|
|
|
Meaning and models | Key ideas
|
| Addition | Subtraction
| Multiplication | Division
| Mixed operations | Order of
Operations |
Meaning
and models quiz answers
A
selection of possible answers:
| 1. |
4
thousands 95 ones |
| |
409
tens 5 ones |
| |
40
hundreds 9 tens 5 ones |
| 2. |
3
ten thousands 17 hundreds 36 ones |
| |
317
hundreds 3 tens 6 ones |
| |
31
thousands 73 tens 6 ones |
| 3.
|
39
hundreds 6 tens |
| |
3
thousands 9 hundreds 60 ones |
| |
396
tens |
| 4. |
1
thousand 21 tens 2 ones |
| |
12
hundreds 12 ones |
| |
121
tens 2 ones |
Answers
to Further questions
| Set A | Set B | Set
C | Set D | Set E
Set
A
1. The
positions from right to left are: ones, tens, hundreds, thousands,
tens of thousands, hundreds of thousands, millions. The thousands
position is fourth from the right, therefore 7.
2.
The ones position is first on the right therefore 4.
3. The hundreds of thousands position is sixth from the right therefore
7.
Set
B
1. The
positions from right to left are: ones, tens, hundreds, thousands,
tens of thousands, hundreds of thousands, millions. The 2 is in the
tens of thousands position so it represents 2 x ten thousand = 20
thousand.
2. The
5 is in the tens of thousands position so it represents 5 tens of
thousands.
3. The
4 is in the thousands position so it represents 4 thousands = 40 hundreds
= 400 tens.
Set
C
1. Thirty
ones is three tens. We have five hundreds, three tens and nine ones.
Answer: 539
2. Nine
thousands and seven ones has no hundreds and no tens. Answer: 9007
3. Forty-two
thousands is four tens of thousands and two thousands, sixteen ones
is one ten and six ones. There are no hundreds. Answer: 42 016
4. Ninety-five
ones is nine tens and five ones. There are six thousands and no hundreds.
Answer: 6095
5. Eight
hundred and five thousand is eight hundred thousands no tens of thousands
and five thousands. There are also two hundreds. Twelve ones is one
ten and two ones. Answer: 805 212
Set
D
1. The
6 is in the hundreds position. The following digit is 9 which is greater
than 5 so round the 6 up to 7. This 7 represents 700, so 3 514 698
to the nearest hundred is 3 514 700
2. The
1 is in the ten thousands position. The following digit is 4 which
is less than 5 so do not change the 1. This 1 represents 10 000, so
3 514 698 to the nearest ten thousand is 3 510 000
3. The
3 is in the millions position. The following digit is 5. For any number
greater than or equal to 5 the convention is to round up, so the 3
becomes 4. This 4 represents 4 000 000, so 3 514 698 to the nearest
million is 4 000 000
Set
E
1. 90
x 30 would be greater than 89 x 29 because 90 is greater than 89 and
30 is greater than 29. For positive numbers, when you multiply two
larger numbers the answer is larger.
2. 140
+ 60 would be greater than 137 + 59 because 140 is greater than 137
and 60 is greater than 59. For positive numbers, when you add two
larger numbers the answer is larger.
3. 50
x 10 would be less than 52 x 16 because 50 is less than 52 and 10
is less than 16. For positive numbers, when you multiply two smaller
numbers the answer is smaller.
4. 160
- 120 would be less than 162 - 119 because 160 is less than
162 and 120 is greater than 119. For positive numbers, when you subtract
a larger number than the given one from a smaller number than the
other given number the answer is smaller.
5. 300
+ 200 would be greater than 295 + 187 because 300 is greater than
295 and 200 is greater than 187. For positive numbers, when you add
two larger numbers the answer is larger.
6. 90
÷ 30 would be greater than 89 ÷ 31 because 90 is greater
than 89 and 30 is smaller than 31. For positive numbers, when you
divide a larger number by a smaller number the answer is larger.
