Whole numbers Recall 1

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1
Whole numbers
Recall 1
Prepare for this chapter by attempting the following questions. If you have difficulty with
a question, go to Pearson Places and download the Recall Worksheet from Pearson Reader.
1 What is the place value of the red digit?
Wo
r ks h e e t R 1. 1
(a) 45 783
A
eight
B tens
C eighty-three
D hundreds
B hundreds
C ten thousands D hundred thousands
(b) 1 264 184
A
two
2 (a) Rearrange the following numbers in ascending order (from smallest to largest).
Wo
567, 4500, 0, 74, 11 100, 6008, 12, 602
r ks h e e t R 1. 2
1200, 204, 987, 2196, 240, 95, 2400, 1010
3 Find:
Wo
r ks h e e t R 1. 3
(a) 50 000 + 6000 + 800 + 90 + 5
4 (a) Round 1245 off to the nearest:
Wo
(i) 10
r ks h e e t R 1. 4
pa
(b) 7 000 000 + 20 000 + 5000 + 70 + 3
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(b) Rearrange the following numbers in descending order (from largest to smallest).
(ii) 100
(iii) 1000
e
(b) Round 8983 off to the nearest:
(ii) 100
5 Find:
r ks h e e t R 1. 5
(a) 3 × 2 × 3
(b) 5 × 3 × 3 × 2
m
Wo
pl
(i) 10
(iii) 1000
(c) 2 × 2 × 2
(d) 10 × 10 × 10 × 10
6 Set out these calculations in your preferred way and work out the answers.
r ks h e e t R 1. 6
(a) 456 + 56
Sa
Wo
(b) 16 + 2047
(c) 90 + 1267 + 341
7 Set out these calculations in your preferred way and work out the answers.
Wo
r ks h e e t R 1. 7
(a) 298 − 123
(b) 854 − 227
(c) 1406 − 249
8 Set out these calculations in your preferred way and work out the answers.
Wo
r ks
h e e t R 1.
8
(a) 45 × 7
(b) 134 × 5
(c) 34 × 95
9 Set out these calculations in your preferred way and work out the answers.
Wo
r ks
h e e t R 1.
9
(a) 844 ÷ 4
(b) 3708 ÷ 9
(c) 897 ÷ 7
After completing this chapter you will be able to:
•
•
•
•
•
•
2
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choose and use a range of mental strategies for calculations
understand how the properties of numbers can be used to calculate efficiently
estimate answers to problems using rounding strategies
interpret and work with numbers in index form
apply the order of operations
solve problems involving whole numbers.
mathematics 7 essentials edition
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1.1
1.1
Mental strategies
Need to Know
The order in which you add or multiply any two numbers does not change the result. For
example, 2 × 3 = 3 × 2 and 4 + 5 = 5 + 4. This is known as the commutative law.
The order in which three or more numbers are added or the order in which they
are multiplied is not important. For example, (2 × 3) × 5 = 2 × (3 × 5) and
(6 + 7) + 8 = 6 + (7 + 8). This is known as the associative law.
When two numbers are multiplied together, the result is called the product.
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To multiply by a large number, split it up into 10s and 1s (or 100s, 10s and 1s). Multiply by
these separately, then add or subtract each of the products. This process works because of
the distributive law.
1
pa
Worked Example 1
Calculate the following using the ‘make easy numbers’ strategy.
2 × 13 × 5
= 2 × 5 × 13
= 10 × 13
= 130
pl
(b)
(c)
m
7 + 32 + 13
= 7 + 13 + 32
= 20 + 32
= 52
(c) 293 + 568
Sa
(a)
(b) 2 × 13 × 5
e
(a) 7 + 32 + 13
Worked Example 2
293 + 568
= 293 + 7 + 561
= 300 + 561
= 861
2
Evaluate the following using the distributive law.
(a) 7 × 22
(a)
7 × 22
= 7 × (20 + 2)
= 7 × 20 + 7 × 2
= 140 + 14
= 154
(b) 15 × 9
(b)
15 × 9
= 15 × (10 − 1)
= 15 × 10 − 15 × 1
= 150 − 15
= 135
1 • Whole numbers
3
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1.1
Answers
page 403
.
