solutions for all
Maths Literacy
Grade 10
Learner’s Book
Schools Development Unit
solutions for all Maths Literacy Grade 10 Learner’s Book
© Schools Development Unit, 2011
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First published 2011
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Published by
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Cover design by: Deevine Design
Cover image from: AAI Fotostock
Illustrations by: Ian Greenop
The publishers have made every effort to trace the copyright holders.
,IWKH\KDYHLQDGYHUWHQWO\RYHUORRNHGDQ\WKH\ZLOOEHSOHDVHGWRPDNHWKHQHFHVVDU\DUUDQJHPHQWVDWWKH¿UVWRSSRUWXQLW\
ISBN: 9781431006373
WIP: 3046M000
It is illegal to photocopy any page of this book
without written permission from the publishers.
e-ISBN: 978-1-4310-1740-9
CONTENTS
Chapter 1 Number skills 1: Numbers and operations ................................... 1
Lesson 1 Numbers all around us ........................................................................ 2
Lesson 2 Revising how to use calculators .......................................................... 4
Lesson 3 Calculations with fractions ................................................................ 10
Lesson 4 Calculations with decimals ................................................................ 13
Lesson 5 Percentages ...................................................................................... 18
Chapter 2 Number skills 2: Negative numbers, ratio and proportion ........ 24
Lesson 1 Thinking about negative numbers ..................................................... 25
Lesson 2 Ratios ................................................................................................ 28
Lesson 3 More about ratio ................................................................................ 35
Lesson 4 Proportion ......................................................................................... 37
Lesson 5 Rate .................................................................................................. 41
Chapter 3 Number skills 3: Extending your number skills ......................... 50
Lesson 1 Rounding off ...................................................................................... 51
Lesson 2 More calculator skills ......................................................................... 56
Lesson 3 Working with exponents and roots .................................................... 63
&KDSWHU.HHSLQJ¿WDQGKHDOWK\ ................................................................ 69
/HVVRQ.HHSLQJ¿W ......................................................................................... 70
Lesson 2 Hiking ................................................................................................ 72
Lesson 3 Planning meals for the hike ............................................................... 74
Lesson 4 Healthy food choices ......................................................................... 75
Lesson 5 Meal planning .................................................................................... 79
Chapter 5 Introduction to graphs .................................................................. 81
Lesson 1 Making sense of graphs that tell a story ........................................... 82
Lesson 2 Straight line graphs ........................................................................... 88
/HVVRQ*UDSKVUHSUHVHQWLQJD¿[HGUHODWLRQVKLS .......................................... 95
Chapter 6 Graphs, tables and equations .................................................... 100
Lesson 1 Graphs and tables ........................................................................... 101
Lesson 2 Thinking about equations ................................................................ 103
Lesson 3 Graphs of inverse proportion .......................................................... 107
Chapter 7 Measurement: conversions and time ........................................ 115
Lesson 1 Basic conversions within the metric system .....................................116
Lesson 2 Telling time ...................................................................................... 121
Lesson 3 Time management .......................................................................... 125
Chapter 8 Financial documents in the home ............................................. 132
Lesson 1 Till slips and Value-Added Tax ........................................................ 133
Lesson 2 Household accounts ....................................................................... 136
Lesson 3 Understanding banks and banking ................................................. 143
Lesson 4 A closer look at a municipal account ............................................... 147
Chapter 9 Measuring length, weight, volume and temperature ............... 155
Lesson 1 Units of measurement ..................................................................... 156
Lesson 2 Different ways of measuring length ................................................. 161
Lesson 3 Measuring weight ............................................................................ 167
Lesson 4 Measuring volume ........................................................................... 169
Lesson 5 Measuring temperature ................................................................... 171
Chapter 10 Scale and mapwork .................................................................. 174
Lesson 1 Understanding scale ....................................................................... 175
Lesson 2 Scale and plans .............................................................................. 179
Lesson 3 Bar scales ....................................................................................... 182
Lesson 4 Directions (left, right, along, up, down, straight) .............................. 185
Chapter 11 Probability .................................................................................. 191
Lesson 1 A scenario for exploring probability ................................................. 192
Lesson 2 The language of chance ................................................................. 192
Lesson 3 Understanding the weather report .................................................. 196
Lesson 4 A practical experiment for predicting the probability of an event ..... 199
Lesson 5 Using two-way tables to work out possible outcomes ..................... 203
Lesson 6 Using tree diagrams to list possible outcomes................................. 206
Chapter 12 Income, expenditure and budgeting ....................................... 214
Lesson 1 Different sources of income ............................................................ 215
Lesson 2 Budgeting ........................................................................................ 220
Lesson 3 More budgeting ............................................................................... 225
Lesson 4 Cutting down to save money ........................................................... 227
Lesson 5 Young entrepreneurs ....................................................................... 228
Chapter 13 Being a wise consumer ............................................................ 232
Lesson 1 Are economy packs always economic? .......................................... 233
Lesson 2 Some more supermarket decisions ................................................ 235
Lesson 3 Tempting the customer .................................................................... 237
Chapter 14 Calculating perimeter and area ............................................... 240
Lesson 1 Measuring perimeter ....................................................................... 241
Lesson 2 Calculating perimeter ...................................................................... 242
Lesson 3 More conversions ............................................................................ 246
Lesson 4 Measuring and calculating area ...................................................... 248
Chapter 15 Models and plans ...................................................................... 256
Lesson 1 Exploring packing space ................................................................. 257
Lesson 2 Boxes on a supermarket shelf ........................................................ 259
Lesson 3 Packing round objects ..................................................................... 263
Lesson 4 Following instructions to build a model ........................................... 267
Lesson 5 Understanding building plans ......................................................... 270
Chapter 16 Interest and banking ................................................................. 277
Lesson 1 Understanding interest .................................................................... 278
Lesson 2 Bank accounts to manage your money .......................................... 281
Lesson 3 Case study – Sam’s bank accounts ................................................ 288
Lesson 4 Financial trouble .............................................................................. 294
/HVVRQ&DVHVWXG\±3HWHU¶V¿QDQFHV ........................................................ 300
Chapter 17 Understanding the research process ..................................... 305
Lesson 1 Posing a research question ............................................................ 306
Lesson 2 Who should we collect the data from? ............................................ 309
Lesson 3 Designing the questionnaire ........................................................... 313
Lesson 4 Recording the responses ................................................................ 316
Lesson 5 Interpreting and reporting on results ............................................... 319
Lesson 6 Collecting data based on an observation ........................................ 321
Lesson 7 Drawing conclusions ....................................................................... 322
Chapter 18 Working with data ..................................................................... 329
Lesson 1 Classifying, summarising and representing data ............................ 330
Lesson 2 Classifying collected data ............................................................... 332
Lesson 3 Summarising data ........................................................................... 337
Lesson 4 Representing data ........................................................................... 341
Lesson 5 Reading data from a broken line graph .......................................... 349
Lesson 6 Analysing a pie chart ....................................................................... 353
Lesson 7 Analysing a table and bar graph ..................................................... 356
Lesson 8 Analysing a frequency table and histogram ..................................... 360
Chapter 19 Looking at data critically .......................................................... 370
Lesson 1 Looking at data critically .................................................................. 371
Lesson 2 Changing the scale of an axis ......................................................... 374
Lesson 3 Understanding the data handling cycle ........................................... 378
Chapter 20 Assignments and investigations ............................................. 385
C
1
er
p
ha t
NUMBER SKILLS 1:
Numbers and operations
This chapter will equip you to use basic mathematical skills to solve real-life
problems. As you work through the activities you will revise the mathematics
you have learnt in previous years.
What you will learn in this chapter
You will:
U Ê work with different number formats, including word formats
U Ê interpret, understand and use different numbering conventions in contexts
and recognise that although these representations look like numbers, some
of them cannot be operated on in the same way
U Ê perform calculations for numbers expressed as whole numbers, fractions,
decimals and percentages
U Ê use the calculator to perform basic calculations.
Talk about
Numbers have different functions in different places. Notice that in this
photograph the numbers on the podium show the positions that the athletes
achieved.
Solutions for all
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1
Lesson 1: Numbers all around us
Numbers are all around us. Look at the examples in the picture below.
Sometimes they are labels.
Sometimes they indicate
position. If a building has
Room 1012, it could mean
the 12th room on the 10th
floor.
Most times numbers indicate value.
The numbers alongside are all made up of the digits 0 and 1.
What is the difference between these numbers?
The numbers 1 and 1,0 and 1,000 all have the same value.
Sometimes 1,0 is written as 1.0.
1
The number 0,1 equals 10 , which is ten times smaller than
one. We say ‘nought comma one’ or ‘zero comma one’.
10 is ten times bigger than one.
1 000 is one thousand times bigger than one.
1:00 represents the time one o’clock. We could also write 01:00 or 01h00.
In South African schools, we use a decimal comma. We use a space to
separate thousands from hundreds to make numbers easier to read.
However, you should also get used to seeing numbers written in other
ways, for example, a decimal point and a comma used as a thousands
separator. Different conventions are used in different countries and in
different situations.
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Solutions for all
Chapter 1
Classwork activity 1.1
1
Use the digits 5 and 0 to write each of these without using words.
a)
c)
e)
g)
i)
2
b)
d)
f)
h)
j)
five o’clock
fifty thousand
five squared
fifty-five thousand
five rand
In each of the following, decide whether the number represents a
name or a value.
a)
b)
c)
d)
3
five thousand
one fifth
five million
minus five degrees
five thousand and fifty
She lives at number 12.
He is Mrs Smith’s 4th child.
There are 80 broken desks.
Bus number 7 is going to Nyanga.
a) Tumisho has a doctor’s appointment at a medical centre. The
doctor is in Room 307. Tumisho takes the lift to the third floor.
How did he know the doctor was on the third floor?
b) The doctor sends him to Room 514 for an X-ray. Tumisho takes
the stairs. Does he go upstairs or downstairs? Explain why you
say so.
Homework exercise 1.1
Write these using number symbols. The digit 8 will be in each one.
a) half past eight
c) three eighths
e) eight hundred and twentyfour rand and eight cents
g) eight degrees below zero
2
b) twenty to nine
d) eighty thousand and eight
f) eight ninths
h) He came eighth in his race.
In each of the following, decide whether the number represents a
name or a value.
a) She owes me R92.
b) He earns R16 000 a month.