7. 400
x 20 would be less than 401 x 22 because 400 is less than 401 and
20 is less than 22. For positive numbers, when you multiply two smaller
numbers the answer is smaller.
Key
ideas quiz answers
| Set
A|
Set B | Set C | Set
D |
Set A
| 1. |
2
x 5 = 10, so 2 x 87 x 5 = 10 x 87 |
| 2. |
5
x 4 = 20, so 5 x 16 x 4 = 20 x 16
|
| 3. |
5
x 16 = 80, so 5 x 16 x 4 = 80 x 4
|
| 4. |
10
x 40 = 400, so 2 x 10 x 40 = 2 x 400
|
| 5. |
2
x 40 = 80, so 2 x 10 x 40 = 80 x 10 |
| 6. |
25
x 4 = 100, so 25 x 5 x 4 = 100 x 5 |
| 7. |
6
x 5 = 30, so 6 x 12 x 5 = 12 x 30
|
| 8. |
2
x 7 = 14, so 2 x 8 x 7 = 14 x 8 |
| 9. |
9
x 10 = 90, so 9 x 3 x 10 = 90 x 3
|
| 10. |
250
x 4 = 1000, so 250 x 6 x 4 = 1000 x 6
|
Set
B
| 1. |
23
= 20 + 3 = 3 + 20, so 27+23 = 27 + 3 +20
|
| 2. |
16
= 14 + 2, so 16+46 = 46 + 16 = 46 + 14 +2 |
| 3. |
18
= 17 + 1, so 18 + 33 = 33 + 18 = 33 + 17 +1 |
| 4. |
28
= 25 + 3, so 75 + 28 = 75 + 25 +3
|
| 5. |
48
= 47 + 1, so 113 + 48 = 113 + 47 +1
|
| 6. |
14
= 13 + 1, so 87 + 14 = 87 + 13 +1 |
| 7. |
9
= 6 + 3, so 1094 + 9 = 1094 + 6 +3
|
| 8. |
23
= 21 + 2, so 449 + 23 = 449 + 21 +2 |
| 9. |
1003
= 1000 + 3 and 1103 = 1000 + 103, so 1003 + 1103 = 1000 + 1000
+3 + 103 = 1000 + 1000 +106 |
| 10. |
3205
= 3200 + 5 and 3207 = 3200 + 7, so 3205 + 3207 = 3200 + 3200
+5 + 7 = 3200 + 3200 +12
|
Set
C
| 1. |
+
6 and - 6 are inverse operations, so 18 + 6 - 6 = 18 + 0
= 18
|
| 2. |
+
(understood) 324 and - 324 are inverse operations, so 324
+ 156 - 324 = 0 + 156 = 156
|
| 3. |
x
29 and ÷29 are inverse operations, so 98 x 29 ÷
29 = 98 x 1 = 98 |
| 4. |
x
184 and ÷184 are inverse operations, so 184 x 236 ÷184
= 1 x 236 = 236 |
| 5. |
256
x 943 ÷ 256 + 149 - 149 = 256 ÷ 256 x 943 +
149 - 149 = 1 x 943 + 0 = 943 |
Set
D
| 1. |
75
+ 135, 135 + 75 Answer: 210 CDs.
|
| 2. |
19
+ 27 + 13, 27 + 19 + 13, 13 + 19 + 27, 13 + 27 + 19. Two other
number sentences 19 + 13 + 27 and 27 + 13 + 19 will give the
same answer but they are not as meaningful in this context.
