the fraction multiplied by its reciprocal, is always equal to 1.
Say Mary has ten pizzas and she wants to know how many of her friends can get __ 23 of a pizza. (She does not know that she can multiply by the reciprocal of __ 23 .) • • •
She will first multiply 10 by 3 to find how many thirds there are in ten. The answer is 10 × 3 = 30. Next, she will divide 30 by 2 to find how many groups of 2 there are in the 30 thirds. The answer is 15. So, 15 of her friends can each get two-thirds of a pizza. She will get the same answer if she does the following in one step: 10 × __23 = 15.
©Optimi
7
3 ÷ __34 __ 13 ÷ 4
Solutions a)
3 ÷ __ 34
= 3 × __ 43
3
Multiply 3 by the reciprocal of __ 4
= ____ 3 ×3 4
The reason why we multiply by the reciprocal of a fraction if we divide by a fraction, can be explained as follows:
•
Change the mixed number to an improper fraction. Change the improper fraction to its reciprocal. Multiply the number that you want to divide by this reciprocal.
= 4 b)
__ 13 ÷ 4 = __ 13 × __ 41 1 = ____ 13 × ×4 1 = __ 12
1
Multiply __ 3 by the reciprocal of 4
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 c)
Worked example 8: Divide fractions by fractions
= __ 72 ÷ __ 57 = __ 72 × __ 75
Calculate the following without a calculator and leave your answers in their simplest form if necessary: b) c)
5 = ____ 12 × ×1
6 __ 35 ÷ __ 35 7 1 __ 25 ÷ 2 __ 10 3 __12 ÷ 1 __25
= 2 __12
7 1 __ 25 ÷ 2 __ 10 7 = __ 25 ÷ __ 21 10 7 = __ 25 × __ 10 21 = __ 15 × __ 23
7 divides into 7 once
Write the answer in its simplest form
Exercise 70: Divide whole numbers by fractions and fractions by fractions
PL
You are not allowed to use a calculator in this exercise. Leave your answers in their simplest form.
Multiply by the reciprocal of the fraction
3 divides into 3 once and into 6 twice, 5 divides into 35 seven times
SA M
b)
6 __ 35 ÷ __ 35 = __ 35 × __ 35 6 = __ 11 × __ 72 = __ 72 = 3 __12
Multiply by the reciprocal of the second fraction
= __ 52
Solutions a)
Convert the mixed numbers to improper fractions
E
a)
3 __12 ÷ 1 __25
Convert the mixed number to an improper fraction
Multiply by the reciprocal of the fraction
7 divides into 21 three times, 5 divides into 25 five times and into 10 twice
1.
Calculate:
1.1 3 ÷ __ 14 1.2 9 ÷ __ 34
1.3 8 ÷ 2 __23 1.4 18 ÷ 1 __35 1.5 10 ÷ 1 __14 1.6 4 __25 ÷ 11
2 = ____ 15 × ×3 2 = __ 15
8
1.7
9__38 ÷ 25
1.8
5__58 ÷ 35
1.9
12__14 ÷ 49
3.7 1 __14 ÷ 2 __12 ÷ 1 __32
1.10 27 ÷ 1 __45
3.8 − 7 __21 ÷ (− 5 __14 ) ÷ ( − 3 __67 )
2.
3.9 7 __12 ÷ 4 __12 ÷ 2 __32
Calculate:
7 2.1 __25 ÷ __ 10
3.10 2 __56 ÷ 3 __79 ÷ 2 __17
5 2.2 __58 ÷ __ 16
1.5
7 2.6 __ × __ 8 ÷ __ 14 16 9 27 9 2.7 __47 ÷ __ 32 ÷ __ 16
2.8 (__ 23 + __ 43 − __ 16 ) ÷ ( __ 14 + __ 12 − __ 13 ) 3.
Calculate:
3.1 1 __13 ÷ 1 __21 3.2 5 __35 ÷ 1 __95
3.3 12 __34 ÷ 1 __87
PL
2.5 __23 ÷ __ 94 ÷ __ 58
We already covered this content in Theme 3. We revise this briefly here.
