MODULE 1: Numbers and Number
Sense
Module 1: Numbers and Number Sense
Unit 1 Place Value and Decimals
1.1 Place Value
The objectives of this section are to
•
revise how to write and speak whole numbers
•
recognise the importance of place value
•
round numbers to the nearest 10, 100, 1000, etc.
First note how we write and speak numbers. We'll look at the number 572 641 839.
6444474444
8
Hundreds Tens Units
of Millions
5
7
6444474444
8
Hundreds Tens Units
of Thousands
2
6
4
1
Hundreds Tens
8
Units
3
9
We say
"Five hundred and seventy two million, six hundred and forty one thousand,
eight hundred and thirty nine."
Now we consider rounding numbers.
7451 is 7450 to the nearest 10, since it is nearer to 7450 than to 7460
(see the number line below).
7450
7455
7451
7460
7451 is 7500 to the nearest 100, since it is nearer to 7500 than to 7400.
7451 is 7000 to the nearest 1000, since it is nearer to 7000 than to 8000.
Note
The convention is that '5' rounds up to the nearest 10,
e.g. 35 to the nearest 10 is 40.
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Module 1: Numbers and Number Sense
1.1
Example 1
Write these numbers in words.
(a)
147
(b)
87
(c)
43 219
Solution
(a)
(b)
(c)
One hundred and forty seven
Eighty seven
Forty three thousand, two hundred and nineteen
Example 2
Write each of the following in figures.
(a)
Fifty nine
(b)
Three hundred and eight
(d)
Two hundred and thirty three thousand, four hundred and one
(c)
Three hundred and eighty
Solution
(a)
59
(b)
308
(c)
380
(d)
233 401
Example 3
Write 2716 to the nearest
(a)
10
Solution
(a)
2720
(b)
100
(c)
1000
(b)
2700
(c)
3000
Example 4
What is the value of the '6' in each of these numbers?
(a)
167
(b)
2006
(c)
6423
Solution
(a)
(b)
(c)
'6' means 6 tens = 60
'6' means 6 units = 6
'6' means 6 thousands = 6000
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Module 1: Numbers and Number Sense
1.1
Exercises
1.
2.
Write these numbers in words.
(a)
6.
(c)
(f)
1463
4 713000
(i)
3991001
(e)
(g)
3000 000
(h)
328
86
Write each of the following in figures.
(a)
Twenty four
(b)
Eighty six
(d)
One hundred and twenty
(f)
One thousand and twenty six
(h)
One thousand and five
(g)
5.
14
124
(e)
4.
(b)
(d)
(c)
3.
32
Nineteen
Three hundred and four
Three million, four hundred thousand
Write each of these numbers to the nearest 10.
(a)
89
(b)
45
(c)
72
(d)
12
(e)
9
(f)
2
(g)
4713
(h)
5629
(i)
4755
Write each of these numbers to the nearest 100.
(a)
376
(b)
1417
(c)
24 699
(d)
101
(e)
149
(f)
251
Write each of these numbers to the nearest 1000.
(a)
1001
(b)
2500
(c)
3999
(d)
132 400
(e)
56 471
(f)
555511
A truck driver delivered 109865 crates of cola in one year.
Write the number of crates to:
(a)
the nearest 100
(b)
the nearest 1000
(c)
the nearest 10
(d)
the nearest 10 000
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1.1
Module 1: Numbers and Number Sense
7.
A school has 1256 students. Write this number to:
(a)
(b)
(c)
8.
the nearest 1000
(a)
19
(b)
91
(e)
19 800
(f)
2190
(g)
(i)
190
(d)
9 100 001
(h)
900 371 423
1971
9 001 111
Place the numbers below in order, with the smallest first.
(a)
147,
222,
32,
3813,
1004
(a)
What is the largest possible number you can make using each of these
digits once only: 4, 6, 3, 2 and 8 ?
60 000,
621,
47,
1472,
6000,
3416,
316,
(b)
(c)
10.
the nearest 100
Explain what the '9' represents in each of these numbers.
(c)
9.
the nearest 10
3000,
1471,
30 000,
15 721
4 000 000
(b)
What is the smallest number you can make using all the digits in (a)?
(d)
How do your answers change if you can use 0 as well?