Can you see why? Answer: 59 people.
|
| 3. |
4
x 23, 23 x 4. Answer: 92 people. Another possible number sentence
is repeated addition, 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4
+ 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4.
|
| 4. |
2
x 25 x 50, 2 x 50 x 25, 50 x 2 x 25, 50 x 25 x 2, 25 x 2 x 50,
25 x 50 x 2. Answer: 2500 words. A number sentence could also
be made using repeated addition - although it would be time consuming.
|
| 5. |
16
x 4 + 12 x 2, 16 x 4 + 2 x 12, 4 x 16 + 12 x 2, 4 x 16 + 2 x 12,
12 x 2 + 16 x 4, 2 x 12 + 16 x 4, 12 x 2 + 4 x 16, 2 x 12 + 4
x 16. Answer: 88 legs. As above, repeated addition could also
be used.
|
| 6. |
14
x 4 + 84 x 4, 4 x 14 + 4 x 84, 4 x 14 + 84 x 4, 14 x 4 + 4 x 84,
4 x (14 + 84), 4 x (84 + 14), (14 + 84) x 4, (84 + 14) x 4. Answer:
392 legs. As above, repeated addition could also be used.
|
| 7. |
5
x 4 x 3 + 7 x 5 x 2, 5 x 4 x 3 + 7 x 2 x 5, 5 x 4 x 3 +2 x 7
x 5, 5 x 4 x 3 + 2 x 5 x 7, 5 x 4 x 3 +5 x 2 x 7, 5 x 4 x 3
+ 5 x 7 x 2,
3 x 5 x 4 + 7 x 5 x 2, 3 x 5 x 4 + 7 x 2 x 5, +3 x 5 x 4 +2
x 7 x 5, 3 x 5 x 4 + 2 x 5 x 7, 3 x 5 x 4 +5 x 2 x 7, 3 x 5
x 4 + 5 x 7 x 2,
3 x 4 x 5 + 7 x 5 x 2, 3 x 4 x 5 + 7 x 2 x 5, 3 x 4 x 5 +2 x
7 x 5, 3 x 4 x 5 + 2 x 5 x 7, 3 x 4 x 5 +5 x 2 x 7, 3 x 4 x
5 + 5 x 7 x 2,
4 x 3 x 5 + 7 x 5 x 2, 4 x 3 x 5 + 7 x 2 x 5, 4 x 3 x 5 +2 x
7 x 5, 4 x 3 x 5 + 2 x 5 x 7, 4 x 3 x 5 +5 x 2 x 7, 4 x 3 x
5+ 5 x 7 x 2,
4 x 5 x 3 + 7 x 5 x 2, 4 x 5 x 3 + 7 x 2 x 5, 4 x 5 x 3 +2 x
7 x 5, 4 x 5 x 3 + 2 x 5 x 7, 4 x 5 x 3 +5 x 2 x 7, 4 x 5 x
3 + 5 x 7 x 2,
Using the commutative law each of the above, a + b can be written
as b + a. Answer: 130 buttons.
|
| 8. |
120
x 80 + 230 x 60, 80 x 120 + 230 x 60, 120 x 80 + 60 x 230, 80
x 120 + 60 x 230. Answer: 23400 pieces of fruit approximately.
|
| 9. |
20
x 4 x 16, 20 x 16 x 4, 4 x 20 x 16, 4 x 16 x 20, 16 x 20 x 4,
16 x 4 x 20. Answer: 1280 songs.
|
| 10. |
9
x 3 + 11 x 2, 9 x 3 + 2 x 11, 3 x 9 + 11 x 2, 3 x 9 + 2 x 11.
Answer: 49 newspapers.
|
Addition
quiz answers
|
1.
|
2.
|
|
3.
|
4.
57 890 is approximately 60 000 and 41 632 is
approximately 40 000. Hence, the answer is approximately
100 000.
|
|
5.
102 337 is close to 100 000, 8 965 is close to 9 000
and 54 921 is close to 55 000, so the estimated total
is 159 000.
The
exact answer is 166 223.
|
Answers
to Further questions:
|
1.
Total number of animals = 256 + 134
Total
number of animals = 390
|
2.
Total number of dogs = 8 + 15 = 23 dogs
|
|
3.
Total mass = 38 + 49
Total
mass = 87 kg
|
4.
Distance ridden by Sally = 29 + 13
Distance
= 42 km
|
|
5.
Number = 1275 + 499
Number
is 1774
|
6.
Number of maths problems solved this month = 192.
Number
of maths problems solved last month = 192 + 29.