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Solutions for all
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3
c) The doctor added 0,2 ml of the antibiotic to the drip.
d) The milk is in aisle 3.
3
Jenna lives in Apartment 312 in an apartment block. Her friend
lives in the apartment right above hers. What is the number of her
friend’s apartment?
Lesson 2: Revising how to use
calculators
Different types of calculators have different keys and different ways of
operating. You need to get to know how your calculator works. Compare
your calculator to the one in the photograph below. Note any differences. Are
there keys that your calculator does not have?
Before you do a calculation on your calculator, it is good practice to estimate
the answer, even if you just do this mentally. This will help you to ensure that
the answer you get on the calculator is a sensible one.
If you press an incorrect key or leave off a zero, you will then notice the
difference between your estimated answer and the calculated answer
immediately.
Calculators are
not for dummies!
4
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Solutions for all
Chapter 1
We can easily make an error when adding up a long list of numbers, as it
often happens that we type in a number incorrectly. There are a few
strategies and good habits to get into so that you pick up errors quickly.
U Ê Do the whole calculation a second time. If your second answer is
different to the first, you have made a mistake. Do it again to see which
one is correct. However, if the answer is the same, you have probably
not made an error. You should get into the habit of doing this whenever
you use a calculator.
U Ê You could also break up the calculation into parts and write down the
answer for each part.
When you are doing a calculation that involves a few steps, try not to round
off any of the numbers until the final answer. This will make the final answer
more accurate.
EXAMPLE 1
19 853 – 692,38
SOLUTION
First estimate:
20 000 – 700 = 19 300
Estimate your answer.
That way you can check
that the answer on the
calculator makes sense.
Now use the calculator:
19 853 – 692,38 = 19 160,62
Check this calculation on your own calculator.
Solutions for all
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5
The term BODMAS is sometimes used to help people remember the order of
operations.
B: First do calculations in brackets.
O: of
D:
M:
Divide and Multiply, working from left to right.
A:
S:
Add and Subtract, working from left to right.
For complex calculations you could do the calculations as a series of steps.
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Solutions for all
Chapter 1
EXAMPLE 2
4(37,2) + (3 × 22) – (58 ÷ 5) + 14
4(37,2) = 4 × 37,2
= 148,8 + 66 – 11,6 + 14
= 217,2
Check this calculation on your own calculator.
Classwork activity 1.2
Estimate each of the following. The first one has been done for you.
a)
b)
c)
d)
e)
2
3 × 212
79 × 187
59 + 26(3)
58 ÷ 5 – 62 × 2
Do the following calculations on your calculator.
a)
b)
c)
d)
e)
3
4 × 37,2 = 4 × 40 = 160
55,8965 + 12,443 × 3
Write your
answer to 1
decimal place.
(6 × 3) – 4 + 26 ÷ 2
44,65 ÷ 3 × 11,11 + 68
24 – 16 + 3 + 5 – 6 + 10 – 2
15 ÷ (2 + 2)
Jabu works at a shoe factory. He has to check the stock in the
warehouse. As he counts the pairs of shoes, he makes the following
notes:
Item code
Price per item in rand
435A
259,99
Number of items
41
435B
319,99
209
277A
199,99
316
277B
249,50
17
277C
99,99
850
569A
79,99
34
569B
49,99
75
233X
169,99
177
184D
185,99
289
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Solutions for all
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a) How many pairs of shoes are there in the warehouse?
b) Calculate the value of the stock. Show all your calculations.
c) What is the average price of one pair of shoes? Show your
calculations.
4
The following table gives details of the number of children at a
children’s home, as well as their sock size.
Size
Total number of children
3–4 years
9
5–6 years
8
7–8 years
5
A factory donates clothing to the home. The items are listed in the
following table:
Item
Size/Age
Total
Socks (pairs)
3–4 years
45
5–6 years
45
7–8 years
30
Sun hats (girls)
All ages
60
Peak caps (boys)
All ages
60
Do not use your calculator for these problems.
a) Five children wear socks of size 7–8 years. How many pairs will
each child get?
b) Nine children wear socks of size 3–4 years. How many pairs will
each child get?
c) There are eight children who wear socks of size 5–6 years.
Suggest a fair way to share the socks.
d) Of the total number of children, 12 are girls. How many are
boys? Show your calculations.
e) Each girl is given one sun hat and each boy is given one peak
cap. How many sun hats are left over? Set your answer out
neatly, starting with a suitable equation.
f) How many peak caps are left over? Show your calculations.
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Solutions for all
Chapter 1
Homework exercise 1.2
1
Estimate each of the following.
a) 91 + 47 + 11
c) 147 ÷ 3
e) 1 003 + 41 × 2
2
Do the following calculations using your calculator.
a) 4 × (3 + 1) – 2 ÷ 2
3
b) 93 × 12
d) 997(28)
b) 16 ÷ 6 ÷ 4
c) 6 × 8 – 4
Jess works at a music store. She has to check the stock of CDs. She
completes a form:
Item Code
Price per item in rand
Number of items
R101
289,95
243
R103
196,50
167
C115
319,99
352
G120
99,99
287
X117
129,99
117
A115
145,50
26
J144
185,59
114
P183
209,90
204
K109
69,99
75
Disc255
29,99
1 548
Disc162
59,99
743
a) How many CDs are there in the store?
b) Calculate the value of the stock. Show all your calculations.
c) What is the average price of one CD? Show your calculations.
4
There are 42 children at a holiday camp. They are going to have
pizza for supper. There are 8 slices in one pizza. How many pizzas
must the chef make for each child to have one slice? Do not use a
calculator to answer the question.
You will learn more functions on the calculator in Chapter 3.
Solutions for all
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Lesson 3: Calculations with
fractions
Few people like doing calculations that involve fractions, but without
fractions we would not be able to talk about parts of an object.
The denominator (bottom number) tells us how many parts the whole
is divided into. The numerator (top number) tells us how many of the
parts we have.
8 parts out
8 @ We have 8 parts
of 8 is one
whole.
8 @ The whole is divided into 8 parts.
3 parts out
of 8 is less
than half.
3 @ We have 3 parts
8 @ The whole is divided into 8 parts
Fractions can be written as:
One number (numerator) over
another number (denominator):
numerator
denominator
3
OR as a decimal fraction: = 0,375
8
OR as a percentage:
This means that
3
= 37,5%
8
3
= 0,375 = 37,5%. Each represents 3 parts out of 8.
8
Let’s revise some strategies and methods for working with fractions.
10
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Solutions for all
Chapter 1
EXAMPLE 1
Write the mixed numbers as improper fractions.
a) 2
4
5
b) 6
3
4
SOLUTION
a) 2
4
means 2 wholes and 4 parts out of 5, of another whole.
5
We can write this as
5 5 4
5 5 5
5 = 1 whole
5
b)
EXAMPLE 2
When we add or subtract fractions, we should always first write them so that
the denominators are the same. To do this, find the lowest common
denominator (LCD) and use it as the new denominator.
a)
7 9
5 15
7 3
9
= 5 3 15
21 9
15 15 (LCD = 15)
30
15
2
5 3
1
7 5
19 8
7 5
19 5
=
7 5
95 56
35 35
39
4
1
35
35
b) 2
8 7
=
5 7
(LCD = 35)
Solutions for all
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11
EXAMPLE 3
When we multiply fractions, we multiply the numerators and multiply the
denominators. We could simplify fractions by dividing the numerator and
denominator by the same factor.
4
1 19 7
a) 3 = 2 =
5
3 5 3
19 = 7
5 =3
133
13
8
15
15
4
19 6
b) 3 = 6 =
5
5 1
19 = 6
5 =1
114
4
22
5
5
2
4
19 10
c) 3 =10 1 =
5
5
1
19 = 2
38
1 =1
EXAMPLE 4
When we divide fractions, we invert the second fraction and then multiply
the fractions.
a)
5 1
5 4 1
8 2=
8 4
8
1
5
2
1
2
2
1
2
12
U
Rewrite as improper fractions.
a) 7 7
8
c) 13 1
3
b) 10 3
4
d) 3 2
5
Calculate.
a) 3 7 2 3
8
8
c) 4 1 3 1
5
4
e) 4 = 3
5 4
g) 9 8 2
5 3
b) 8 1 6
2
d) 9 2 2 1
3
2
f) 8 = 5
9
h) 2 8 3
3 2
Solutions for all
Ð
Classwork activity 1.3
Chapter 1
3
Sonny delivers the newspaper to people in his area. He gets 1 of
5
the sales as commission.
a) Sonny sells 122 newspapers at R2,45 each. What was the value of
his sales?
b) Calculate Sonny’s commission. Show all your calculations.
Homework exercise 1.3
1
2
Rewrite as improper fractions.
a) 2 2
5
b) 6 1
3
c) 5 3
8
d) 4 1
2
Simplify.
a) 2 2 1 1
5 5
c)
3
2=1
9 6
b) 3 1 2 1
4
2
d) 3 8 1
8 9
Mrs Sonke runs a laundry service. She charges R75 for a load of
washing. Her costs take up 3 of the money.
4
a) How much does Mrs Sonke charge for 5 loads of washing? Show
your calculations.
b) How much profit does she make? Show your calculations.
Lesson 4: Calculations with
decimals
Adding and subtracting decimals
When we add or subtract decimals:
U Ê it is useful to write the numbers underneath one another so that the
decimal commas are underneath one another
U Ê in your answer the number of digits after the decimal comma will be the
same as the number of digits after the decimal comma in the numbers you
are calculating.
Solutions for all
U
13
EXAMPLE 1
12,362
a)
+
68,18
b)
7,149
–
19,511
33,464
34,716
Multiplying decimals
When we multiply decimals, the answer has the same number of digits after
the comma as the sum of the number of digits after the comma in the two
terms.
Let’s understand how this rule works.
EXAMPLE 2
We know that 12 × 2 = 24
a)
1,2 × 2 = 2,4
b)
1,2 × 0,2 = 0,24
c)
1,2 × 0,02 = 0,024
Each term has one digit after the
comma. The answer has 2 digits
after the comma.
EXAMPLE 3
When we multiply by a power of 10 (e.g. 10, 100 or 1 000) count the number of
zeros and move the decimal comma the same number of places to the right.
EXAMPLE 4
When we divide by a power of 10 (e.g. 10, 100 or 1 000) count the number of
zeros and move the decimal comma the same number of places to the left.