(c)
What do you notice about the order of the digits in your answers to
(a) and (b)?
11.
Rashan says that there are 120 students in his grade at school. If he has
rounded the number of students in his grade to the nearest 10, how many
students could there be in his grade? (Write all the possible answers.)
12.
A newspaper report states that 32 000 people watched a football match at
the National Stadium in Kingston. The actual number has been rounded to
the nearest 1000.
(a)
(b)
CCSLC
What is the largest possible number of people that watched the
match?
What is the smallest possible number of people that watched the
match?
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Module 1: Numbers and Number Sense
1.1
13.
The table gives the results and attendances for some football matches in the
English Premier League. Answer these questions using the table.
(a)
Which match had the
largest attendance?
(b)
Find the total attendance
at all the matches to the
nearest 1000.
(c)
How many more people
watched Fulham than
watched Cardiff City, to
the nearest 100?
BLACKBURN
28 212
1
MANCHESTER
UNITED
4
WEST HAM
29 126
4
HULL CITY
0
LIVERPOOL
43 007
2
EVERTON
1
FULHAM
36 534
2
ARSENAL
1
ASTON VILLA
28 036
1
NEWCASTLE
1
NORWICH
15 131
0
CHELSEA
1
CARDIFF CITY
33 463
1
STOKE CITY
0
1.2 Decimals and Place Value
The objectives of this section are to
•
recognise the importance of place value in decimals
•
order decimals
•
round decimals to a given number of decimal places.
Note that the number
means
1�.�7�4�3
1. 7 4 3
1 unit
7 tenths
4 hundredths
3 thousandths
Example 1
What is the value of '8' in each of these numbers?
(a)
0.812
Solution
(a)
8 tenths
(b)
8.107
(c)
0.085
(b)
8 units
(c)
8 hundredths
Example 2
Write these numbers in order, smallest first.
0.5,
0.95,
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0.905,
0.59,
0.509,
0.6,
5
0.9
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Module 1: Numbers and Number Sense
1.2
Solution
0.5, 0.509, 0.59, 0.6, 0.9, 0.905, 0.95
Example 3
Write 8.4751 correct to
(a)
3 decimal places
(b)
2 decimal places
(c)
1 decimal place
Solution
(a)
8.475, since 8.4751 is nearer to 8.475 than to 8.476
(b)
8.48, since 8.4751 is nearer to 8.48 than to 8.47
(c)
8.5, since 8.4751 is nearer to 8.5 than to 8.4
Exercises
1.
What is the value of the '5' in each of these numbers?
(a)
(d)
0.45
(b)
3.415
(e)
0.54
(c)
4.258
(f)
2.
Write the numbers in order, smallest first.
3.
Write each of these numbers correct to 1 decimal place.
0.85,
(a)
(d)
4.
1.47
3.751
0.8,
0.58,
0.6,
(b)
(e)
0.5,
3.68
4.08
(a)
3.444
4.7612
(b)
(e)
8.555
0.3002
3.502
0.87
(c)
(f)
Write each of these numbers correct to 2 decimal places.
(d)
5.
0.9,
5.74
(c)
(f)
0.45
5.005
0.321
4.1050
Shanice is given a number correct to 3 decimal places. She writes it to
2 decimal places as 4.71.
Write down a list of the numbers she could have been given.
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Module 1: Numbers and Number Sense
1.2
6.
Write these numbers in figures.
(a)
Four and six tenths
(b)
Five and four hundredths
(d)
One hundred and five hundredths
(c)
(e)
Sixteen, three tenths and four hundredths
One thousand and twenty six and five thousandths
7.
Write these numbers in words.
8.
What is the difference between four tenths and forty hundredths? Explain
your answer.
9.
Write these numbers in order, largest first.
10.
You are given the digits 3, 4, 0, 7 and a decimal point. Using each
number only once, what is
(a)
0.7,
(a)
(b)
5.7
(b)
0.2991,
1.05,
1.508,
5.006
0.58,
(c)
3.02
2.4
the largest number you can make
the smallest number you can make?
1.3 Addition and Subtraction
The objectives of this section are to
•
revise addition and subtraction of whole numbers
•
extend addition and subtraction to decimal numbers.