Number
of maths problems solved in the two months = 413
|
|
7.
Total population = 131 000 000 + 18 000 000
Total
population = 149 000 000 people.
|
8.
Total area = 1 700 000
+ 1 400 000 + 985 000
Total
area = 4 085 000 km2
|
Subtraction
quiz answers
|
1.
(a)
(b)
|
2.
|
|
3.
|
|
4.
|
5.
Distance from Tarcutta to Sydney = 819 - 450
Distance
= 369 km
|
Answers
to Further questions:
|
1.
Number of books owned by Kara = 364 - 93
Kara
owns 271 books
|
2.
Extra stickers required = 2520 - 1398
Extra
stickers required = 1122
|
|
3.
Number cups remaining = 160 - 37
Number
cups remaining = 123
|
4.
Difference = larger number - smaller number = 151 - 112
Difference
= 39 cm
|
|
5.
Distance left = 200 - 137
Distance
left = 63 m
|
6.
Other number = 159 - 17
The
other number is 142
|
|
7.
1 092 127 - 426 300 = 665 827
|
8.
Extra people in India = 913 000 000 - 21 000 000
= 892 000 000
|
Multiplication
quiz answers
|
1.
(a)
(b)
|
2.
|
|
3.
(a) 34 x 70 = 34 x 7 x 10
34 x 7 = 238
238 x 10 = 2 380
(b)
96 x 7 = 672
(c)
10 241 x 5 = 51 205
(d)
56 x 200 = 56 x 2 x 100
56 x 2 = 112
112 x 100 = 11 200
|
|
4.
Distance travelled to work each week = 14 x 3 = 42 km/week
Distance travelled to work each year = 52 x 42
Distance travelled to work each year = 2184 km
|
5.
See
the completed multiplication tables below.
|
x
|
12
|
8
|
10
|
4
|
|
5
|
60
|
40
|
50
|
20
|
|
3
|
36
|
24
|
30
|
12
|
|
4
|
48
|
32
|
40
|
16
|
|
8
|
96
|
64
|
80
|
32
|
|
x
|
9
|
3
|
11
|
6
|
|
7
|
63
|
21
|
77
|
42
|
|
9
|
81
|
27
|
99
|
54
|
|
6
|
54
|
18
|
66
|
36
|
|
4
|
36
|
12
|
44
|
24
|
|
x
|
8
|
4
|
7
|
3
|
|
12
|
96
|
48
|
84
|
36
|
|
3
|
24
|
12
|
21
|
9
|
|
8
|
64
|
32
|
56
|
24
|
|
9
|
72
|
36
|
63
|
27
|
|
x
|
9
|
12
|
8
|
5
|
|
4
|
36
|
48
|
32
|
20
|
|
10
|
90
|
120
|
80
|
50
|
|
6
|
54
|
72
|
48
|
30
|
|
11
|
99
|
132
|
88
|
55
|
|
x
|
5
|
7
|
6
|
12
|
|
12
|
60
|
84
|
72
|
144
|
|
8
|
40
|
56
|
48
|
96
|
|
2
|
10
|
14
|
12
|
24
|
|
7
|
35
|
49
|
42
|
84
|
Answers
to Further questions:
| 1.
Distance = 9 x 45 = 405 km. |
|
2.
Number = 15 x 38
Number
= 570
|
|
3.
Number of cans = 36 x 125
Number of cans = 4500
|
|
4.
Number of seats = 26 x 130
Number
of seats = 3380
|
|
5.
Brenda’s age = 4 x 7 = 28
Tony’s
age = 3 x 28 = 84
Tony
is 84 years of age.
|
|
6.
Time
for light to reach earth
= 8 minutes = 8 x 60 = 480 seconds.
Distance
traveled y light in 480 seconds
=
480 x 300 000 = 48 x 3 x 1 000 000
=
144 000 000
Distance
from the earth to sun = 144 000 000 km
|
| 7.