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Solutions for all
Chapter 1
Classwork activity 1.4
Do these calculations without using your calculator.
a)
c)
e)
g)
i)
k)
m)
2
37,5 + 23,4
26,4 – 25,1
2,3 × 0,2
3,4 × 0,04
24,3 × 100
375,86 ÷ 1 000
b)
d)
f)
h)
j)
l)
Hint:
40,5 = 40,50
40,5 + 60,35
72,9 – 65,6
1,2 × 2,0
24,3 × 10
5,298 × 100
464,57 ÷ 100
62,5 ÷ 1 000
For each of the following, your answer will have two steps. See the
example that follows.
a) Estimate the answer. Show your calculations.
b) Use your calculator to find an accurate answer.
Now do these.
i) (34,82 + 166,75) – (36,9 × 2,5)
ii) (51,36 + 48,4) – 25,2 × 3,6
iii) 51,36 + 48,4 – 25,2 × 3,6
iv) (28,23 – 6,4) – (1,1 × 11,2)
Ð
1
EXAMPLE 5
Calculate 58 × 5
a)
60 × 5 = 300
b)
58 × 5 = 290
Solutions for all
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15
Mr Potelwa has five boxes to deliver. Two boxes weigh 33,8 kg
each, one weighs 57,5 kg and two weigh 15,7 kg each.
3
a) What is the total weight of the boxes?
b) What is their average weight?
Do questions 4 to 10 without using your calculator.
4
a) 0,032 × 10
d) 3 275 × 10
b) 0,032 × 100
e) 3 275 × 100
c) 0,032 × 1 000
f) 3 275 × 1 000
5
a) 46,5 ÷ 10
d) 9 982,4 ÷ 10
b) 46,5 ÷ 100
e) 9 982,4 ÷ 100
c) 46,5 ÷ 1 000
f) 9 982,4 ÷ 1 000
6
a) 46,5 ÷ 10
d) 9 982,4 ÷ 10
b) 46,5 ÷ 100
e) 9 982,4 ÷ 100
c) 46,5 ÷ 1 000
f) 9 982,4 ÷ 1 000
7
a) 45,5 × 10
d) 45,5 × 20
b) 45,5 × 100
e) 45,5 × 200
c) 45,5 × 1 000
f) 45,5 × 2 000
8
a) 23,07 × 10
d) 23,07 × 40
b) 23,07 × 100
e) 23,07 × 400
c) 23,07 × 1 000
f) 23,07 × 4 000
9
a) 627,3 ÷ 10
d) 627,3 ÷ 30
b) 627,3 ÷ 100
e) 627,3 ÷ 300
c) 627,3 ÷ 1 000
f) 627,3 ÷ 3 000
10
a) 5 980 ÷ 10
d) 5 980 ÷ 50
b) 5 980 ÷ 100
e) 5 980 ÷ 500
c) 5 980 ÷ 1 000
f) 5 980 ÷ 5 000
Homework exercise 1.4
1
Do these calculations without using your calculator.
a)
c)
e)
g)
2
37,5 + 23,4
2,3 × 3
188,2 × 100
211,15 ÷ 10
b)
d)
f)
h)
84,4 – 61,6
2,5 × 0,4
210,25 × 1 000
31 364 ÷ 100
For each of the following, your answer will have two steps.
a) Estimate the answer. Show your calculations.
b) Use your calculator to find an accurate answer.
i)
(64,4 × 3) – (125 – 25,4)
ii) (64,4 × 3) – 125 – 25,4
iii) 64,4 × 3 – (125 + 25,4)
iv) (64,4 × 3) – 125 + 25,4
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Solutions for all
Chapter 1
3
Nurse Mary looks after 6 new-born babies in the high care unit at
the hospital. For each 200 g of weight a baby gets 3 ml of a special
nutrient formula.
For a baby that weighs 900 g, Nurse Mary does the following
calculation:
900 g ÷ 200 g = 4,5
4,5 × 3 ml = 13,5 ml
She will give the baby 13,5 ml of the formula.
a) A baby weighs 840 g. How much formula must she get? Show
your calculations.
b) Another baby weighs 965 g. How much formula must he get?
c) The total weight of the six babies is 5 412 g. How much formula
does Nurse Mary use?
d) If the formula comes in 20 ml tubes, how many whole tubes of
formula should Nurse Mary order?
Solutions for all
U
17
Lesson 5: Percentages
Percentage means part of a hundred. Percentages, fractions and decimals are
equivalent representations of the same value.
20 parts of 100 can be expressed as:
A fraction:
20
100
0,2
A decimal:
A percentage: 20%
Calculations with percentage
To convert a decimal to a percentage, multiply by 100.
EXAMPLE 1
1
0,75 = 75%
2
0,345 = 34,5%
To convert a fraction to a percentage, multiply by 100.
EXAMPLE 2
1
18
2
5 100 500
62,5%
=
8 1
8
3
4 100 400
=
44, 4%
9 1
9
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Solutions for all
Chapter 1
EXAMPLE 3
For each example, the calculation without the calculator is shown, followed by
the key sequence for the calculator.
1
Calculate 15% of
120
Without a calculator
15 120 1800
=
18
100 1
100
Using the calculator
2
If R20 is 6% of a
total, how much
is 100%?
Without a calculator
6% = R20
‘1% = 20 =R3,3333
6
‘100% = R333,33
Using the calculator
3
Increase 135 kg
by 30%
Without a calculator
100% 135 kg
‘ 130% = 130 = 135
100 1
17 550
=
100
= 175,5 kg
Using the calculator
4
Decrease 135 kg
by 30%
Without a calculator
100% = 135 kg
70 135
=
100 1
= 94,5 kg
70% =
Using the calculator
Solutions for all
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5
Without a calculator
An amount of
money was
increased by
20%. The result
was R135,00.
How much
money did the
person have
before the
increase?
120% = 135
9
135
8
120
9 100
100% = =
8 1
= R112,50
‘ 1% =
Using the calculator
Classwork activity 1.5
1
Write as a percentage.
a) 0,43
1
e) 8
2
b)
0,538
c)
f)
5
7
g)
3
16
b) 90% of 550
d) 11% of 135
Find 100% if:
a) 22 is 4%
4
d)
Calculate.
a) 16% of 1 000
c) 28% of 250
3
2
25
42
30
b) 1 705 is 55%
c) 311,5 is 89%.
a) Increase 590 by 20%
b) Decrease 358 by 10%
c) Increase R12 350 by 15%
5
Martin improves his Mathematics mark by 23%. If his original mark
was 13, what is his new mark? Show your calculations.
6
Mary loses 7% of her weight. If she weighed 72 kg, how much weight
did she lose? What does she weigh now?
7
R150 is shared among 9 people.
a) How much would each person get?
b) If 4 people have collected their share, what percentage of the R150
remains?
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Solutions for all
Chapter 1
Homework exercise 1.5
1
Write as a percentage.
a) 0,56
2
b) 36% of 135
c) 80% of 440
b) 180 is 35%
c) 215,5 is 98%
Find 100% if:
a) 28 is 10%
4
c)
Calculate.
a) 10% of 1 200
3
5
6
b) 0,636
a) Increase 545 by 10%.
b) Decrease 320 by 25%.
c) Increase R23,40 by 6,5%.
5
Martin scores 18% more for a Life Sciences test. If his original mark was 34, what is his
new mark? Show your calculations.
6
Mary is pregnant. Her weight has increased by 32%. If she weighs 76 kg now, what
was her weight before the pregnancy?
Summary practice exercise
1
Choose the number that is the odd one out. Give a reason for your
answer.
a) 2,0; 2,00; 0,2; 63
21 1
b) 0,7; 70%; 30 ; 7
Use the number 7 to write each of the following without words.
a) seven hundredths
b) seven o’clock
c) seven tenths
d) one seventh
3
The Department of Labour is in a big building. You have to go to
Room 820. Which floor must you go to? Explain your answer.
4
You buy a chicken burger at Burger Delite. You place your order
and pay, then you join the other people waiting for their orders.
The till operator gives you your till slip and says ‘You are number
97.' Does 97 represent a value or a label? Explain your answer.
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Solutions for all
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5
6
7
In the game of cricket there are six balls in an over. So 5.3 overs
means that five full overs and 3 balls of the sixth over have been
bowled. (Note that this is different to the decimal system.)
a) If 5.3 overs have been played, how many balls have been
bowled?
b) If there are 50 overs, how many overs must still be played?
Do the following without using your calculator.
a) 25 × 4
b) 100 ÷ 5
c) 22 × 3
d) 238 × 100
e) 25% of 200
f) 50% of 850
g) 548,9 × 100
h) 8 590,11 ÷ 10
For each of the following, your answer will have two steps.
a) Estimate the answer. Show your calculations.
b) Use your calculator to find an accurate answer.
i) 491 ÷ 5
ii) (18 × 4,7) – (38,6 ÷ 7,7)
iii) 299 × 11,7
iv) 783 + 192 – 189
v) 613 × 5 – 416 – 148
8
a) Jumani wrote a Life Sciences test out of 85. He got 13%. How
many marks did he score?
b) Pumza scored 18 in a Mathematical Literacy test. Calculate her
50
percentage.
c) A dress is marked down on sale by 15%. If the dress originally
cost R450.00, what is the new price?
d) A music shop has a special offer. DVDs that normally cost
R236,00 are selling at one third of the price.
i) What do the DVDs cost during the special offer?
ii) How much cheaper are the DVDs during the special offer?
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Solutions for all
Chapter 1
Word bank
antibiotic:
apartment block:
commission:
invert:
medicine that cures infection and illness caused
by bacteria
block of flats
when a sales person sells an item such as a car or
house and receives a percentage of the money
when we invert a fraction, we turn it upsidedown, e.g.:
4
is inverted to become 5
5
4
label:
LCD:
sum:
product
something that identifies or names a person or
thing
Lowest Common Denominator
the result of an addition
the result of a multiplication
Chapter summary
UÊ
UÊ
Numbers can be labels or values.
UÊ
When we put a number into a calculator, we do not leave any space:
19853
UÊ
When we write numbers, we use a decimal comma to indicate
fractions: 22,79
UÊ
When we put the number into a calculator, we use a decimal point to
indicate fractions: 22.79
UÊ
When we multiply by 10, 100 or 1 000, we move the decimal comma to
the right, and when we divide by 10, 100 or 1 000, we move the decimal
comma to the left.