Example 1
(a)
3
(b)
18 − 4
(c)
12 − 4 − 2
CCSLC
6
2
7
3 8
11
since 6
2
8
18 − 11
7
since 4
7
11
12 − 2
10
since 4 − 2
7
2
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Module 1: Numbers and Number Sense
1.3
(d)
8−2
6
12 − 4 − 2
since 12 − 4
8
Example 2
Calculate:
(a)
102.8
(b)
92.69 − 10.4
15.21
Solution
(a)
To find 102.8
15.21, line up the decimal points:
102.80
+ 15.21
118.01
(b)
To find 92.69 − 10.4 , line up the decimal points:
92.69
10.40
–
82.29
Exercises
1.
Find:
(a)
3
(d)
7
(j)
21
(g)
14
5
8
2.
8
(h)
18
(n)
82
(e)
22
22
(m) 47
(b)
(k)
9
3
7
(c)
6
(f)
9
18
9
7
5
(i)
16
(o)
72
(l)
6
7
14
9
15
31
17
Is each of these statements true or false?
(a)
3
9
9
3
(c)
8
2
9
9
(e)
3
16 − 3
(g)
4
16
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9
(b)
3−1 1− 3
(d)
14
16
(f)
17 − 10
11 16
(h)
14
8
2
8
7
8
6
7
20
10 − 17
8
14
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Module 1: Numbers and Number Sense
1.3
3.
Find:
(a)
8 −5
(b)
9−7
(c)
28 − 15
(h)
48 − 26
(i)
12 − 9
(n)
92 − 47
(o)
57 − 39
(d)
8 −6
(j)
16 − 7
(g)
(e)
(k)
(m) 122 − 86
4.
5.
6.
Find:
(a)
14 − 5
18 − 5
(l)
32 − 24
6−2
(b)
5− 8−7
6 −8
(d)
15 − 4
3
(e)
17 − 1 − 4
(f)
23 − 4 − 2
(g)
5
(h)
4
(i)
8− 3−2
(j)
16 − 8 − 7 − 5
14 − 7 − 3
5
2
71 − 1
1
Copy these sums and put brackets into each one, so that they are correct.
(a)
5 − 8 − 7
(c)
5
4
7 − 2 − 1 11
Find the sum of each set of numbers.
(a)
18 and 15
42, 33 and 62
(b)
6 − 3
(d)
14 − 7 − 3 − 2
(b)
6, 10 and 24
(d)
2
1
8
47, 82 and 37
Find the difference between each pair of numbers.
(a)
(c)
8.
(f)
(c)
(c)
7.
3
15 − 3
7 −4
18 and 15
(b)
92 and 46
(d)
22 and 47
57 and 84
Miss Sharp teaches 2 classes in one morning. There are 32 children in the
first class and 28 in the second.
(a)
(b)
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How many children does she teach altogether?
How many more children are there in the first class than in the
second?
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Module 1: Numbers and Number Sense
1.3
9.
There are 22 people on a bus.
(a)
(b)
10.
At the next stop 5 people get off and 12 people get on. How many are
now on the bus?
At the next stop nobody gets off and the bus leaves with 35 people on
board. How many people got on at this stop?
This season Alex has scored 12 goals for his team. Last season he scored
18 goals for his team.
(a)
(b)
How many goals did he score in total in the two seasons?
What is the difference between the number of goals scored in each
season?
11.
Jade has 42 DVDs. Shona has 8 fewer than Jade. Nicole has 13 more than
Shona. How many DVDs does Nicole have?
12.
In one school there are 3 classes in Grade 7. One class has 38 children, one
has 39 children and the other has 41 children. How many children are there
in Grade 7?
13.
There are 216 cars in a car park. In the next hour, 82 cars arrive and 73 cars
leave. How many cars are in the car park at the end of the hour?
14.
Alison goes on holiday on her motorbike. She keeps a record of how far she
rides each day.
Day
1
2
3
4
5
Km
120
38
59
62
119
What is the total distance she rides?
15.
Use a quick method for each of these sums.
(a)
18 7 12
(b)
108
(c)
99
17
11
(d)
17
(e)
46
23 − 16
(f)
128 − 15 − 13
(g)
72
11 38
(h)
19
6−9
(i)
52
23 − 12
(j)
16
18 − 6
(k)
37
42 − 2
(l)
68
19
1
(m)
33 − 7
(n)
67
18
13
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17
10
19
19
12
13
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Module 1: Numbers and Number Sense
1.3
16.