Number
when divided by 8 gives an answer 12 = 8 x 12 = 96
Number
required gave a remainder of 1
Therefore, required number = 96 + 1 = 97
|
|
8.
Number in group = 24 x 16 + 8
Number in group = 384 + 8 = 392 people.
|
| 9.
Distance traveled = 100 x 12 = 1 200 km. |
|
10.
Number
of seconds in 1 hour = 60 x 60 = 3 600
Number
beats in 1 hour = 4 x 3 600 = 36 x 4 x 100 = 14 400
beats
|
Division
quiz answers
|
1.(a)
|
1.(b)
|
|
2.
Number of groups = 1020 ÷ 15
There
are 68 groups.
|
3.(a)
|
|
3.(b)
This
answer can also be written as 152 + remainder 12. 12 is a half
of 24 or 0.5
|
Answers
to Further questions:
|
1.
Each
person gets $668
|
2.
26
people.
|
|
3.
The
other number is 45
|
4.
Undo the multiplication by dividing 54 096 by 14.
Now
divide 3 864 by 14 to get the required answer.
The
answer should have been 276
|
|
5.
13
groups of 4 can be made.
|
6.
416
+ remainder 6
|
|
7. 70 000 ÷ 2 500 = 700 ÷ 25
Tasmania
is 28 times larger than the Australian Capital Territory.
|
8.
Number minutes to empty 13 500 L is 13 500 ÷
250
13 500 ÷ 250 = 1 350 ÷ 25
It
will take 54 minutes to empty the tank.
|
|
9.
Number minutes for 25 000 copies = 25 000 ÷
100 =
250 minutes.
Number hours in 250 minutes = 250 ÷ 60 = 25 ÷
6
Because
there is a remainder it will take more than 4 hours. Hence,
the photocopier cannot complete the order in 4 hours.
|
10.
The number of spaces between 4 blocks of wood = 3
Each
space is 8 cm long.
Distance
for spaces = 3 x 8 = 24 cm
Distance
to be covered by logs = 148 - 24 = 124 cm
Length
of each block of wood = 124 ÷ 4
Each
block of wood is 31 cm long.
|
Mixed
Operations quiz answers
The
mixed operations quiz can be found at the end of the division section.
To go there now, click here.
|
1.
Number of eggs now = 20
Number eggs before 12 were sold = 20 + 12 = 32
Number eggs before 14 were dropped = 32 + 14 = 46
Number eggs before hen laid 15 = 46 - 15 = 31
Number eggs before 8 were sold = 31 + 8 = 39
There
were 39 eggs in the beginning.
|
2.
Make a group of sweets for Angela and two groups of sweets for
Xuping because she has to get twice as many sweets as Angela.
This makes 3 groups. We want 36 how many 3s. 36 ÷ 3 = 12
Angela
gets one group of 12 and Xuping gets 2 groups of 12 i.e. 2 x
12 = 24.
Angela
gets 12 sweets and Xuping gets 24 sweets.
|
|
3.(a)
Number red and green balloons = 56 + 27 = 83
Number yellow balloons = 142 - 83 = 59
The
number of yellow balloons was 59.
|
3.(b)
Number of balloons given away = 12 x 2 + 12 x 3 + 12 x 4 = 24
+ 36 + 48 = 108
Number
of balloons left = 142 - 108 = 34
The
number of balloons left was 34.
|
|
4.
11020 how many 58s?
Number
steps = 11 020 ÷ 58
Claudine
will need to take 190 steps.
|
5.
(35
x 8) ÷ 10 + 602 = 280 ÷ 10 + 602
= 28 + 602
= 630
You
get the number 630
|
|
6.
Sum of 12 and 18 = 12 + 18 = 30
Product
of 12 and 18 = 18 x 12 = 216
Difference
= 216 - 30 = 186
Difference
between the sum and the product = 186
|
7.(a)
Number of jelly beans = 9 x 5 = 45
Number of smarties = 4 x 12 = 48
There
are more smarties.
