| 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. 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

 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

 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

 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,

 3 - 8

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)

 3 - 8 = M+ CE
-5 (My calculator does not have brackets)

 4 x 4 = CE

16 (My calculator does not have a 'raise to the power of ' key)

 32 ÷ 16 = x MR

10

(b)

81 ÷ (4 - 7)3

 4 - 7 = M+
-3 (My calculator does not have brackets)

 x MR x MR =

-27 (My calculator does not have a 'raise to the power of ' key)

 81 ÷ 27 +/- =

3

(c)

 4 M+ CE
2

 11 ÷ MR = x 3 =

16.5

(d)

 8 x 2 = M+ CE
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.

 8 ÷ 2 = ÷ MR =

0.25

(e)

 4 x 7 ÷ 2 = M+ CE
I have worked out the denominator first and stored it in memory so that I can then divide the numerator by the memory value.

 14 ÷ MR =

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.