1 1 Mental strategies
Fluency
2
1 Calculate the following using the ‘make easy numbers’ strategy.
(a) 8 + 23 + 42
(b) 15 + 57 + 35
(c) 64 + 79 + 56
(d) 5 × 6 × 2
(e) 4 × 6 × 5
(f) 2 × 42 × 5
(g) 5 × 7 × 6
(h) 5 × 3 × 8
(i) 5 × 14 × 4
(j) 47 + 73
(k) 124 + 56
(l) 211 + 169
(m) 37 + 128 + 63
(n) 77 + 78 + 23
(o) 89 + 116 + 11
2 Evaluate the following using the distributive law.
(a) 17 × 9
(b) 19 × 8
(c) 49 × 6
(d) 6 × 31
(e) 7 × 52
(f) 5 × 43
(g) 99 × 9
(h) 77 × 3
(i) 57 × 8
(j) 14 × 11
(k) 15 × 13
(m) 101 × 8
(n) 113 × 5
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1
(l) 16 × 12
(o) 124 × 11
3 Use any appropriate mental strategy to work out the following.
(b) 347 + 156
(d) 8 × 9 × 5
(e) 14 × 7
(f) 21 × 9
(g) 103 × 6
(h) 22 × 11
(i) 3 × 194
(j) 147 + 213
(k) 19 × 14
(l) 4 × 7 × 15
23 × 7 could be calculated by:
A
m
When two numbers
are multiplied
together, the
answer is called
the ‘product’!
4 Choose the correct answer to the following question.
multiplying 3 and 7, then adding 20
B
multiplying 20 and 7, then adding 3
C
multiplying 20 and 7, multiplying 3 and 7, then adding the products together
D
multiplying 20, 3 and 7 all together.
Sa
Hint
(c) 335 − 170
pl
Understanding
e
pa
(a) 23 + 41 + 57
5 Bilal has completed the first three stages of a bike rally. He rode 87 km in Stage 1, 95 km
in Stage 2, and 63 km in Stage 3. Use mental strategies to calculate:
(a) the total distance that Bilal has ridden so far
(b) how far Bilal still has to ride, if the total rally distance is 480 km.
6 The Year 7s at Mountain View Secondary College are undertaking a project to improve
their environment. Each student will plant 5 seedlings of a native plant. Use a mental
strategy to calculate how many seedlings 8 classes of 25 students will need.
7 Jason is saving $8 every week for some new cricket gear. Use mental strategies to calculate:
(a) how much Jason has saved after 17 weeks
(b) how much he still has to save if the cricket gear he wants costs $189.
4
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mathematics 7 essentials edition
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1.1
8 Carlos is monitoring traffic on a busy road. Twelve cars go past him in 1 minute.
(a) Use a mental strategy to calculate how many cars Carlos can expect to go past in
1 hour, based on his 1 minute count.
(b) List two reasons why the actual number of cars might be less than your answer to (a).
9 Jessica earns $5 every time she walks her neighbour’s dog. If she walks the dog 3 times a
week, how much will she earn in 6 months?
10 Alicia is ordering stationery for her office cupboard. Use mental strategies to calculate the
cost of each of the following, in dollars.
(a) 8 notepads at 98 cents each
(b) 3 gluesticks at 77 cents each
(c) 12 pens at 59 cents each
(d) 5 boxes of paperclips at 82 cents each
Hint
There are 26 weeks
in 6 months.
Reasoning
11 Below are some mistakes made by students on a test, and their explanation of the method
they used. Write what each student has done incorrectly and what the answer should be.
(a) 21 × 7 = 161. Kate: ‘I multiplied 7 by 20, and this gave me one less lot of 21 than
ge
s
I needed. So, then I added 21.’
(b) 35 × 3 = 140. Sam: ‘I doubled 35, then doubled my answer to get 140.’
(c) 256 − 65 = 209. Leah: ‘I first subtracted 56 to get back to 200, and then added the
remaining 9.’