UÊ
When doing calculations, it is important to follow the correct order of
operations: BODMAS
When we write numbers, we leave a space to indicate thousands:
19 853
Solutions for all
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C
2
er
p
ha t
NUMBER SKILLS 2:
Negative numbers,
ratio and proportion
What you will learn in this chapter
You will:
U Ê work with positive and negative numbers as directional indicators
U Ê perform operations with negative numbers
U Ê understand basic ratio concepts and ratio calculations
U Ê convert between different forms of ratios
U Ê determine missing numbers in a ratio
U Ê divide or share an amount in a given ratio
U Ê develop equivalent ratios
U Ê write ratios in simplest form
U Ê write ratios in unit form
U Ê perform calculations involving direct and inverse proportion
U Ê work with various types of rates including cost rates, consumption rates
and exchange rates.
Talk about
When you buy paint for a project, especially when you buy a mix of colours, it
is important to buy more than enough of the paint. It is very difficult to get the
exact shade again later, even though there are special machines that mix paint in
the required ratios. This is an example of the use of ratios.
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Solutions for all
Chapter 2
Lesson 1: Thinking about
negative numbers
Some history
People first used negative numbers to represent debt. If you had no money it
was written as zero. If you owed an amount of money it was written as a
negative amount.
To owe money is worse than having
no money because zero has greater
value than a negative number.
Later negative numbers were used to indicate a downward trend.
The lower the temperature, the smaller
the number. A negative number is smaller
than zero, so –2 °C is colder than 0 °C.
Yes, and the level of the sea is regarded as 0 m.
The height of a mountain is written as a positive
number because a mountain is higher than the
sea, but the depth of the ocean can be written as a
negative number because it is lower than the sea.
Which is bigger: –2 or –8?
The answer is –2. The number 8 is bigger than 2, but –8 is smaller than –2
because –8 is further from 0 than –2.
Solutions for all
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Look at the thermometer. Which is colder: 1 °C or –3 °C?
–3 °C is colder because –3 is smaller than 1.
Who owes more money?
Mo’s bank balance is –R3 421,00 and Jabula’s bank balance is –R1 253,00. Who
owes more money?
Mo owes more money because –3 421 is further away from zero than –1 253.
This means that Mo has a bigger debt.
Classwork activity 2.1
1
Write these numbers in descending order (from the biggest to the
smallest).
a) 6; 15; –4; 1 428; –19; –2 311; –1 540; –9; 0; 53
E ï225ï416; 416ï1 000ï21; 22
c) 3 336; 647ï332ï10; 65; 78ï101; 5
2
Which is coldest:
a) – 8 °C; –12 °C or 0 °C?
E ï1ƒ&ï10 °C or 10 °C?
c) 23 °C; 16ƒ&RUï16 °C?
3
Which level is lower in each case?
a) –3 m or –7 m
b) 6PRUï3 m
F ï15PRUï25 m
4
According to these bank statements, who is richest?
a) Harry’s bank balance is R12,43.
b) Jerry’s bank balance is –R80,44.
c) Benny’s bank balance is –R8 453,94.
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Solutions for all
Chapter 2
Homework exercise 2.1
1
Write these numbers in ascending order (from the smallest to the
biggest).
D ï
121; 121; 0; 100ï10; 35
b) 0ƒ&ï10 °C; 25 °C; 15 °C; 13ƒ&ï3 °C
F ï564,43; R100,01; R25,52ï581,49
2
Which of these ocean depths is deeper in each case?
D ï456PRUï352P
3
E ï5PRUï15P
F ï21 m or 22 m
According to these bank statements, who is poorest?
a) Simon’s R4 562,32
b) Puoane’Vï54 562,32
c) Jannie’Vï545,62
4
Patricia owes R531,27 on her FashionWorths account (in other
ZRUGVKHUEDODQFHLVï5531,27). Her mom says that she will pay
R225 off her account as part of Patricia’s 21st birthday present.
a) What will Patricia’s new balance be?
b) If she then buys a skirt for R156, what is the balance on her
account?
c) Patricia is paid R100 at the end of every week for babysitting. She
uses this money to pay off her FashionWorths account. How
many weeks will it take her to pay off her account?
Solutions for all
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Lesson 2: Ratios
A ratio is a relationship between two or more quantities that allows you to
compare the quantities. This is a ratio: 20 : 8 : 1. We do not use units when we
write a ratio.
EXAMPLE 1
A baker prepares a basic dough mix of
flour, sugar and baking powder in the
ratio 20 : 8 : 1. This means that for every
20 parts of flour the baker needs 8 parts
of sugar and 1 part of baking powder. If
he made five times the basic dough
mixture, he would use 100 parts flour, 40
parts sugar and 5 parts baking powder.
(He would multiply each unit by 5.)
We do not use units, because the ratio
could apply to any units.
The baker could use a spoon to measure
the quantities: 20 spoons of flour, 8
spoons of sugar and one spoon of baking
powder.
The baker could use a measuring cup to
measure the quantities:
20 ml of flour, 8 ml of sugar and 1 ml of
baking powder.
The baker could use a scale to weigh the
quantities: 20 kg of flour, 8 kg of sugar
and 1 kg of baking powder.
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Solutions for all
Chapter 2
The baker could make bigger or smaller quantities of the mixture:
For 100 ml of flour he uses 40 ml of sugar and
5 ml of baking powder.
For 2 kg of flour he uses 0,8 kg (800 g) of sugar
and 0,1 kg (100 g) of baking powder.
Multiply each
quantity by 5.
Multiply each
quantity by 0,1.
From this we see that
20 : 8 : 1 = 100 : 40 : 5 = 2 : 0,8 : 0,1
We say they are equivalent ratios because the quantities are in the same ratio.
We can use equivalent ratios to determine missing quantities.
EXAMPLE 2
The baker makes a filling for the cake, using sugar, butter and coconut in the
ratio 5 : 3 : 2. If he uses 30 parts of sugar, how many parts of butter and
coconut should he use?
SOLUTION
The baker used six times the amount of sugar because 5 × 6 = 30.
5 : 3 : 2 = 30 : 18 : 12
Multiply each quantity by 6.
He used 18 parts butter and 12 parts coconut.
Interesting facts about ratios
A ratio is another way of writing a fraction: 6 : 8 is a ratio of 6 parts to 8 parts.
So there are 14 parts in total. We can write the ratio 6 : 8 as the fraction 6
14
(or 6 parts out of a total of 14 parts).
6 is the same as 3 .
14
7
And the ratio 6 : 14 is the
same as the ratio 3 : 7.
Solutions for all
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EXAMPLE 3
An artist mixes 3 parts blue paint with 5 parts yellow paint. Write this as a
ratio in two different ways.
SOLUTION
3
The ratio 3 : 5 has 8 parts in total, so it can be written as 8 . This is in in
simplest form.
A ratio is in its simplest form when there are no common factors between the
quantities.
EXAMPLE 4
Write the following ratio in its simplest form: 24 : 9 : 15
SOLUTION
3 is the highest common factor. The simplest form of the ratio is 8 : 3 : 5.
A ratio is in unit form when one of its terms is 1.
EXAMPLE 5
Write the following ratio in unit form: 24 : 8 : 15
SOLUTION
8 is the smallest term in the ratio, so we divide each term by 8 therefore
3 : 1 : 1,875
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Solutions for all
Chapter 2
EXAMPLE 6
a)
Measurement
The width and length of the
rectangle are in the ratio 6 : 8. In a
similar rectangle whose sides have
the same ratio, the width is 2 cm.
What is the length?
SOLUTION
The width of the original rectangle is
6 cm. The length must be three times
smaller so that the ratio is equal to 6 : 8.
You do not need to convert
metres to centimetres. The
ratios are equivalent and apply
to any units.
6:8
8
3
2
=2: 2
3
2 :
The length of the rectangle is 2 2 cm.
3
b)
Objects
The ratio of pink
marshmallows to white
marshmallows is 1: 3. This
means that for every one
pink marshmallow there are
3 white ones. If there are 10
pink marshmallows how
many white ones are there?
SOLUTION
1 : 3 = 10 : 30
There are 30 white marshmallows.
Multiply both
terms by 10.
Solutions for all
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c)
People
For every 4
female teachers
1
there are 1
2
male teachers.
The ratio of female teachers to male teachers in this diagram is 4 : 1,5.
If there were 900 female teachers how many male teachers would
there be?
SOLUTION
Multiply each
term by 225.
900 ÷ 4 = 225
‘4 : 1,5 = 900 : 337,5
You can’t have 337,5 teachers. In reality
there is no such thing as half a person.
There would be 337 or 338 male teachers.
Classwork activity 2.2
Write a ratio for each of the following in its simplest form.
a)
b)
c)
d)
2
There are 7 blue sweets for every 3 pink sweets in a packet.
There are 34 blue sweets and 18 red sweets in the bowl.
There are 85 boys and 155 girls in Grade 2 at Park Primary.
The cook used 6 onions and 27 tomatoes.
Complete each set of equivalent ratios.
a) 5 : 2 = 10 : ____
c) 3 : 8 : 7 = ____ : 24 : ____
e) 48 : 50 = 12 : ____
3
Write each of these ratios in unit form.
a) 5 : 2
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b) 5 : 2 = ____ : 16
d) 1 : 2 = ____ : 50
Solutions for all
b) 4 : 8 : 7
c)
12 : 15 : 21 d) 96 : 42 : 12
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Chapter 2
4
The ratio of red apples to green apples in a bag is 1 : 2. If there are
12 red apples,
a) how many green apples are there?
b) how many apples are there altogether in the bag?
5
6
7
There are 12 boys and 15 girls in an athletic team. Complete the
sentences.
a) There are _____ athletes in the team.
b) The ratio of boys to girls is 4 : ____.
c) The ratio of boys to the total number of athletes is 12 : ____.
At a hospital, the ratio of female nurses to male nurses is 5 : 2.
a) Write the ratio of male nurses to the total number of nurses.
b) What fraction of the nurses is male?
c) What fraction of the nurses is female?
d) There are 70 nurses at the hospital. How many are female?
Instructions on a bottle of plant fertiliser: Mix 20 ml to 1 Ɛ of water.
a) 1 000 ml = 1 Ɛ. What is the ratio of fertiliser to water?