(b)
Extra smarties = 48 - 45 = 3
There
are 3 more smarties than jelly beans.
|
|
8.(a)
Harry’s age = 6
Mother’s age = 5 x 6 = 30
Father’s age = 6 x 6 = 36
3 years younger than mother’s age = 30 - 3 = 27
Bob’s age = 27 ÷ 3 = 9
Bob
is 9 years old.
(b) Total ages = 6 + 30 + 36 + 9 = 81
|
9.
Total points = 193
Number
points for tries = 19
Number
points for goals = 193 - 19 = 174
Number
goals = 174 ÷ 6
There
were 29 goals kicked in the match.
|
|
10.(a)
Mass
of cat food = 125 x 24
Mass of dog food = 175 x 16
Total
mass of food = 3 000 + 2 800 = 5 800 g
10.(b)
Amount food eaten per day = 2 800 ÷ 14
Dog eats 200 g each day.
|
11. 17 must divide evenly into the number
Because
there is no remainder when 153 is divided by 17 there must be
153 videos. Answer D
|
|
12.
Number books in larger set = 85
Number
books in smaller set = 85 ÷ 5 = 17
Total
number of books = 85 + 17 = 102
Number
books required per shelf = 102 ÷ 2 = 51
Extra
books required for smaller set = 51 - 17 = 34
34
books would have to be moved to the shelf with the smaller number
of books.
|
13.
The larger the numbers multiplied together the larger their
product so one number will have 9 for its first digit and the
other number will have 8 for its first digit. The numbers could
be 96 and 81 or 91 and 86. The answer would be larger if you
multiplied 91 by 6 than if you multiplied 96 by 1.
Numbers
that would give the biggest product are 91 and 86.
|
|
14.
Number of groups = 218 ÷ 6
This
means there would be 36 groups of 6 and 2 students left over.
Since there can be no more than 6 to a group, these 2 could
form a group by themselves. This would give 37 groups.
|
15.
No.
points for two trips to Brisbane = 13075 x 2 =
26150
No.
of points for three trips to Hobart = 4027 x 3 = 12081
Total
points used for these trips is 26150 + 12081 = 38231
No.
points left = 46655 - 38231 = 8424
There
are enough points to go to Adelaide but not to Sydney.
|
|
16.
Total cost paying per lesson = 12 x 10 x 4 = $480
Amount
saved paying per year = 480 - 418 = 62
She
would save $62 with a yearly subscription.
|
17.(a)
Number of children per bus = 990 ÷ 24
There
are 6 children left over.
(b)
41 children per bus initially
1
extra makes 42
Some
buses carried 42 children
|
|
18.
Cost
of books = $23 x 12 = $276
Cost
of CDs = 16 x $21 = $336
CDs cost more.
Excess
cost of CDs = 336 - 276 = $60
CDs
cost $60 more than the books
|
19.
Two
digit multiples of 5, plus 2 are: 12,
17, 22, 27, 32, 37, 42 etc.
Two
digit multiples of 6, plus 1 are:13, 19, 25, 37
37
is the smallest number to appear in both groups
Required
number is 37.
|
| 20.
If there are three birds per cage, two are left over.
Number
birds = multiples of 3 plus 2 =
5, 8, 11, 14, 17,20, 23 etc.
If
there are 4 birds per cage, one is left over.
Number
birds = multiple of 4 plus 1 =
5, 9, 13, 17
Number
of cages when 3 birds per cage is one more than number of cages
used when 4 birds per cage.
Hence,
number of birds cannot be 5 because only one cage used in both
instances.
Number
of birds = 17
Number
of cages = 5.
|
Order
of operations quiz answers
| 1. |
10
- 1 - 2. We can insert brackets to show which part of the expression
we are working on first. Although this expression only contains
subtraction, it is still important to follow the correct order
of operations, that is, work from left to right.
(10
- 1)
- 2 = 9 - 2 = 7. Here we have worked from left to right as per
the correct order of operations.