Here are their suggestions:
Indrah: 10 × 29 − 9
Lucy: 9 × 30 − 30
(a) Who has written a correct strategy?
e
Alex: 10 × 29 − 29
Khalid: 9 × 30 − 9
pa
12 Four students were asked to write down a mental maths strategy for calculating 9 × 29.
pl
(b) Why do two different strategies give the same answer?
Open-ended
m
(c) Explain what is wrong with the other strategies.
Sa
13 Tranh is in the hardware shop buying some supplies. He has 5 picture hooks, which are
28 cents each, 12 curtain rings, which are 15 cents each, and a small hammer costing $6.
As he walks to the checkout, Tranh wonders if he has enough money, as he only has $10
in his pocket. Describe the mental calculations Tranh could do to be sure he has enough
money to pay for his items. Is $10 enough?
14 Describe how you could use your calculator to work out 11 × 23 if the ‘1’ button was broken.
15 Brendan has conducted a survey of the number of pets owned by each member of his
class. His results are: 2, 2, 2, 1, 1, 2, 3, 3, 2, 2, 1, 4, 1, 3, 5, 2, 4, 1, 2, 1, 3, 2, 1.
Brendan has been trying to use his calculator to add up his list of numbers, but he keeps
losing his place in the list. Suggest a method that Brendan could use to organise his
numbers and use his calculator more efficiently. Use your method to add up Brendan’s list.
1 • Whole numbers
5
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1.2
1.2
Indices
Need to Know
To ‘square’ a number, multiply it by itself:
62 = 6 × 6
= 36
A perfect square is a whole number that is obtained by squaring a number.
To ‘cube’ a number, multiply it by itself and itself again, so that the number is written
three times in all:
43 = 4 × 4 × 4
= 64
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s
Do not confuse 43 with 4 × 3.
A perfect cube is a whole number that is found by cubing a number.
pa
Finding the square root or cube root of a number is the reverse of squaring or cubing
a number.
Index form:
4
index (power)
5
e
base
pl
Expanded form: 5 × 5 × 5 × 5
Say: ‘5 to the power of 4’, or ‘5 to the fourth’ or ‘base 5, index 4’.
m
The value of 54 is 625.
Sa
Any number to the power of 1 is itself. 51 = 5.
Worked Example 3
3
Write each of the following numbers in expanded form, then find its value.
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(a) 43
(b) 85
(a) 43 = 4 × 4 × 4
= 64
(b) 85 = 8 × 8 × 8 × 8 × 8
= 32 768
mathematics 7 essentials edition
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1.2
.
1 2 Indices
Answers
page 403
Equipment required: Calculator for Questions 1(j)–(l), 3, 10, 12, 17, 19(a), 21(a)
Fluency
1 Write each of the following numbers in expanded form, then find its value.
(a) 23
(b) 24
(c) 26
(d) 18
(e) 07
(f) 103
(g) 64
(h) 55
(i) 86
(j) 113
(k) 124
(l) 145
3
2 Write each of the following in index form.
(b) 4 × 4 × 4 × 4 × 4 × 4
(c) 12 × 12 × 12 × 12 × 12
(d) 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16
(e) seventeen cubed
(f) nineteen squared
(g) eight to the power of 4
(h) thirteen to the power of seven
(i) base 11, index 7
(j) base 9, index 6
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(a) 8 × 8 × 8
3 Find the value of each of the following.
(b) 66
(c) 94
(d) 83
(e) 75
(g) seventeen squared
(h) fifty cubed
(j) 5 × 102
(k) 4 × 103
(l) 103 × 2
(m) 2 to the power of 10
(n) 3 to the power of 9
(o) base 1, index 8
pa
(a) 47
(f) 153
pl
e
(i) the cube of 4
4 Write in words how we could say each of the following.
(b) 312
(e) 45
(f) 96
m
(a) 52
(c) 33
(d) 273
(g) 14
(h) 77
Sa
5 If 1 is the first square number and 4 is the second, write the 5th, 6th and 7th square numbers.
6 If 1 is the first cube number and 8 is the second, write the 4th, 5th and 6th cube numbers.
7 Evaluate:
(a)
(e)
(i)
3
(m)
3
25
(b)
1
(f)
8
(j)
3
(n)
3
0
49
(c)
0
(g)
64
(k) 3 1000
125
(o)
3
100
(d)
4900
(h)
8000
121
400
(l)
3
1
(p)
3
27 000
Hint
3
5 means the
cube root of 5.