Write your answer in the form 1 : ___.
b) How much fertiliser must be added to 5 Ɛ of water?
Homework exercise 2.2
Write a ratio for each of the following in its simplest form.
a)
b)
c)
d)
2
There are 17 black balls for every 15 white ones.
There are 51 black balls and 17 white ones in the basket.
There are 99 boys and 156 girls in the swimming club.
The decorator used 22 blue, 18 red and 10 yellow lights.
Complete each set of equivalent ratios.
a)
b)
c)
d)
e)
1 : 3 : 3 = 2 : ___ : ___
2 : 5 : 8 = ___ : 15 : ___
2 : 23 = 1 : ___
11 : 13 = ___ : 39
___ : 24 = 48 : 96
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Solutions for all
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3
Write each of these ratios in unit form.
a) 18 : 8 : 28
b) 24 : 54 : 120
c) 85 : 17 : 153
4
The ratio of red balloons to yellow balloons at a carnival is 2 : 7.
a) How many red balloons are there if there are 21 yellow ones?
b) At one stall the ratio of red : yellow is 8 : 12. If there are 60 red
balloons:
i) how many yellow balloons are there?
ii) how many balloons are there altogether?
5
There are 12 girls and 20 boys in the swimming team. Complete the
sentences.
a) There are _____ swimmers in the team.
b) The ratio of boys to girls is 5 :_____.
c) The ratio of boys to the total number of swimmers is 20 :_____.
6
At a supermarket, the ratio of till operators to shelf packers is 7 : 3.
a) Write the ratio of shelf packers to the total number of workers.
b) What fraction of the workers is operating the tills?
c) There are 90 workers. How many are till operators?
7
Instructions on a bottle of juice concentrate read: Mix 10 ml to 1
cup (250 ml) of water.
a) What is the ratio of juice concentrate to water?
b) Complete the sentence: Mix 1 part juice concentrate to ____ parts
water.
c) How much concentrate must be added to 1 Ɛ of water? Show
your calculations.
(1 Ɛ = 1 000 ml)
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Solutions for all
Chapter 2
Lesson 3: More about ratio
We are able to calculate the individual quantities in a ratio if we know the
total quantity of the parts.
EXAMPLE 1
Busi and Daniel mix sand and cement to make mortar for
bricklaying. The sand and cement are in the ratio 5 : 2.
Busi needs 840 cm3 of the mortar. How much sand and
cement does she need?
SOLUTION
Thinking about the question:
5 parts sand and 2 parts cement mean that there are 7
parts altogether. The ratio of sand to cement is 5 : 2.
5
This means the sand is 5 parts of 7 i.e. of the mixture and the cement is 2
7
2
parts of 7 i.e. of the mixture.
7
Sand =
5 840
=
600 cm3
7 1
Cement =
2 840
=
240 cm3
7 1
Classwork activity 2.3
1
An artist mixes blue and yellow paint in the ratio 2 : 7. Calculate
how many millilitres of blue and yellow paint she needs for:
a) 360 ml of the mixture
b) 250 ml (1 cup) of the mixture.
For one cup of coffee Mr Zuma uses approximately 1 teaspoon of
coffee, 2 teaspoons of sugar, 75 ml of milk and 150 ml of water.
a) 1 teaspoon = 5 ml. Write a ratio for the four ingredients.
b) Use the ratio to calculate how much coffee, sugar, milk and water
he needs to fill a 960 ml flask.
(Show all of your calculations.)
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Solutions for all
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3
4
5
Four people invest some money together. Person A contributes
R35 000. Person B contributes R80 000. Person C contributes R22 000
and Person D contributes R85 000. After a few years, their
investment is worth R229 432.
a) How much did they invest originally?
b) Write a ratio in its simplest form, to compare each person’s
investment.
c) They share the R229 432 proportionally. Use the ratio in b) to
calculate how much each person gets.
Plant fertiliser made up of nitrogen, phosphorus and potassium is
mixed in the ratio 3 : 4 : 1. Calculate how much nitrogen,
phosphorus and potassium there is in a 5 kg bag.
At a school the ratio of Maths learners to Maths Literacy learners is
1 : 3. There are 1 152 learners. How many do Maths?
Homework exercise 2.3
1
2
3
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An artist mixes red and white paint in the ratio 1 : 9. Calculate how
many millilitres of red and white paint she needs for:
a) 700 ml of the mixture
b) 420 ml of the mixture.
Three children, Methu, Simon and Boet club their money together
to buy sweets. Methu puts in R2, Simon puts in R3,50 and Boet puts
in R1,50.
a) How much money do they have to buy sweets?
b) Complete the statement: Methu, Simon and Boet put in money
in the ratio ____ : 7 : ____
c) They buy 28 sweets. How will they share the sweets
proportionally?
One 250 mg tablet is made up of chemical A, chemical B and
chemical C in the ratio 3 : 2 : 5.
a) i) Calculate how many mg of each chemical is in one tablet.
ii) Calculate how much of each chemical is needed for 100
tablets.
b) A pharmacist has 60 mg of chemical B left. How much of
chemicals A and C does she need if she uses the 60 mg of B?
c) How many tablets would she be able to make from the mixture
in b)?
Solutions for all
Chapter 2
Lesson 4: Proportion
When two ratios are equivalent, we find that the
quantities in one ratio have been increased or
decreased in the same proportion to give the
second ratio.
Direct proportion: as one quantity increases, the other increases OR as one
quantity decreases, the other decreases.
Inverse proportion: as one quantity decreases, the other increases OR as one
quantity increases, the other decreases.
EXAMPLE 1
1
Work out whether these ratios are equivalent:
15 : 27 : 225 and 5 : 9 : 75
2
In a hospital ward, a baby who weighs 3 kg uses 5 nappies in one day
and a baby who weighs 8 kg uses 7 nappies. Is the ratio the same?
SOLUTION
1
15 : 27 : 225 = 5 : 9 : 75 [divide each term in first ratio by 3]
The two ratios are equivalent.
2
3:8≠5:7
The two ratios are not equivalent.
Solutions for all
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EXAMPLE 2
1
Peter uses 1 2 cups of flour and 2 eggs to make 12 muffins. How much flour
and how many eggs will he use to make 48 muffins?
SOLUTION
Peter needs to convert the recipe for 12 muffins to a recipe for 48 muffins.
12 : 48 = 1 : 4
12 × 4
12 muffins
1
1
cups of flour
2
1
1
×4
2
48 muffins
6 cups of flour
2×4
2 eggs
So Peter needs 4 × 1
8 eggs
1
= 6 cups of flour and 4 × 2 = 8 eggs.
2
EXAMPLE 3
The governing body of Adelaide Tambo High allocates R20 000 a month to
employ teacher assistants. Complete the table to show the relationship
between the number of assistants and their monthly salary.
SOLUTION
1
2
3
8
10
20 000
1
20 000
2
20 000
8
20 000
10
R20 000
R10 000
20 000
3
R6 666,67
R2 500
R2 000
Number of
assistants
Monthly
salary
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Solutions for all
Chapter 2
Classwork activity 2.4
1
Decide whether each of the following pairs of ratios are equivalent.
a)
b)
c)
d)
e)
2
10 : 7 : 12 and 45 : 1 : 3
1 : 2 : 9 and 2 : 4 : 8
52 : 13 and 4 : 1
9 : 7 and 18 : 28
Insert the missing terms so that these quantities are in proportion.
a)
b)
c)
d)
e)
3
10 : 7 : 12 and 50 : 35 : 60
84 : 4 : 16 and 21 : ____ : ____
84 : 4 : 16 and ____ : 1 : ____
26 : 52 : 13 and ____ : ____ : 1
____ : 11 : ____ and 11 : 121 : 11
7 : 5 : 6 and ____ : 25 : ____
A travel agent advertises a holiday package for 4 people at a cost of
R8 520.
a) How much would each person pay?
b) How much would 12 people pay for the holiday?
4
A chef has to cook a meal for 200 people. He uses a recipe for 4
people and adjusts the ingredients.
a) What ratio should he use to calculate the ingredients if his recipe
is for 4 people and he needs to serve 200 people?
b) Complete the table.
Recipe for 4 people
Recipe for 200 people
2 eggs
150 g flour
_____
g = _____
20 ml milk
_____
ml =BBBBB
kg (1 000 g = 1 kg)
Ɛ1 000 ml = 1Ɛ
5 ml sugar
2 ml vanilla essence
5
A Maths Literacy tutor charges R200 for an hour lesson.
a) How much would it cost for a private lesson?
b) What would it cost for two learners to share a lesson?
c) If a learner could only afford to pay R50, how many people
would she need to share the lesson with?
Solutions for all
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Homework exercise 2.4
1
Decide whether each of the following pairs of ratios is equivalent.
a) 1 : 2 : 3 and 6 : 12 : 18
c) 12 : 13 : 15 and 24 : 26 : 32
e) 3 : 6 and 1 : 2
2
Add the missing terms so that these quantities are in proportion.
a)
b)
c)
d)
e)
3
b) 5 : 75 : 10 and 1: 25 : 2
d) 8 : 12 : 16 and 4 : 3 : 4
63 : 252 : 189 and 1 : ____ : ____
7 : 6 : 12 and 49 : ____ : ____
25 : 50 : 150 and ____ : ____ : 6
9 : 27 : 54 and ____ : 3 : ____
81: 6 561 : 531 441 and ____ : ____ : 6 561
The tutor runs holiday tutorials for groups of learners. The cost is
R800 for 4 learners for 5 sessions.
a) How much would 20 learners pay?
b) What is the cost per learner?
c) What is the cost per lesson?
4
A box of cereal has the following
nutritional information on the
package: carbohydrates 81 per 100 g
a) How many carbohydrates are
there in a 40 g serving?
b) How many carbohydrates in a
500 g box of cereal?
5
a) A school needs to hire transport to take learners on an outing.
They can hire a 50-seater bus for R2 000 per day. Copy and
complete the following table.
Number of learners
on outing
10
20
30
40
50
40
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Solutions for all
Cost per learner
(50-seater bus)
Chapter 2
b) A local company offers them the use of a 26-seater vehicle for
only R500 per day. Copy and complete the following table.