Find:
(a)
0.3
(e)
1.46
(c)
0.42
(g)
(i)
0.04
1.521
Find:
(b)
0.8
0.1
(f)
5.7
2.4
(j)
5.6
(n)
16.5
(d)
3.42
3.6
(m) 18.14
17.
0.11
6.7
17.2
(k)
0.6
1.2
(h)
0.42
5.12
(l)
3.2
6.3
3.7
8.99
3.21
4.72
3.218
(a)
0.7 − 0.2
(b)
0.9 − 0.6
(e)
6.9 − 3.5
(f)
8.9 − 7.3
(j)
18.62 − 1.7
(n)
0.88 − 0.49
(c)
(g)
(i)
(k)
1.3 − 0.1
(d)
7.2 − 5.3
4.2 − 3.1
(h)
19.24 − 8.3
15.2 − 3.46
6.6 − 4.8
(l)
(m) 0.7 − 0.04
11.4 − 3.12
1.4 Multiplication of Whole Numbers
The objectives of this section are to
•
revise and practise multiplication of whole numbers
•
use multiplication of whole numbers in practical problems.
We start with multiplication
of whole numbers, which is
a useful technique for many
problems.
1
2
3
4
You should know your
multiplication tables up to
10 10 , but for revision,
we include these here.
5
6
7
8
9
1
2
3
4
2
4
6
8 10 12 14 16 18 20
1
2
3
6
4
3
4
5
5
6
6
7
7
8
8
9 10
9 10
9 12 15 18 21 24 27 30
8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100
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Module 1: Numbers and Number Sense
1.4
Example 1
Jai spends $3 on sweets each week for 7 weeks. Calculate how much he spends
altogether.
Solution
He spends (in $) 3
3
3
3
3
3
3
7
3
21 , but it is easier to calculate
21 is $21.
Exercises
1.
Find
(a)
2
3
(b)
5
7
(c)
8
5
(h)
6
6
(i)
(n)
9
(d)
3
(j)
8
(g)
(m) 6
2.
(e)
7
(k)
7
5
4
9
8
9
Is each of these statements true or false?
(a)
(c)
3.
7
5
8
4
9
4
4
5
36
(d)
3
9
4
(f)
9
(l)
7
(o)
(b)
6
6
21
8
5
5
2
9
6
6
7
7
15
Jamil saves $5 per month from his pocket money.
(a)
(b)
How much does he save in 4 months?
How long will it take him to save $30?
4.
How many bottles are there in
this crate?
5.
Emma, Rachel, Sarah and Hannah go to a disco. It costs $3 each to get in.
How much do they pay altogether?
6.
The picture shows the tiles on one wall in
Sophia's bathroom. How many tiles are
on this wall?
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Module 1: Numbers and Number Sense
1.4
7.
Packets of chewing gum are packed in a box. In a box there are 8 layers
with 9 packets of chewing gum in each layer. How many packets are there
in the box?
8.
A hotel has 9 floors. On each floor there are 7 windows. How many
windows are there in the hotel?
9.
Find, by any method:
(a)
3
(d)
9
(j)
43
(g)
84
(m) 184
10.
42
43
(b)
8
(h)
19
(n)
392
(e)
22
62
(k)
192
Use the box method to find:
(a)
12
(d)
45
15
(b)
57
(e)
12
172
32
62
35
(c)
62
(f)
48
42
21
62
(o)
494
(c)
91
15
(i)
(l)
412
6
(f)
461
89
112
22
32
18
78
72
42
428
1.5 Multiplying with Decimals
The objectives of this section are to
•
revise and practise multiplication with decimals
•
use multiplication with decimals in practical problems.
Example 1
You know that 35
19
Deduce the value of
(a) 3.5 19
(b)
665 .
3.5
(c)
1.9
350
1.9
(d)
350
190
Solution
(a)
3.5 19
35
10
19
35 19
10
665
10
CCSLC
66.5
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Module 1: Numbers and Number Sense
1.5
(b)
3.5 1.9
35
10
19
10
35 19
100
665
100
6.65
(c)
350
1.9
35
10
35
10
10
665
(d)
350
190
19
10
19
35
10
19 10
35
19
10 10
665
100
66 500
Of course, in practice you do not need to write out the calculations in full like this,
but simply write down the answers.