It does not mean
3 × 5.
8 (a) In 23 = 8 the base number is:
A
2
B
3
C
6
D
8
B
10
C
25
D
125
27
C
243
D
2187
(b) The square of 5 is:
A
2.2
(c) The cube root of 729 is:
A
9
B
1 • Whole numbers
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1.2
Understanding
9 Evaluate the following without using a calculator.
Hint
Ascending order
means from
smallest to largest.
Descending order
means from largest
to smallest.
(a) 14 + 22
(b) 23 − 16
(c) 34 − 24
(d) 23 × 22
(e) 22 × 24
(f) 33 × 32
(g) 25 + 52 − 62
(h) 102 + 32 − 43
(i) 52 − 16 + 33
10 (a) Arrange these numbers in ascending order:
45, 54, 1200, 103, 46, 55
(b) Arrange these numbers in descending order:
1002, 105, 11000, 0100, 32, 23
11 (a) Write down two numbers between 5 and 40 that are both even and square.
(b) Write down two numbers between 30 and 90 that are both odd and square.
12 Evaluate these with your calculator, using the most efficient method possible:
(b) 214 − 4481
(c) 154 − 5625
(d) 363 × 53
(e) 144 × 14
(f) 195 × 21
(g) 212 + 218
(h) 310 + 124
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(a) 163 − 4096
(i) 49 + 311
13 Only the square roots of perfect squares are whole numbers. Square roots of other
10
(b)
5
(e)
99
(f)
2
e
(a)
pa
numbers lie in between the perfect square roots (they are decimal numbers). For example,
16 = 4 and 25 = 5, so 20 would lie in between 4 and 5. Find which two consecutive
whole numbers the following lie between. (Consecutive numbers come one after the
other; e.g. 8 and 9.)
(c)
20
(d)
62
(g)
70
(h)
108
(a) 24, 3 64 , 5,
8 , 10,
90, 4
2
(b) 72, 3 125 , 9,
(d)
m
(c)
3
49
pl
14 Arrange the following in ascending order.
80, 8,
60
105, 33
Sa
15 (a) Complete the following table of the powers of 10.
Index form
Expanded form
Value
101
10
10
2
10
103
10
10 × 10
1000
4
(b) For each row of the table, compare the index numbers in the first column with the
number of zeroes in the third column. Describe the pattern.
16 For each of the following, find the missing power of 10.
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(a) 20 000 = 2 × 10
(b) 400 = 4 × 10
(c) 5000 = 5 × 10
(d) 300 000 = 3 × 10
(e) 80 = 8 × 10
(f) 7 000 000 = 7 × 10
(g) 150 000 = 15 × 10
(h) 91 500 = 915 × 10
(i) 2 340 000 = 234 × 10
mathematics 7 essentials edition
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1.2
Reasoning
17 (a) Use your calculator to work out 52 × 53 and 5 × 5 × 5 × 5 × 5.
(b) Now, try it with 23 × 26 and 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.
(c) What can you conclude?
18 How many whole numbers have the value of their square root in between:
(a) 2 and 3
(b) 5 and 6
(c) 8 and 9?
19 (a) Use your calculator to answer TRUE or FALSE to each of the following statements.
(i) 46 is bigger than 64
(ii) 210 is bigger than 102
(iii) 39 is bigger than 93
(iv) 192 is bigger than 219
(b) Look at your answers for part (a) and, without using your calculator, answer
TRUE or FALSE to each of the following statements.
(i) 98 is bigger than 89
(ii) 2100 is bigger than 1002
20 (a) The number 10100 was called a googol by the mathematician Edward Kasner.
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(i) How many 10s are multiplied together to give a googol?
(ii) How many zeroes would follow the 1 in a googol?
(b) If you raise the number 10 to the power of a googol, you get a number called
a googolplex.
pa
(i) How many 10s are multiplied together to give a googolplex?
(ii) How many zeroes would follow the 1 in a googolplex? How much time do you
think you save by writing a googolplex in index form?