Number of learners
on outing
Cost per learner
(26-seater)
10
20
c) Why does the table in question b) stop at 20 learners?
d) State the most sensible transport option to hire if the following
numbers of learners were going on the outing:
i) 10 learners
ii) 20 learners
iii) 50 learners
e) i) How much would it cost to hire two 26-seater vehicles?
ii) How much would it cost per learner to transport 50 learners
if two 26-seater vehicles were used?
Lesson 5: Rate
A rate is a ratio between two quantities measured in different units. It
describes how the one quantity changes in relation to the other.
The exchange rate determines how one currency is related to another.
EXAMPLE 1
The rand-dollar exchange rate is R7,50. How much would you get for $85 if
you exchange it for rands?
SOLUTION
$1,00 = R7,50
‘ $85 = 85(7,50)
= R637,50
The interest you pay on a loan or the interest you earn on an investment
depends on the interest rate.
Solutions for all
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41
EXAMPLE 2
You buy a CD player from a store on hire purchase. The cash price is
R1 495,00.
You are charged an interest rate of 8,5%. How much do you pay in total for
the CD player?
SOLUTION
Calculate the interest:
1 495 = 8,5
1
100
127,075
R127,08
Total amount paid = R127,08 + R1 495,00
= R1 622,08
Speed is also a rate, although we seldom drive at exactly the same speed all
the time. We slow down and speed up as we travel. Therefore speed is taken
to be the average rate for covering a distance.
EXAMPLE 3
The speed limit on a road is 60 km/h. If you travel at
an average speed of 60 km/h how long would it take
you to travel a distance of 100 km?
SOLUTION
60 km takes 1 hour.
1
‘ 1 km takes 1 hour ÷ 60 =
60
1
hours ×100
‘ 100 km takes
60
= 1,666667 hours
(=1,666667 × 60 min = 100 min)
If you work with this answer
as 1,67 (rounding off to 2
decimal places) you end up
with 100,2 minutes. Do the
same calculation using 60
minutes instead of 1 hour and
you will see that it works out
to exactly 100 minutes.
You would take 1,67 hours, i.e. 1 hour 40 minutes.
Many services such as phone calls, electricity and water are charged at a rate
per unit.
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Solutions for all
Chapter 2
EXAMPLE 4
A telecommunications company charges R1,21 per minute for phone calls.
Use this information to complete the table.
Length of call
(in minutes)
3
5
8
12
Cost of call (R)
a)
b)
c)
d)
SOLUTION
a)
R1,21 × 3 = R3,63
b)
R1,21 × 5 = R6,05
c)
R1,21 × 8 = R9,68
d)
R1,21 × 12 = R14,52
Classwork activity 2.5
1
Potatoes are on special at a store for R1,40 per kg. How much
would you pay for 3,5 kg?
2
Bread costs R4,25 per loaf. How much would 5 loaves cost?
3
The rand–euro exchange rate is R9,05 to the euro (€).
a) How much would €150 cost?
b) You have €48 to exchange. How many rands do you get?
c) You change R500 to euro. How many euro would you get?
4
The speed limit on the national road is 120 km/h. If you
maintained an average speed of 120 km/h, how long would a
journey of 420 km take?
5
A tennis coach charges R205 per hour.
a) How much would a 3-hour lesson cost?
1
4
6
A taxi charges R3,15 per kilometre. How much would a customer
pay for a 27,5 km journey?
Solutions for all
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Ð
b) How much would a 1 -hour lesson cost?
43
7
A hawker sells 4 bananas for R10.
a) How much would 1 banana cost?
b) How much would 22 bananas cost?
8
Three litres of milk cost R25,71.
a) How much does one litre of milk cost?
b) How much would five litres of milk cost?
9
A journey in a taxi came to R53.50. The cost was made up of a R10
basic charge and then R10 per kilometre. How many kilometres did
you travel?
A holiday resort advertises that it costs R5 400 per couple for one
week.
a) How much does it cost per couple for one day?
b) How much does it cost per person for the week?
10
Homework exercise 2.5
44
Milk costs R9,50 for a 2 Ɛ bottle. How much would you pay for 16
litres?
2
Six eggs costs R4,95. How much would a tray of 30 eggs cost?
3
The speed limit on a stretch of road is 80 km/h. If you maintained
an average speed of 80 km/h, how long would a journey of 48 km
take?
4
You pay R195 per hour to hire a venue. How much would it cost to
hire the venue for 7 hours?
5
A dance studio advertises hip-hop lessons at R350 per couple for 8
lessons.
a) How much would each person pay for the 8 lessons?
b) How much does each lesson cost:
i) per couple?
ii) per person?
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Solutions for all
Ð
1
Chapter 2
6
The South African rand (ZAR) to Malawian kwacha (MWK)
exchange rate is 1 : 20.
a) How many kwacha would you get for R240?
b) How much is 1 MWK worth in South African currency (ZAR)?
c) If you have MWK9 876, how much would you have in rands
(ZAR)?
7
The rand-dollar (US$) exchange rate is R7,21 to US$1.
a) What would it cost to buy US$50 with rands?
b) What would you get if you exchange US$100 for rands?
c) If your holiday allowance is R5 000 when you visit the United
States, how much money will you have in dollars?
8
Electricity is charged at 76c per unit. There is also a daily surcharge
of R1,27. You buy R300 worth of electricity. Surcharge is deducted
for 27 days and you are allocated units for the balance of the
money.
a) Calculate the surcharge amount.
b) Calculate the amount of money allocated for units.
c) How many units are you allocated?
Solutions for all
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Summary practice exercise
1
Write these numbers in descending order (from the biggest to the
smallest).
ï99ï199; 99; 59; 909ï999; 1 099
2
Copy and complete the following:
a)
b)
c)
d)
3
___ : ___ : 5 = 5 : 125 : 25
___ : 4 = 29 : 58
1 : 2 : 7 = ___ : 92 : ___
10 : 9 = ___ : 3
At a pre-school there are 5 teachers and 84 children.
a) Write a ratio, in its simplest form, for the number of teachers to
children.
b) The ratio of girls to boys is 5 : 2. How many boys are at the preschool?
c) Write a ratio, in its simplest form, of the number of boys to the
total number of children.
4
Mbulelo, Tami and Jill have a meal at a restaurant. They agree to
split the bill in the ratio 2 : 1 : 1. Calculate how much each paid if
the bill was R176,56.
5
A chef has to cook a meal for 58 people. He uses a recipe for 4
people and adjusts the ingredients.
a) What ratio should he use to calculate the ingredients if his recipe
is for 4 people and he needs to serve 58 people?
b) Complete the table.
Recipe for 4 people
Recipe for 58 people
2 eggs
150 g flour
20 ml milk
5 ml sugar
Ð
2 ml vanilla essence
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Solutions for all
Chapter 2
6
A baker mixes butter and flour in the ratio 1 : 3 to make dough.
a) If she has 50 g of butter, how much flour will she need?
b) How much butter is required to make 200 g dough?
c) How much flour is needed for 1 000 g (1 kg) of dough?
7
A packet of chocolate-coated nuts weighs 200 g. Information on the
box indicates that there is 20 g of fat for every 100 g. How many
grams of fat for the contents of the packet?
8
The rand-dollar exchange rate is R7,59 to the US dollar ($).
a) How much would $500 cost?
b) You have $20 to exchange. How many rands do you get?
9
Diesel costs R8,67 per litre.
10
a) How much would 42 litres cost?
b) It costs R523 to fill a car’s tank. How many litres of diesel are
there in the tank?
A Maths tutor advertises her lessons at a fee of R500 a month for 2
lessons per week.
a) How much does it cost for one week?
b) How much does it cost per lesson?
11
Some learners are wrapping chocolates to sell at their school fundraising market. It takes one learner 2 hours to wrap all the
chocolates.
a) How long would it take 2 learners to wrap the chocolates?
b) How long would it take 3 learners to wrap the chocolates?
(Give your answers in minutes.)
Solutions for all
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Word bank
bank balance:
mortar:
the amount of money you have in a bank
account
a mixture of cement, sand and water which is
used by builders for bricklaying.
equivalent:
equal in value
fertiliser:
a substance to make the soil more fertile for
plants
investment:
spending money to make more money, putting
your money in the bank in order to earn
interest, or buying a property and selling it
later at a profit in order to make money on your
original investment
juice concentrate:
when fruit juice has most of the water removed,
the juice is much stronger and takes up less
volume and less space (small carton of juice
concentrate makes a big quantity of juice when
water is added to it)
nutritional information: information about the nutritional content of
food
48
surcharge:
an additional amount added to the fee, to cover
administration and other costs
telecommunications:
the science and technology of communicating
by telephone, internet, radio and TV
travel agent:
a person who organises travel arrangements for
others
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Solutions for all
Chapter 2
Chapter summary
UÊ
A ratio is a relationship that allows you to compare quantities of the
same kind and of the same units.
UÊ
UÊ
In ratio notation we do not use units.
UÊ
UÊ
UÊ
UÊ
When two ratios are equivalent, we say they are in proportion.
We simplify a ratio by dividing each quantity by the same number
(factor).
Direct proportion: both quantities increase or decrease proportionally.
Indirect proportion: as one quantity increases the other decreases.
A rate is a ratio between two different types of quantities. The units of
the two quantities are different.
Solutions for all
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C
3
er
p
ha t
NUMBER SKILLS 3:
Extending your
number skills
In this chapter you practise the skill of rounding off and you extend your
calculator skills.
What you will learn in this chapter
You will:
U Ê learn to use the different functions on a basic calculator
U Ê round numbers up, down, or off (to an appropriate number of decimal
places) depending on the requirements of the context
U Ê recognise that a small change in rounding can make a large difference to an
answer if the error or change is compounded over many calculations or
through a large multiplication
U Ê clearly state workings and methods used for solving a problem
U Ê justify comparisons and opinions with calculations or with information
provided in the context
U Ê modify the solution as required by the context of the problem
U Ê apply the principle of the correct order of operations (BODMAS) and
brackets
U Ê learn about addition and multiplication facts (distributive and associative
properties)
U Ê learn about squaring, cubing and square rooting
U Ê learn about income, expenditure and budgets
U Ê learn about calculating perimeter and area.
Talk about
Have you ever felt like this lady? Things get very complicated when you need
to do real-life calculations with a lot of different amounts to consider. For
example, if you do any rounding off along the way in a long calculation, be very
careful, because this can have an unexpected effect on the final answer. It can
also be tricky to keep track of the bits and pieces in your calculations. This
chapter gives you some efficient methods that can make life easier, for example
by using basic calculators. Basic calculators have some very useful functions
that can help you keep things accurate.