Example 2
In a train there are 6 coaches each with 68 seats and two coaches each with
42 seats. What is the total seating capacity of the train?
Solution
The total number of seats
6
408
68
2
42
84
492 seats
Example 3
Find the cost of 12 lunches, each costing $3.29.
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Module 1: Numbers and Number Sense
1.5
Solution
You can use long multiplication to get the answer.
3.29
12
3290
+ 658
$39.48
Exercises
1.
Find:
(a)
3
(d)
9
(j)
9.2
(g)
2.1
(m) 22
2.
1.2
(b)
5
(h)
5.6
(n)
62
(e)
3.2
1.8
(k)
9.4
0.7
6
1.2
(c)
1.5
3
(f)
7.2
6.2
8
(i)
8.4
(o)
74
(l)
7.1
2.6
7.9
2.1
15
7.3
5.3
Work out the following, using a quick method if possible.
(a)
6
10
(e)
2
3.2
(c)
(g)
(i)
3.
0.8
Find:
(a)
(d)
12.2
1.47
100
1000
100
2.47
1.6
7.24
5.16
0.7
(f)
2
(j)
200
(d)
5
365
(b)
(h)
(b)
(e)
3.25
8.21
11.1
15.1
112
10
10
62
18.41
50
10
7200
(c)
(f)
5
3.42
32.1
6.19
0.47
4.
It costs $9 to go on a school trip. A class of 28 children all go on the trip.
How much do they pay in total?
5.
Chocolate bars are packed in boxes. Each box contains 24 bars. Mrs Patel
buys 8 boxes for the tuck shop. How many bars does she buy?
6.
A train has 8 carriages. There are 52 seats in each carriage. How many
seats are there on the train?
7.
A crate contains 24 bottles of juice. There are 32 crates on a truck. How
many bottles are on the truck?
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Module 1: Numbers and Number Sense
1.5
8.
Matthew organises a trip to a concert. He buys 32 tickets which cost $35
each. How much does he spend on the tickets?
9.
Sean helps his parents build a patio. It is rectangular. There are 12 slabs
along one side and 18 along the other side. How many slabs are there in the
patio?
10.
A burger costs $1.29. Find the cost of 10 burgers.
11.
Andre earns $2.54 each day for his paper round. How much does he earn in
6 days?
12.
A meal for an adult costs $4.99 and a meal for a child costs $2.25. Find the
total cost of 2 adult and 4 child meals.
13.
Rope is sold for $1.28 per metre. Find the cost of 10 metres of rope.
14.
The price of a carpet is $4.99 per square metre. Find the cost of 8 square
metres of carpet.
15.
Chain is sold for $2.44 per metre. Find the cost of 3.2 metres of chain.
16.
Apples are sold for $1.06 per kilogram. Find the cost of 2.4 kilograms of
apples.
1.6 Order of Operations: BODMAS
The objectives of this section are to
•
revise division of whole numbers
•
understand that the order of operations is important.
The process of division is multiplication in reverse. So, since 4 3 12 , then
12 4 3 and 12 3 4 but you also need to remember the order in which
operations must be carried out, which can be summarised by BODMAS:
Brackets first
O
Divide
Multiply
Add
Subtract
Example 1
Calculate (a)
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16
2
3,
(b)
16
16
2
3.
CXC
Module 1: Numbers and Number Sense
1.6
Solution
(a)
16
(b)
16
2
3
2
32
3
35
16
3
(Multiplication before Addition)
5
80
(Brackets before Multiplication)
Example 2
Use the BODMAS convention to calculate
(a)
4
2
12
(b)
3
2
4
3
3
Solution
(a)
4
2
12
3
4
2
8
4
4
(D)
(M)
12
(b)
3
2
4
3
6
(A)
12
(M)
18
(A)
Exercises
1.
Find
(a)
16
4
(b)
12
6
(c)
36
9
(h)
15
3
(n)
56
(d)
20
(j)
42
(g)
4
(e)
7
(m) 49
(k)
7
18
9
24
3.