21 (a) Complete the following.
1112 =
11112 =
e
112 =
pl
(b) Look at the pattern in part (a) and, without using a calculator, copy and complete
the following.
111 1112 =
1 111 1112 =
Sa
Open-ended
m
11 1112 =
22 Use the digits 1, 2 and 3 to create at least three numbers that are greater than 500.
Hint
23 (a) A number has two digits when it is squared and three digits when it is cubed.
Use index
numbers in
Question 22.
What might be the number?
(b) A number has three digits when it is squared and four digits when it is cubed.
What might be the number?
24
(a) What is wrong with Mina’s explanation? Write down a better explanation that Mina
could give to Jo.
(b) What question did Jo get right? Why was this the case?
1 • Whole numbers
9
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1.2
Problem
Problem
oblem solving
Teacher’s age
Cheryl, who is a teenager, was curious
about her maths teacher’s age.
than an odd number that has been
squared.’
‘Well’, said the teacher, ‘Right now the
second digit of my age is a square number
and the product of my digits is the cube of
a number. In 10 years’ time my age will be
a square number and 25 years from now
my age will be a number that has been
cubed.’
How old is the teacher?
• Guess and check.
• Make a table.
• Work backwards.
• Test all possible combinations.
1.2
e
pa
More strategies for
multiplication and
division
Sa
m
pl
1.3
Strategy options
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‘I know your age,’ Cheryl said with
confidence. ‘It’s also interesting that the
difference between our ages is one more
How old is Cheryl?
Need to Know
18
multiplier
×
32
÷
dividend
27
multiplier
=
10
=
divisor
486
product
3
rem 2
quotient
remainder
Multiply or divide in stages by breaking the multiplier or divisor down and doing a series
of simpler multiplications or divisions. This is the ‘work in stages’ strategy.
To multiply whole numbers by numbers that are multiples of 10:
10
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1
write each multiple of 10 as the product of a number and a power of 10
2
rewrite the multiplication, grouping the powers of 10 together
3
multiply the other numbers, using the total number of zeroes in the powers of 10 to
write the correct number of zeroes in the answer.
mathematics 7 essentials edition
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1.3
To divide by whole numbers that are multiples of 10:
1
divide both numbers by 10 until one number is no longer a multiple of 10. Show this
by cancelling zeroes
2
perform the simplified division.
Worked Example 4
4
Calculate the following using the ‘work in stages’ strategy.
(a) 34 × 8
34 × 8
= 34 × 4 × 2
= 34 × 2 × 2 × 2
= 68 × 2 × 2
= 136 × 2
= 272
900 ÷ 15
= 900 ÷ 3 ÷ 5
= 300 ÷ 5
= 60
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(b)
pa
(a)
(b) 900 ÷ 15
Worked Example 5
5
40
7
920
+ 16 1
108 1
3
20
3
40
800
120
7
140
21
Sa
20
47 = 40 + 7
m
23 = 20 + 3
pl
e
Calculate 23 × 47 using the ‘use an array’ strategy.
20
3
40
800
120
920
7
140
21
161
Worked Example 6
6
Calculate the following.
(a) 7 × 900
(b) 6000 × 30
(c) 4500 ÷ 90
1 • Whole numbers
11
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1.3
(a)
Answers
page 404
7 × 900
= 7 × 9 × 100
= 63 × 100
= 6300
(b)
6000 × 30
= 6 × 1000 × 3 × 10
= 6 × 3 × 1000 × 10
= 18 × 10 000
= 180 000
(c)
4500 ÷ 90
= 450 ÷ 9
= 50
.