Worked example 9: Calculate squares and cubes of common fractions
SA M
7 2.4 __49 ÷ __ 18
Squares, cubes, square roots and cube roots of common fractions
E
9 2.3 __35 ÷ __ 10
a)
b) c) d)
2
(__ 78 )
2
(− 1 __ 14 ) 3
(− __ 23 ) 3
(1 __ 23 )
Solutions a)
2
(__ 78 ) 2
= __ 7 2 8
3.4 − 2 __58 ÷ 4 __21
Square both the numerator and the denominator (variation of 4th exponential law)
= __ 49 64
3.5 − 3 __14 ÷ (− 8 __23 ) 5 3.6 3 __38 ÷ 1 __ ÷ 2 16
©Optimi
9
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 2
1 2 __ (− 1 4 )
2
= (− __54 ) (− 5) 2 4
= _____ 2
1.2 (__ 58 )
2
1.5 − (− __38 )
( − 5) × ( − 5) = 25
3
9 = 1 __ 16
1.6 (− __32 )
2 3 __ (− 3 ) (− 2) 3 = _____ 3 5
1.7 − (− __23 )
3
(variation of 4th exponential law)
3
1.10 − (__ 54 )
3 = __ 5 3 3
= ___ 125 27
17 4 __ = 27
SA M
3
= ( __ 53 )
3
3 1.9 − (− __ 10 )
(−2) × (−2) × (−2) = 4 × (−2) = − 8
3
2 __ (1 3 )
3
1.8 (− __53 )
Find the cube of both the numerator and the denominator
PL
−8 = ___ 125
d)
2
1.4 (− __83 )
Square both the numerator and the denominator (variation of 4th exponential law)
= __ 25 16
c)
2
1.3 − (__ 83 )
Convert the mixed number as an improper fraction
E
b)
Convert the mixed number to an improper fraction
Find the cube of both the numerator and the denominator (variation of 4th exponential law)
Exercise 71: Squares and cubes of common fractions
2.
Calculate: 2
2.1 (1 __23 ) 2
2.2 (− 1 __43 ) 2
2.3 − (1 __83 ) 2
2.4 − (− 2 __14 ) 2
2.5 (− 2 __32 ) 2
You are not allowed to use a calculator in this exercise.
2.6 − (2 __21 )
1.
2.7 (1 __13 )
2
Calculate: 2
1.1 (__ 35 )
2
2.8 (__ 34 − __ 13 ) 10
2
2.9 (__ 34 − __ 32 )
Solutions ___ a)
2
2.10 (3 + 1 __12 ) 3.
3
3.1 − (− 1 __12 ) 3
3.2 (− 1 __12 )
b)
E PL
3
3
3.7 (1 __23 ) − (1 __23 )
SA M
3.6 (2 __12 )
Worked example 10: Calculate square roots and cube roots of common fractions Calculate___ the following without using a calculator: √ 5 __49
c)
3
d)
3
c)
____
27 √ − __ 64 ____
Write the improper fraction as a mixed number
____
3
√
_
√ 64 −3 ___ = 4
Find the cube roots of both the numerator and the denominator The cube root of a negative number is negative
____
10 √ 2 __ 27 3
___
= √ __ 64 27 3
3
_
64 = ___ 3 _ √
√ 27 __ = 43 = 1 __13
___
Find the square roots of both the numerator and the denominator
27 √ − __ 64 3
27 = ____ 3 −_
d)
Convert the mixed number to an improper fraction
_
√ 9 7 __ = 3 = 2 __13
3
b)
___
√
3.5 (1 __15 )
9 √ __ 25
___
_ = ___ 49
3
3.4 (3 __13 )
a)
Find the square roots of both the numerator and the denominator
√ 5 __49
= √ __ 49 9
3 3.3 − (1 __12 )
3 3.8 (__ 43 + __ 12 )
_
√9 = ___ _ √ 25 = __ 35
Calculate:
2
9 √ __ 25
Find the cube roots of both the numerator and the denominator
10 √ 2 __ 27
©Optimi
11
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 ____
61 2.6 3√ 1 __ 64
Exercise 72: Square roots and cube roots of common fractions 1.
______
2.7 √ __ 14 + 1 __19 ____
Calculate the following without using a calculator:
_____
10 17 2.8 3√ 2 __ + 3√ − 4 __ 27 27
___
9 49 1.1 √ __
_____
64 1.2 3√ − ___ 125
2.
___
E
1.3 √ __ 16 25
Now that we know more about fractions, we can also solve problems in context using our knowledge of fractions.
_____
PL
1.4 3√ _____ 2 16 000 ___
2.1
1.5 √ __ 25 36
______
_______
9 8 1.7 √ ___ 100 + __ 50
1.8 √ 2.
______
__ 13 − __ 1 36 9
SA M
34 __ 1.6 √ __ 54 + 10 27 3
Calculate the following without using a calculator:
2.1 √ 2.2 √ 3
___
1 __79
___
3 __38
Problems involving fractions of several objects or fractions of a whole Worked example 11: Use fractions to calculate the number of people
There are 32 learners in a Grade 8 class. Nine-sixteenths of the learners are girls. Determine the number of: a)
girls in the class
b)
boys in the class.