50
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Solutions for all
Every time I try
to work out my
finances I get a
different answer!
Chapter 3
Lesson 1: Rounding off
Looking at the following examples, how do we round off? Identify the last
significant digit, e.g. if you are rounding off to 1 decimal place, check the
VHFRQGGLJLW5RXQGXSLILWLV•5 (greater than or equal to 5). Don’t change
the last significant digit of the rounded answer if the next digit is < 5 (less
than 5).
EXAMPLE 1
Round off 34,7601 to
2 decimal places.
SOLUTION
34,7601 ≈ 34,76
Check the 3rd digit; change the 2nd
digit if necessary. We use ≈ because
the rounded off value is not exact.
It is approximately the same as the
actual value.
EXAMPLE 2
Round off 34,7601 to
1 decimal place.
SOLUTION
The 2nd digit is > 5
so we round up.
34,7601 ≈ 34,8
EXAMPLE 3
The tariff for a local phone call is R0,00724 per second.
Calculate the cost of a 6 minute call.
SOLUTION
6 × 60 × 0,00724
= 2,6064
≈ R2,61
6 minutes = (6 × 60) seconds
= 360 seconds
Money is always rounded to
2 decimal places.
Solutions for all
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51
Sipho:
Modupi:
Sipho:
Modupi:
Sipho:
Why is the tariff so
complicated? Why not round
it off to the nearest cent?
You mean charge R0,01 per
second?
Will it make a difference?
Do the calculation, bro. It
comes to R3,50 for a 6-minute
call.
Wow! Who would have
thought it would make such a
difference?
What is the difference between rounding up,
rounding down and rounding off?
In examples 1 and 2, we rounded off. That means we wrote a number closest
in value to the actual number, using a set of guidelines. We round off to
simplify the number. This is useful when we want to limit the number of
decimal places which could include writing a value as a whole number.
When we round down, we write a number smaller in value than the actual
number, using a set of guidelines. This is useful when we have to make
complete sets and there is a remainder.
EXAMPLE 4
1
2
Molweni is filling bottles with milk. Each
bottle holds 2 Ɛ. How many bottles can he fill
if he has 237,5 Ɛ of milk?
The bottles are packed into containers. Each container holds 20 bottles.
How many containers will he fill?
SOLUTION
1
237,5
118, 75
2
1,5 Ɛ = 0,75 of 2 Ɛ
Molweni must round down the answer to 118. There will be 1,5 Ɛ of
milk left over.
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Solutions for all
Chapter 3
2
118
5,9
20
18 bottles = 0,9 of 20 bottles
Molweni must round down the answer to 5.
18 bottles of milk will remain.
When we round up, we write a number bigger in value than the actual
number, using a set of guidelines.
EXAMPLE 5
A lift holds a maximum of 16 people. How many times would the lift have to
operate if 38 people want to go to the 37th floor of a building?
SOLUTION
38
2,375
16
The lift operator must round up the answer to 3. There would be 6 people left
behind if the lift made only 2 trips.
6 people = 0,375 of 16 people
Rounding down to 5c
In South Africa, the smallest coin has the value of 5c. This means that if
someone pays cash, shops need to either charge amounts that end in a whole
5c, or they need to round off the totals to the nearest 5c. Shops round down,
even if the nearest 5c is higher. For example, a shop would charge R1,95 for a
total of R1,99. Customers need to pay the full amount if they are paying by
debit card or credit card.
Solutions for all
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53
Classwork activity 3.1
1
Round off the following values as indicated.
a)
b)
c)
d)
e)
f)
g)
2
2,06 (1 dec. place)
399,999 (round to a whole number)
68,758 (2 dec. places)
1,0019 (3 dec. places)
789,346 (round to the nearest ten)
789,344 (round to the nearest whole number)
49,795 (2 dec. places)
Round these up or down, depending on the situation. In each case,
explain why you rounded up or down.
a) Mandla is packing peaches. There are 6 peaches in a punnet.
He has 170 peaches to pack. How many punnets does he need?
b) A hotel has to transport 76 tourists to the airport. Each shuttle
can take 12 passengers. How many shuttles should the hotel
send to the airport?
c) Jesse collects stamps. Each page of his stamp collection book
holds 15 stamps. How many pages would he use if he has to
paste in all 87 stamps?
d) Des paints patterns on tiles. Each tile takes 22 minutes to finish.
He hires his work space by the hour. How many hours would he
pay for if he painted 40 tiles?
e) Peter’s Plumbing Services charge R200 per hour or part thereof
for their services. Peter sets a timer when he starts a job. How
much would he charge for a job that took 223 minutes?
f) A ferry boat across a river only operates when there are 25
people on board. 68 people want a ride. How many trips does
the ferry boat do?
g) Xola is putting up washing lines in his yard. The distance from
one pole to the other is 3,2 m. How many lines can he put up if
he has 18 m of washing line?
h) Marcia makes a local phone call that lasts 10 minutes 18 seconds.
If the tariff is R0,00724 per second, calculate the cost of the call.
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Solutions for all
Chapter 3
Homework exercise 3.1
1
Round off the following values as indicated.
a)
b)
c)
d)
e)
2
28,4 (round off to the nearest whole number)
0,072 (round off to 1 dec. place)
654 (round off to the nearest 100)
5,987 (round off to 2 dec. places)
34,85 (round off to 1 dec. place)
Round these up or down, depending on the situation. In each case,
explain why you rounded up or down.
a)
10 mm = 1 cm,
100 cm = 1 m and
1 000 mm = 1 m
Mack manufactures kitchen
cupboards. Small doors measure
400 mm × 300 mm. He has a piece
of wood that measures
300 cm × 3 m. How many small
doors can he cut from the wood?
b) A Gauteng truck driver takes 3 days to do a delivery to Cape
Town. He rests for 1 day, then does another 3-day trip delivering
goods from Cape Town to Gauteng, where he rests for another
day. How many deliveries can he make in 25 days?
c) A bread factory packs 60 loaves into a basket. How many baskets
does the factory need for an order of 220 loaves?
d) Alison makes 5 international calls in one month. The total time
spent on the calls is 7 minutes 24 seconds. The tariff is R0,01325
per second.
i) Calculate the cost of the calls.
ii) The tariff has 5 decimal places. What would the tariff have
been if it was rounded to two decimal places?
iii) What would Alison’s international calls have cost if the tariff
had been rounded off?
Solutions for all
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Lesson 2: More calculator skills
The memory function on a calculator is very useful when you are doing
multiple operations. Instead of performing each operation, writing down the
answer, then adding or subtracting the answers, you can store the answers in
the calculator’s memory appropriately and recall the final answer.
What do the keys allow you to do?
EXAMPLE 1
A carpenter is measuring up a room for a job.
He has written down the following note for himself:
(16 × 8,2 + 12 + 14 × 6,1) ÷ 3
SOLUTION
Think about the calculations: He must use the correct
order of operations.
First complete calculations in the brackets:
16 × 8,2 + 12 + 14 × 6,1
= 131,2 + 12 + 85,4
Then add the answers:
131,2 + 12 + 85,4
= 228,6
Then divide the answer by 3:
228,6 ÷ 3 = 76,2
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Solutions for all
You could also use the
memory function on the
calculator to make it simpler:
(Don’t forget to first press
MRC button twice, to clear
the memory.)
Chapter 3
EXAMPLE 2
The following sketch represents a garden. The pale yellow area is paved.
Calculate the area of the paved surface.
SOLUTION
Hint: Ensure the memory in your calculator has been cleared before starting a
new calculation.
Write a number sentence.
Area of paved area = (area of big rectangle) – (area of flower bed) –
(area of fish pond) – 3(area of tree planter)
Area of rectangle = length × breadth
Area of circle =›U2
A = (250 × 145) – (22 × 110) – 3,142(70)(70) – 3,142(25)(25)(3)
= (250 × 145) – (22 × 110) – 3,142(70)(70) – (3)(3,142)(25)(25)
= 12 542,95 cm2
Solutions for all
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57
Here is the key sequence:
To clear the memory, press MRC again.
Classwork activity 3.2
1
There are 46 learners in Grade 8A, and 57 learners in Grade 8B. Half
the learners in Grade 8A and a third of the learners in 8B took part
in a school tournament, along with 63 Grade 9 learners. Write a
number sentence, then use your calculator to work out how many
learners took part in the tournament.
2
The brown area in the drawing shows an area in the garden that
has to be paved.
a) What is the width of the path?
Show your calculations.
Ð
b) The path can be divided into
three sections. Copy the sketch
into your book and divide the
path into the three sections,
filling in the measurements per
section.
c) Write a number sentence to
calculate the area of the paving,
then solve the problem using
your calculator.
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Solutions for all
Chapter 3
3
Paul runs a hiring service for catering equipment. Someone
places the following order:
20 tablecloths @ R5,20
100 cups @ 75c
100 saucers @ 55c
100 teaspoons @ 70c
100 cake forks @ 70c
75c = R0,75
(20 × 5,20) + (20 × 8,55) + (20 × 7,80)
is the same as
20(5,20 + 8,55 + 7,80)
20 teapots @ R8,55
20 milk jugs @ R7,80
a) Write a number sentence to show how he would calculate his
costs.
b) What was his final cost for the order? Use your calculator
sensibly to make the working easier.
4
Mrs Smith wants to make curtains for
two rooms. She measures the width of
the window, then doubles it and adds
on allowances for four seams. These
are her notes:
a) Do the calculation again to show
that Mrs Smith has made a
mistake. Show that window 1
needs 3,4 m and window 2 needs
8,2 m.
b) Explain why Mrs Smith’s
calculations were incorrect.
Solutions for all
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Homework exercise 3.2
1
Refiloe has a big catering contract. She has prepared a shopping
list, with prices.
a) Explain why it would be useful for Refiloe to use the memory
function on her calculator.
b) Calculate the cost of the items.
2
A builder measures the perimeter of the rooms of a house to
determine how much wood he needs for skirting. He writes down
the measurements:
Room 1: 1,4 × 2,1
Room 2: 2,3 × 2,7
Room 3: 3,5 × 4,2
Room 4: 2,3 × 3,8
a) Each set of measurements represents the length and breadth of
the room. Remembering that Perimeter = 2(length + breadth),
write down a suitable number sentence to work out how much
wood the builder needs.
b) Use your calculator to solve the problem. Remember to use your
memory key effectively.