1 3 More strategies for
multiplication and division
Fluency
(b) 41 × 6
(d) 600 ÷ 4
(e) 153 ÷ 9
(g) 35 × 9
(h) 390 ÷ 6
(j) 15 × 12
(k) 450 ÷ 25
(c) 62 × 8
(f) 864 ÷ 16
(i) 70 × 15
(l) 360 ÷ 24
2 Calculate the following using the ‘use an array’ strategy.
(a) 12 × 41
(b) 17 × 26
(c) 19 × 68
(d) 23 × 61
(e) 31 × 49
(f) 39 × 56
(h) 49 × 53
(i) 55 × 55
(k) 72 × 77
(l) 96 × 51
(b) 400 × 5
(c) 2000 × 13
(d) 360 ÷ 20
(e) 5400 ÷ 200
(f) 8000 ÷ 500
(g) 1200 × 400
(h) 400 × 32 000
(i) 7000 × 20 000
(j) 12 000 ÷ 20
(k) 350 000 ÷ 700
(l) 8 400 000 ÷ 4000
(g) 42 × 47
pl
(j) 61 × 82
(a) 700 × 3
m
3 Calculate the following.
Sa
6
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s
(a) 27 × 4
pa
5
1 Calculate the following using the ‘work in stages’ strategy.
e
4
4 Calculate the following using any suitable strategy.
(a) 180 ÷ 20
(b) 450 ÷ 90
(c) 800 ÷ 25
(d) 40 × 19
(e) 53 × 20
(f) 102 × 18
(g) 2700 ÷ 18
(h) 1075 ÷ 25
(i) 2250 ÷ 15
(j) 120 × 12
(k) 59 × 72
(l) 98 × 25
Understanding
5 Charlotte is a distance runner in training. She runs 20 laps of a 400 metre race track 6 times
a week. How many metres does Charlotte run in a week?
6 Year 7 students are practising for sports day. They need to be divided into five equal
groups. There are 130 students in the year. Explain a quick way the teacher could use
to work out how many students will be in each group, and find the answer.
7 Marie’s monthly mobile phone bill is $52. If she continued to pay this amount every
month, use a mental or written strategy to calculate how much Marie will have paid in
phone bills by the end of her 2-year contract.
12
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mathematics 7 essentials edition
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1.3
8 A group of eight friends have won $2000 and want to divide it equally between them. How
much will each receive?
9 Rehan is paying back a bank loan that he used to buy a house. His repayments are set at
$1800 per month for 30 years. How much money will Rehan have paid back at the end of
this period of time?
10 Dharma has printed 3700 brochures advertising his business. He wants to place them
in the letterboxes of houses in the local area. Dharma estimates that there are about
40 houses in each street. How many streets will he be able to cover with 3700 brochures?
Reasoning
11 A travelling salesperson has lost his calculator. He needs to work out the individual cost
of an item that is packaged into groups of 12 items, and costs $660 for the pack.
(a) He decides to approximate by dividing by 10. Find the answer he obtains.
(b) Calculate the actual cost of each individual item.
(c) He sells five items at his approximated cost. How much did he lose or gain?
12 The ‘use an array’ method of multiplication can be extended
Use an extended grid to calculate the following.
(b) 356 × 412
10
3
100
(c) 107 × 560
40
5
pa
(a) 162 × 246
200
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s
from two-digit to three-digit numbers by adding an extra row
and an extra column to the grid. For example, 213 × 145 can
be calculated using the following array.
13 Which of the following is an incorrect method for calculating 240 ÷ 4?
Halve 240, then halve the answer.
B
Calculate 2 ÷ 4, 4 ÷ 4 and 0 ÷ 4, then add the results together.
C
Calculate 200 ÷ 4 and 40 ÷ 4, then add the quotients.
D
Calculate 24 ÷ 4, then multiply the answer by 10.
pl
e
A
m
Open-ended
Sa
14 ◆ × ● = 1620
(a) What could ◆ and ● be? Find at least seven different combinations of whole numbers.
(b) Find at least three combinations where both ◆ and ● are two-digit numbers.
15 Here is part of Kim’s maths homework. She had been learning about splitting up numbers
to be multiplied, so she decided to use that strategy to complete the following.
Multiplication
(1)
9 × 24
=9×2+9×4
= 18 + 36
= 54
(2)
12 × 43
= 12 × 4 + 12 × 3
= 48
+ 36
= 84
(a) Kim is a bit worried that her answers don’t look big enough. What are the correct
answers?
(b) Explain to Kim the mistake she has made in both questions.
(c) Give Kim some advice so that she can avoid similar mistakes in the future.
1 • Whole numbers
13