Solutions a)
____
9 Number of girls = __ × 32 16
= 9 × 2 = 18
24 2.3 √ 1 __ 25
b)
________
9 2.4 √ 1 __ + 2 __79 16 ____
PROBLEM SOLVING INVOLVING FRACTIONS
___
9 2.5 √ 1 __ + √ 2 __79 16
12
Number of boys = 32 − 18 = 14
Worked example 12: Use fractions to calculate amounts
Exercise 73: Find a fraction of a number
Susan receives R300 pocket money per month. She spends the money as follows: She uses two-fifths to buy airtime. She uses three-tenths to buy food at the tuck shop at school. She uses one-tenth to buy sweets. She saves the remainder of the money.
b)
uses to buy food
c)
saves.
Solutions a)
1.1
Calculate the distance he travelled if he covered four-sevenths of the distance.
1.2
Now calculate the distance he must still travel to reach this town.
2.
There are 84 spectators at a school soccer match. A third of them are parents, one-sixth are girls and the rest are boys.
2.1
How many spectators are:
2.1.1 parents 2.1.2 girls?
SA M
uses to buy airtime
John drives a distance of 28 km from his farm to the nearest town.
PL
Calculate the amount she: a)
1.
E
• • • •
R300 Amount used to buy airtime = __25 × ____ 1
= 2 × R60 = R120
b)
3 Amount used to buy food = __ × R300 10
c)
3 1 The fraction she spent = __25 + __ + __ 10 10 2 × 2 __ = ____ + 3 + __ 1 5 × 2 10 10 4 + __ = __ 3 + __ 1 = __ 8 = __ 4 10 10 10 10 5 The fraction she saves = 1 − __ 54 = __ 55 − __ 45 = __ 15 The amount she saves = __ 15 × ____ R300 = R60 1
2.2
What fraction of the spectators are boys?
2.3
Calculate the number of boys.
3.
Mr Smith has R1 500. He gives one-third of the money to his daughter. He gives two-fifths of the remainder to his elder son. His other son receives the money that is left. What fraction of the money does the second son receive from his father?
4.
Mr Samuels is an angora goat farmer. He had 1 800 goats. One winter, onefifth of the goats died of the cold.
4.1
Calculate the number of goats he had left after some of them died of cold.
4.2
He sold four-ninths of the goats to his neighbour and three-tenths of the goats to his sister. How many goats did he sell to:
= 3 × R30 = R90
4.2.1 his neighbour 4.2.2 his sister?
©Optimi
13
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2 5.
A cross-country race takes place over a distance of 20 km. Julia completed __53 of the distance.
5.1
Calculate the distance Julia ran.
5.2
During this time, Joan has already completed three-quarters of the distance. Calculate how far she is ahead of Julia.
= 36 × 2 = 72oranges Check your answer: 48oranges + 72oranges = 120oranges. Correct.
E
Exercise 74: Divide a number of objects in a certain ratio 1.
Andrew and Johan want to buy a container of oranges. The total price of the container is R60 and the number of oranges in the container is 120.
a)
Andrew received
b)
Johan received.
Solution a)
Andrew contributed R24 of the R60.
SA M
Andrew contributes R24 and Johan R36. Each of them receives a number of oranges according to the amount of money each of them contributed. Calculate the number of oranges:
So, the fraction of the total price that he contributed = __ 36 . 60 Number of oranges that Johan received = __ 36 × 120 60
Problems involving division of objects in a certain ratio Worked example 13: Division of oranges according to each person’s contribution
Johan contributed R36 of the R60.
Mary, Anne and Jessica want to buy a box of sweets. The box contains 120 separate packets of sweets. The price for the whole box is R400. The three girls contribute the following amounts of money to buy the box:
PL
2.2
b)
Mary:
R120
Anne:
R180
Jessica: R100
1.1
Calculate the fraction of sweets each of them is entitled to. The fraction must be in its simplest form.
1.2
Calculate the number of packets each of them then gets.
2.
John, Peter and Jacques want to buy four packets of pens. Each packet contains 12 pens and each packet costs R120. They contribute as follows:
So, the fraction of the total price that he contributed = __ 24 . 60
John: R220
= 24 × 2 = 48oranges
Jacques: R140
Peter: R120
Number of oranges that Andrew received = __ 24 × 120 60
14
2.1
Which fraction of the total amount did each person contribute?
2.2
Calculate the number of pencils each person receives.
3.