10
- (1 - 2) = 10 - (- 1) =
- 11. Here we have not worked from left to right and have obtained
a different, incorrect, answer.
|
| 2. |
The
answers
to the following expressions are as follows:
|
| (a) |
| 11
x (3 + 2) x 4 ÷
2 |
We
solve the brackets first.
|
| =
11 x 5 x 4 ÷
2 |
We
only have division and multiplication, so we work from left
to right.
|
| =
55
x 4 ÷ 2 |
We
only have division and multiplication, so we work from left
to right.
|
| =
220 ÷
2 |
|
| =
110 |
|
|
| (b) |
| 7
- 18 ÷ 2 x 3
+ 5 |
Do
multiplication and division first, working from left to
right.
|
| =
7 - 9 x 3 + 5 |
Do
multiplication and division first, working from left to
right.
|
| =
7 - 27 + 5 |
Do
addition and subtraction, working from left to right.
|
| =
- 20 + 5 |
|
| =
- 15 |
|
|
| (c) |
42
÷ 3 x 7
| 42
÷ 3
x 7 |
We
only have division and multiplication, so we work from left
to right |
| =
14 x 7 |
|
| =
98 |
|
|
|
|
| 3. |
9
+ 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x 2 + 8 ÷ 2
+ 3 x 3
|
|
= 9 + (4 ÷ 2 x 7) - (6 ÷ 3) - (4 x 2) +
(8 ÷ 2) + (3 x 3)
|
| =
9 + 14 - 2 - 8 + 4 + 9 |
To
make it clearer, we have inserted brackets above around
all the multiplication and division. We can then work through
the multiplication and division, from left to right. We
have not changed the meaning of the expression because we
have only inserted brackets around all multiplication and
division which must be treated equally - from left to right.
|
| =
26 |
we
have done addition and subtraction, working from left to
right |
|
|
|
| 4. |
Some
examples are; |
|
9
+ 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x ((2
+ 8) ÷ 2) + 3 x 3
=
9 + 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x (10
÷ 2) + 3 x 3
=
9 + 4 ÷ 2 x 7 - 6 ÷ 3 - 4 x 5 + 3 x 3
= 9 + 14 - 2 - 20 + 9
= 10
9
+ 4 ÷ 2 x 7 - 6 ÷ (3
- 4) x (2 + 8) ÷ 2 + 3 x 3
=
9
+ 4 ÷ 2 x 7 - 6 ÷ -1 x 10 ÷ 2 + 3 x 3
=
9 + 14 - - 30 + 9
=
23 + 39
=
62
9
+ 4 ÷ 2 x 7 - (6 ÷ 3
- 4) x 2 + 8 ÷ 2 + 3 x 3
=
9 + 4 ÷ 2 x 7 - (-2) x 2 + 8 ÷ 2 + 3 x 3
=
9 + 14 - - 4 + 4 + 9
=
40
NOTE:
The placement of brackets in these examples was deliberately
designed to keep the numbers simple. If you have positioned
brackets elsewhere your workings may involve fractions or decimals.
|
| 5. |
The
colour blue has been used in the following expression to indicate
which part of the expression is being worked on in each step.
The
colour scheme on the right hand side reflects the table used here.
Any scheme can be used to keep track of the operations being performed.
Arrows, which have not been used here, are easy to draw when handwriting.
|
Brackets
first
|
|
Brackets
first
|
|
Tidying
up
|
|
Fractions
|
|
Multiplication
|
|
Subtraction
|
|
| 6. |
|
| (a) |
| 32
÷ 42 x (3 - 8) |
| =
32 ÷ 42
x - 5 |
|
| =
32 ÷ 16 x -
5 |
|
| =
2 x - 5 |
|
| =
-10 |
|
|
| (b) |
| 81
÷ (4 - 7)3 |
| =
81 ÷ (-3)3 |
|
| =
81 ÷ - 27 |
|
| =
- 3 |
|
|
| (c) |
 |
Work out the square root first.
|
| =
11 ÷ 2 x 3 |
Then do the division and multiplication, working from left
to right.
|
= 5.5 x 3
|
|
= 16.5
|
|
| |
|
|
| (d) |
|
|
|
Treat
the numerator and the denominator as if they are in brackets. |
 |
Treat
the fraction bar as division. |
| |
|
|
| (e) |
|
|
|
Treat
the numerator and the denominator as if they are in brackets. |
|
Treat
the fraction bar as division. |
|
|
|
| 7. |
These
are the answers I got on my very simple calculator. The calculator
keystrokes are indicated by the grey boxes,
Your calculator may have required a different order of keystrokes.