Find:
(a)
(c)
(a)
10 2
3 + 12
3
(d)
12
(g)
3
(j)
CCSLC
16
4
2
4
8
2
2
4
10
6
(b)
(e)
5
8
4
24
6
(h)
(k)
8
3
5 − 12
(b)
(d)
64
8
(o)
45
(c)
3
6 − 15
8
2
5
40
8
32
8
5
12 + 8 2 10
6 2+3 6
6
3
17
(i)
(l)
8
Is each of these statements true or false?
5
(f)
6
2.
15
3
3
8
6−4
(f)
14
2
(i)
42
7 3
8
6 − 45
(l)
8
5
CXC
Module 1: Numbers and Number Sense
1.6
4.
A student works out 200
4 by this method:
200
100
Use similar methods to find:
(a) 500 4
(b)
(d) 128 4
(e)
(g) 80 16
(h)
2
2
100
50
52 4
224 4
112 16
(c)
(f)
(i)
68 4
104 8
128 8
1.7 Division Methods
The objectives of this section are to
•
revise and extend methods of division of whole numbers and decimals.
Care must be taken when handling divisions, particularly when they involve
decimals.
Example 1
Find
(a)
1300
100
(b)
1.75
5
(c)
6.31
4
Solution
(a)
1300
100
1300
100
13
(b)
(c)
1.75
631
5
4
0.35
gives
0.35
since
157
2 3
4 631
5 1.75
r3
, i.e. 157 with remainder 3
Alternatively, to get the answer in decimal form, write
157.75
2 3 3 2
4 631.00
CCSLC
i.e. 157.75
18
CXC
Module 1: Numbers and Number Sense
1.7
Example 2
45 sweets are divided equally between 9 children. How many do they each get?
Solution
Each child gets 45
9
5 sweets .
Exercises
1.
2.
3.
Find:
(a)
12
(c)
600 000
(e)
5728
(g)
7000
(i)
750
(k)
8412
(b)
4200
10
(d)
3714
10
10
(f)
6000
100
1000
(h)
75000
(j)
3714
(l)
642130
10
100
100
Carry out the following divisions.
(a)
69
(d)
2947
(g)
2208
(j)
10 530
3
(a)
(g)
(c)
6612
7
(e)
7404
6
(f)
37050
12
(h)
13488
(i)
1792
(k)
4284
18
(l)
10 496
41
3
(c)
10.24
4
5
388.5
10 000
9
2
87.5
100
4545
45
2.54
100
(b)
Carry out the following divisions.
(d)
4.
10
15
(b)
21.63
(h)
123.84
(e)
918.4
24
7
12
(f)
(i)
3
5
32
49.24
4
714.84
6
Carry out the following divisions, giving your answers as decimals.
(a)
(d)
21
263
4
4
(b)
(e)
81
84
2
8
(c)
(f)
162
241
4
8
5.
A multistorey car park has 4 levels, each taking the same number of cars.
When full it holds 124 cars. How many cars can park at each level?
6.
Randall borrows $50 from his Dad. He pays it back in 10 equal weekly
instalments. How much does he pay back each week?
CCSLC
19
CXC
Module 1: Numbers and Number Sense
1.7
7.
8.
$375.69 is raised at a jumble sale. This is divided equally between
3 charities. How much does each of the charities get?
Chloe has 24 sweets. She shares them out equally between herself and her
3 friends. How many sweets do they get each?
9.
Three children are paid $15 for working in a garden. They share the money
equally between them. How much do they get each?
10.
Kate buys 6 tickets, each costing the same, for the theatre. She pays a total
of $54 for the tickets. How much does each ticket cost?
11.
A rope is 22.48 m long. It is cut into 4 parts of equal length. How long is
each part?
12.
A baker mixes 1944 grams of dough. It is used to make 12 small loaves of
equal weight. How much dough is used in each loaf?
13.
Rachel, Ben, Emma and Hannah are given $5.50 to share equally between
them. Describe the problem they have.
14.
40 children want to go on a school trip to an athletics competition. They
will be taken in minibuses that each hold 13 passengers. How many
minibuses will be needed for the trip?
15.
How many chocolate bars costing 23 cents each can I buy with $2?
16.