To earn money, three friends, Helen, Mark and Leonard work as waiters at the same restaurant. They are paid per hour. The number of hours that each of them works is as follows:
Exercise 75: Solve problems involving addition and subtraction of fractions
Helen: 4 hours
1.
A survey shows that each learner in a class of 30 Grade 8 learners has either one pet, two pets or no pets at home. Two-fifths have one pet and one-third of the class have two pets.
1.1
How many learners in this class have one pet, and how many have two pets?
1.2
How many learners have no pets at home?
2.
Claudia scores four-fifths and Anna three-eighths of the total marks of a mathematics competition.
Mark: 3 hours Calculate the fraction of the total amount that each of them is entitled to. The fraction must be in its simplest form.
3.2
Calculate the total amount that the three friends earn if Leonard earns R400.
3.3
Calculate the amount that Helen and Mark are paid each.
2.3
Problems involving adding and subtracting fractions
SA M
PL
3.1
E
Leonard: 5 hours
Worked example 14: A problem involving adding and subtracting fractions
On a plot a house uses one-third of the area, two-fifths of the plot is covered with grass and one-sixth of the area is taken up by a pool. What fraction of the area of the plot is left to plant flowers and shrubs?
Solution Total fraction of plot used = __ 13 + __ 25 + __ 16
1 × 10 ____ = _____ + 2 × 6 + ____ 1 × 5 3 × 10 5 × 6 6 × 5
The LCM of 3, 5 and 6 is 30
Calculate the fraction of the total marks that Claudia scores more than Anna.
2.2
Calculate the total marks of the competition if Claudia scores 34 more marks than Anna.
2.3
Calculate the marks scored by:
2.3.1 Claudia 2.3.2 Anna. 3.
In an orchard, __41 of the trees are peach trees, __25 are apricot trees and __18 are fig trees. The remaining 63 trees are orange trees. Calculate the number of trees in the orchard.
2.4
Solving problems by dividing or multiplying fractions Worked example 15: Solving a problem using division
10 __ = __ + 12 + __ 5 30 30 30 27 __ __ = =9 30 10
Calculate the number of pieces of rope that can be cut from a string of rope with a 35 m. length of 120 mif the length of each piece is 1 __
10 __ 1 − 9 = __ The fraction of the plot that can be used to plant flowers and shrubs = __ 10 10 10
©Optimi
2.1
15
G08 – Mathematics
Theme 13: Common fractions
G08 – Mathematics – Study Guide 2/2
Solution Number of pieces = 120 ÷ 1 _35_ = 120 ÷ _ 85_ = 120 × _ 58_
Exercise 76: Solving problems involving multiplication or division of fractions Convert the mixed number to an improper fraction
1.
Joey cuts a pizza into five slices. She divides each of the five pieces into four smaller slices. What fraction of the whole pizza is each of the smaller slices?
2.
Matt has __58 of a cake. He eats __35 of this. Calculate the fraction of the whole cake that he ate.
3.
Susan fills smaller containers with rice. She has 15 kg of rice. Calculate the number of smaller containers she can fill if each container can take __34 kg of rice.
Multiply by the reciprocal of the fraction
= 15 × 5 = 75
E
Worked example 16: Solving a problem using multiplication town. Then he spends three times as many hours shopping in town. Calculate how long he spends shopping.
b)
At what time will he reach his farm if he arrived in town at 12:15?
Solutions a)
SA M
a)
The fuel tank of a lawn mower contains 11 litres of fuel. Each time Thomas mows his lawn, he uses 1__38 litres of fuel. Calculate the number of times Thomas can mow his lawn before the fuel tank is empty.
5.
Five friends share a cake that is cut into 20 pieces. They eat approximately 2_52_ pieces each. Calculate the fraction of the left-over cake.
6.
Jimmy is painting a wall with an area of 6 __34 m2. He has five cans of paint. Each can contains 440 ml of paint. A can of paint can cover 1 __38 m 2 on the wall. Calculate the volume of paint that will be left in ml.
3.
PERCENTAGES
Time he spends shopping = 3 × 2 __14 hours = __ 31 × __ 94 hours
3×9 = ____ hours 4 27 = __ hours 4 = 6 __34 hours
b)
4.
PL
Mr Smith travels from his farm to the nearest town. He takes 2 __14 hours to reach
Convert the mixed number to an improper fraction Multiply the numerators
Change the improper fraction to a mixed number
Total time away from the farm = 6 __43 + 2 __14 hours = 9 hours
Time he will arrive at his farm = 12 : 15 + 9 = 21 : 15 16
21 . Percentages are fractions with 100 as numerator. For example, 21% = ___ 100