All operations on my calculator had to be processed in the order
in which they were required, except where I could use the memory
to store and recall some values.
|
| (a) |
32
÷ 42 x (3 - 8)
|
|
-5
(My calculator does not have brackets) |
|
16
(My calculator does not have a 'raise to the power of '
key) |
|
10
|
| |
|
|
| (b) |
81
÷ (4 - 7)3
|
|
-3
(My calculator does not have brackets) |
|
-27
(My
calculator does not have a 'raise to the power of ' key) |
|
3 |
| |
|
|
| (c) |
|
| (d) |
|
|
I
have worked out the denominator first and stored it in memory
so that I can then work out the numerator and then divide
it by the memory value. |
|
0.25
|
|
| (e) |
|
|
I
have worked out the denominator first and stored it in memory
so that I can then divide the numerator by the memory value. |
|
1 |
|
| 8. |
Bernie
is in the process of landscaping the gardens of two new townhouses.
If he buys 30 bundles of 12 wooden planks for the fence for
each house and 15 bundles of 10 hardwood planks for the decking
for each house, write an expression for the total number of
planks bought and then work it out. If Bernie then returned
2 bundles of the wooden fence planks but bought 5 extra bundles
of the hardwood planks, write a new expression and then work
out the answer.
The
original purchase, expressed in mathematical notation.
| 2
x (30 x 12 + 15 x 10) |
|
| =
2 x (360 + 150) |
|
| =
2 x 510 |
|
| =
1020 |
|
The
original purchase with the returned bundles and extra bundles
taken into consideration.
| 2
x (30 x 12 + 15 x 10) - 2 x 12 + 5 x 10 |
| = 2
x (30 x 12 + 15 x 10) - (2 x 12) + (5 x 10) |
|
| =
1020 - 24 + 50 |
|
| =
1046 |
|
|
| 9. |
Two
thirds of all Year 8 students, one quarter of all Year 9 students,
only 30 Year 10 students and two fifths of Year 11 and 12 students
combined ride their bike to school. If there are 99 Year 8 students,
124 Year 9 students, 111 Year 10 students, 65 Year 11 students
and 50
Year 12 students attending the school, how many students ride
their bike to school.
|
|
|
= 66 + 31 + 30 + 46
|
|
= 173
|
|
|
| 10. |
(300
÷ (10 x 2)) x 4
Making
the problem:
300
couches have to be delivered to each of the 4 Coucharama
stores in Fitzroy, Nunawading, Richmond and Moorabbin today.
If each delivery truck can carry 10 boxes of 2 couches and only
makes 1 delivery run, how many delivery trucks are needed?
Working
out the problem:
|
300
÷ (10 x 2) trucks required for each store
(300
÷ (10 x 2)) x 4 trucks required for the 4 stores
|
This
is a quotition division problem: we know the number in
each group and we want to find out the number of groups.
(see division section)
300
couches are to be delivered to each store in groups (trucks)
of 10 x 2. So 300 needs to be divided by (10 x 2), to
get the number of trucks required to carry the 300 couches
to each store.
We
know there are 4 stores, so the whole expression needs
to be multiplied by 4.
|
|
|
| (300
÷ (10 x 2)) x
4 |
Do
innermost brackets first |
| =
(300 ÷ 20) x
4 |
Do
brackets first |
| =
15 x 4 |
|
| =
60 |
|
| 60
trucks are required to deliver the couches, 15 for each
store. |
|
|
|
|
|
©
University of Melbourne
2003 |