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Solutions for all
Chapter 3
The following two activities give you more practice in rounding and using a
calculator.
Classwork activity 3.3
Work with a partner for this activity. Akosua travels by car to work every
day. The distance is 13,7 km.
1
What is a sensible rounded-off distance for Akosua to use, if she
wishes to calculate the distance she travels to work and back in a
five-day week?
2
Use the rounded off distance to estimate the distance she travels to
work and back for the week.
3
The gauge on her car’s dashboard shows her petrol usage per
100 km. When she leaves home at 06:50, the journey takes 20
minutes and her petrol consumption is 6,9 Ɛ per 100 km.
a) What time does Akosua get to work if she leaves at 06:50?
b) Round off Akosua’s petrol consumption per 100 km.
c) Use your rounded off figures to calculate how much petrol she
uses to go to work each week.
4
If she leaves home 30 minutes later, the journey takes 55 minutes
and her petrol consumption is 9,2 Ɛ per 100 km.
a) What time does Akosua leave home if she leaves 30 minutes
later?
b) Why do you think the journey takes so much longer?
c) Why would her petrol consumption be higher?
d) What time does she get to work?
e) Round off Akosua’s petrol consumption per 100 km.
f) Use your rounded off figures to calculate how much petrol she
uses to get to work each week.
5
Akosua should have used accurate values, not estimated ones, in
her calculations. Do you agree? Discuss this with your partner, then
write down your answer. Provide a reason for your answer.
Solutions for all
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61
Classwork activity 3.4
1
2
3
4
5
What essential item is missing from
Granny’s budget?
How much money is left in her
budget for this item?
What costs cannot be reduced in her
budget?
Granny goes to the shops once a
week. The taxi fare is R6,50 one way.
She goes to the Day Hospital twice a
month. The taxi fare is R8,50 one
way. On Sundays she catches the taxi
to go to church. The fare is R6,50 one
way. Calculate her actual transport
costs.
Granny buys cleaning materials and toiletries
every month. See the list on the right.
a) Calculate the actual cost of her toiletries and
medication.
b) Calculate the actual cost of her cleaning
materials.
6
Revise Granny’s budget so that it reflects actual
costs.
7
What proportion of her budget does she spend on
each of the following? Use the actual costs.
a)
b)
c)
d)
62
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rent and electricity
transport
toiletries
cleaning materials
Solutions for all
For your calculations,
1 month = 4 weeks.
Washing powder @ R14,99
2 bars of soap @ R3,45 each
1 tube of toothpaste @ R5,99
Shampoo @ R15,99
Body lotion @ R12,99
Painkiller tablets @ R8,99
Dishwashing liquid @ R13,99
Detergent @ R12,99
Bleach @ R9,99
Ð
Money decisions
Mandla’s grandmother has to live on a state
pension of R1 080,00 per month. This is her
budget alongside:
Chapter 3
8
Granny buys the following
items of food every week:
Once a month she buys:
1 loaf brown bread @ R6,59
Sugar @ R9,49
Margarine @ R8,99
Peanut butter @ R7,99
6 rolls of toilet paper @ R2,39 each
Teabags @ R8,99
Oats @ R12,99
3 tins of pilchards @ R6,99 each
2 Ɛ milk @ R13,99
Cheese @ R10,00
Vegetables @ R10,00
2 packs of chicken pieces or
meat @ R15,00 each.
a) Calculate her monthly food bill.
b) Her son gives her some money every month. How much should
he give her to cover the shortfall in her expenses?
c) What other needs could Granny have that are not in her budget?
d) How could Granny cut down on her budget so that she is able to
manage on her monthly state pension? Discuss this in your
groups.
Lesson 3: Working with
exponents and roots
In this lesson, you will practise working with exponents and roots.
It is important to be able to do this when you work with formulae later
in this course.
Exponents
When a number is raised to a power, it means that it is multiplied by itself a
certain number of times. For example, 32 or three squared means 3 × 3 = 9.
Numbers can be raised to any power.
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The number that is being raised to the power is called the base. The number
that the base is raised to is called the exponent.
EXAMPLES
32 = 3 × 3 = 9
52 = 5 × 5 = 25
72 = 7 × 7 = 49
73 = 7 × 7 × 7 = 343
Using your calculator to work out powers
The constant function on your calculator repeats the previous operation.
Try this key sequence:
Press 5 × 5 = Your answer should be 25. You have worked out 52.
Press = again 125. You have worked out 53.
Press = again 625. You have worked out 54.
Press = again 3 125. You have worked out 55.
(Note: This constant function may work differently on different kinds of basic
calculators. Find out how your calculator works by experimenting. It is a
quick and easy way to repeat the same operation. You do not need to work
out powers higher than 3, but it is useful to know this short-cut!)
Squares and cubes
Two special kinds of powers are squares and cubes. Any number multiplied
by itself is called the square of the number. We can represent squares of
numbers with a series of diagrams. These numbers are all called perfect
squares:
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Chapter 3
You can count the number of
circles in each diagram to work
out the square of each number.
In each case, the number that is
squared is the square root.
1
2
2
2
3
4
2
So the square root of the
diagram 42 is one side of the
diagram representing 42 and is
equal to 4.
5
2
2
We can write this as 16 4.
It is easy to work out square roots on your calculator: Simply enter the
number and then press the square root key.
In the same way, any number to the power of three is called the cube of the
number. So 33 is three cubed and is equal to 27. These numbers are all called
perfect cubes:
13 = 1
23 = 8
33 = 27
Classwork activity 3.5
1
Use your calculator to calculate the following powers. You can use
the constant function, or you can use any other method you prefer.
a) (5,65)2
d) (6,14)3
2
b) (1,15)2
e) (2,38)3
c) (4,32)2
f) (5,13)3
Use your calculator to work out these square roots:
a)
400
b)
289
c)
676
d)
1764
e)
6 400
f)
6 724
g)
12100
h)
225
You will apply the skills of squaring, cubing and finding square roots in
later chapters.
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Homework exercise 3.3
1
Find the area of the following squares.
a)
2
c)
b)
Find the volumes of the following cubes.
a)
b)
c)
Summary practice exercise
1
Round the numbers up or down, depending on the situation. In each
case, explain why you rounded up or down.
Ð
a) Anna packs chocolates into small boxes to sell for Mother’s Day. She
packs 8 into each box. There are 108 chocolates. How many boxes does
she need?
b) Anna cuts 10 cm of ribbon to tie around each box of chocolates. Will
she be able to put ribbon around all of her boxes if she has 124 cm of
ribbon?
c) A taxi can carry 12 passengers. If there are 44 tourists to transport, how
many taxis should the taxi service send?
d) A phone call costs R2,20 per minute or part thereof. Thandi spends 5
minutes and 20 seconds talking to her sister. How many minutes must
she pay for?
e) A shopping total comes to R34,96. What will the customer
have to pay?
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2
Use a calculator to solve these problems. First estimate the answers.
Write down the key sequences you use.
a) Esther buys 3 bunches of carrots for R6,99 each; one box of tea bags for
R16,99; four bottles of milk for R7,99 each; and six notebooks for
R18,60 each. What does she pay in total?
b) (7 × 7) – (84 ÷ 2)
c) 2 280 + (1 000 ÷ 5) – 60 × 8 + 3 600
d) (2 100 ÷ 7) – (253 × 4)
3
Calculate the areas of squares with sides of the following lengths:
a) 1,4 m
b) 28 cm
c)
10,5 mm
d) 8,1 km
4
The blocks of stone that make up the bottom layer of a pyramid are
cubes with a side of 2,5 m while those at the top are cubes with a
side of 1,75 m. Calculate the volume of each type of block.
5
Calculate the length of sides of square tiles with the following
areas:
a) 121 cm2
b) 625 m2
c)
400 cm2
d) 14 400 mm2
Word bank
consumption:
dashboard:
gauge:
medication:
punnet:
seam:
shuttle:
significant digit:
tariff:
usage
the panel behind the steering wheel of a vehicle,
with all the gauges
measuring instrument, for example a petrol gauge
shows how much petrol is left in the petrol tank of
the car
tablets, medicine and ointment
a small plastic or polystyrene box in which fruit or
vegetables are sold
the line where two pieces of material are joined
together
a vehicle for transporting passengers
the non-zero digits in a number; when we round
off a number the number is shorter and therefore
has fewer significant digits
the rate charged for a service
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4
Chapter summary
UÊ
UÊ
UÊ
UÊ
UÊ
UÊ
UÊ
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Rounding off a number means we reduce the number of significant
digits. For example 39,046732 can be rounded off:
Q
to a whole number:
39
40
Q
to the next ten:
Q
to one decimal place:
39,0
Q
to two decimal places:
39,05
Q
to three decimal places:
39,047
When we round off a number, the new value is an approximation. It is
not as accurate as the original number. We only round off final
answers, otherwise we would be using approximate values in our
calculations and the answer would be even less accurate.
We round up or down, depending on the context.
When we have to do calculations involving many steps, it is useful to
use the memory function on the calculator. This allows us to follow the
correct order of operations, with fewer steps.
When a number is raised to a power, it means that it is multiplied by
itself a certain number of times. For example, 32 or three squared
means 3 × 3 = 9.
Use the square key on your calculator to calculate the square of a
number, or simply multiply the number by itself twice. A cube is
found by multiplying a number by itself three times,
e.g. 33 = 3 × 3 × 3 = 27.
Use the square root key on your calculator to find the square root of a
number.
Solutions for all
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4
er
p
ha t
Keeping fit and healthy
A healthy lifestyle includes exercise and an awareness of what we eat. In this
chapter you revise the skills you developed in the previous chapters. Each activity
relates to a different context that teaches you something about keeping fit and
healthy.
What you will learn in this chapter
You will:
U Ê communicate solutions using appropriate terminology, symbols and units
U Ê clearly state workings and methods used for solving a problem
U Ê justify comparisons and opinions with calculations or with information
provided in the context
U Ê perform calculations with numbers, including fractions, decimals,
percentages, ratios and proportions.
Talk about
What are some ways of applying what you have learnt in the previous chapters?
People are often obsessed with their health and appearance. Mathematical
Literacy can help you to make sense of articles in the media about health and
fitness, and to separate fact from fiction!
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