A teacher has 240 grams of clay. She cuts off lumps of mass 35 grams each.
(a)
(b)
17.
How many lumps can she make?
How much clay is left over?
A text book costs $7.50. A teacher has $149 to spend on books. How many
copies of this text book can she buy?
CCSLC
20
CXC
Module 1: Numbers and Number Sense
1.8 Roman Numerals
The objectives of this section are to
•
understand Roman numerals
•
convert Roman numerals to and from base 10 (Arabic) numbers.
The number system that we use in daily life is the base 10 (Arabic) system.
Roman numerals are an equivalent system of representing numbers.
In the Roman system we use
I
for
1
V
for
5
X
for
10
L
for
50
C
for
100
D
for
500
M
for
1000
The values of consecutive letters are ADDED unless a letter with a lower value
appears in front of a letter with a higher value. When this is the case, the lower
value letter is SUBTRACTED from the higher value, as shown in the following
examples:
•
•
•
placing I before V or before X to make the numbers 4 (IV) or 9 (IX)
placing X before L or C to make the numbers 40 (XL) or 90 (XC)
placing C before D or M to make 400 (CD) or 900 (CM).
Examples of the effects of the order in which letters are placed:
VII
XXV
CCSLC
5
1 1
10
10
7
5
IV
5−1
IC
100 − 1
99
VL
50 − 5
45
25
4
21
CXC
Module 1: Numbers and Number Sense
1.8
Example 1
Complete these number lines using Roman numerals.
(a)
I
0
(b)
V
1
2
3
4
5
10
0
(b)
0
7
8
9
10
11
12
C
20
30
40
50
60
70
80
90
100
110
120
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
1
2
3
4
5
6
7
8
9
10
11
12
X
XX
L
LX
LXX LXXX XC
C
CX
CXX
10
20
50
60
100
110
120
Solution
(a)
6
L
X
0
X
XXX XL
30
40
70
80
90
Example 2
Convert the following Roman numerals to base 10.
(a)
II
(b)
IX
(c)
DI
(d)
LIX
Solution
(a)
11
1 1
2
(b)
IX
10 − 1
(c)
DI
500
1
(d)
LIX
50
10 − 1
9
501
59
Example 3
Convert the following base 10 numbers to Roman numerals.
(a)
14
CCSLC
(b)
84
(c)
27
22
(d)
1010
CXC
Module 1: Numbers and Number Sense
1.8
Solution
(a)
14
10
4
84
50
30
27
20
5
1010
1000
10
4
2
10
5−1
50
XIV
30
5−1
LXXXIV
XXVII
MX
Exercises
1.
Convert the following Roman numerals to base 10.
2.
Convert the following base 10 numbers to Roman numerals.
3.
Complete the number line using Roman numerals.
(a)
VIII
(a)
(b)
6
(b)
CD
109
C
0
4.
(c)
DVI
56
(d)
(d)
CDXC
1499
D
200
300
400
M
500
600
700
800
900
1000 1100
Change the Roman numerals to base 10 numbers.
(a)
(b)
5.
100
(c)
(i)
DIX
(i)
List the numbers in (a) above in decreasing order.
(iv) CDXVI
(ii)
(ii)
(v)
MCMXLV
MCXI
(iii) CMIV
(vi) CMXCIX
Divide the second number in your list by 11 and write the
answer in Roman numerals.
Above the entrance to a church there is a Roman number
MDCCXCI
When do you think the church was built?
The crypt was built 153 years before the main church.
What Roman number is carved in the wall of the crypt?
CCSLC
23
CXC
Module 1: Numbers and Number Sense
1.8
6.
Which Roman numeral can written instead of the shapes to make the
statements true?
(a)
CDLXXIX <
< CDLXXXIII
(b)
CMXCVIII <
< MIV
Give your answers in Roman numerals.
7.
8.
Continue the sequences, giving the next 3 terms, using Roman numerals.
(a)
XLVII, LXVII, LXXXVII, .............., .............., ..............,
(b)
CMI, DCCCI, DCCI, .............., .............., ..............,
The sum of any two adjacent numbers is the number directly above them.
Fill in the missing numbers.
(a)
XL
XC
L
XX
(b)
CCCL
LXXV
CCSLC
L
C
24
CXC