exercise1.1 whole number problem solving - hi.com.au · 1 whole numbers 3 until now you would no...

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Until now you would no doubt have spent many hours practising the four basic operations of mathematics—addition, subtraction, multiplication and division. When problem solving, you need to decide which operation applies, write down the expression, then use your calculator to obtain the answer. The solution is often written in words.

1

The highest mountain in the world, measured from sea level, is the Himalayan peak of Mount Everest. It is 8848 m above sea level. If we measure mountains that start under the ocean, the highest mountain in the world from base to tip is Mauna Kea on the island of Hawaii. Its total height is 10 203 m, of which 4205 m is above sea level.

(a)

How much higher is Mauna Kea than Mount Everest?

(b)

How much of Mauna Kea is below sea level?

(c)

If we don’t count the part of Mauna Kea that is under water, how much higher is Mount Everest?

On 1 January the population of Summertown was 55 234. During the year, 1987 people died, 1245 babies were born, 4324 people left the town, and 3876 moved in. Find the total number of people in the town at the end of the year.

Steps Solution

1. Add together the number of people who died and the number of people who left.

Total decrease:1987 + 4324= 6311

2. Add together the number of people who were born and the number of people who moved into the town.

Total increase:1245 + 3876= 5121

3. Subtract the total who died and left from the original population and add the total of babies born and people who moved in.

Final population:55 234 − 6311 + 5121= 54 044

4. Write the solution in words. The final population was 54 044.

worked example 1

exercise 1.1 Whole number problem solving

Preparation: Prep Zone Q1

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The two longest rivers in the world are the Amazon (6448 km) and the Nile (6670 km). The longest river in Australia is the Darling (2739 km).

(a)

How much longer is the Nile than the Amazon?

(b)

How much longer is the Nile than the Darling?

(c)

How much longer is the Amazon than the Darling?

3

If a car uses 8 L of petrol for every 100 km that it travels, how many litres would it use for a trip of:

(a)

900 km?

(b)

1200 km?

(c)

1500 km?

(d)

1050 km?

(e)

725 km?

(f)

620 km?

4

Harvey ‘Scoop’ Roberts, a journalist with the

Monthly Farm News

, can type 50 words a minute. How long does it take him to type an article of 1800 words?

5

Arnie the body builder stands on a set of scales while he is holding two 3 kg dumb-bells. The scales show a weight of 102 kg. How much does Arnie weigh?

6

Stavros wants to buy some marmalade. A jar in the supermarket is 250 g and costs $1.50; another is 500 g and costs $2.70. Which is better value?

7

Little Lucy is only 4 weeks old. How many minutes old is she?

8

Jules Verne wrote about travelling around the world in 80 days. About how many weeks is that?

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he length of a painting, including the frame, is 85 cm. If the frame is 6 cm wide all the way around, what is the length of the unframed painting?

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The Pizza Pit-Stop employs five people. The two cooks work 36 hours each per week for $12 an hour, and the three waiters work 30 hours each per week for $11 an hour. What does the Pizza Pit-Stop pay its five employees in total per week?

11

Wendy is training to be an Olympic swimmer. Every morning she swims 3600 m in a 50 m pool. How many laps is that?

12

For the numbers 108 and 9, find:

(a)

the sum

(b)

the difference

(c)

the product

(d)

the quotient

13

What is the sum of 42 and 76 added onto the product of 42 and 76?

14

What is the result when the difference between 9864 and 8 is added onto the quotient of 9864 and 8?

15 Describe a problem whose solution involves the calculation 15

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6

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An estimate that you can do in your head or with very little work willoften be all that is required in real-life situations.

Rounding off to the leading digit

—the digit with the highest place value—is a convenient way to begin.

Estimate 368 × 52.

Steps Solution1. Round off both numbers to the first

(i.e. leading) digit.368 ≈ 400 52 ≈ 50

2. Multiply the first digits. 4 × 5 = 20

3. Count the number of zeros in step 1. three 0s (two in 400 and one in 50)4. Write the number from step 2 and put the

number of zeros from step 3 after it.400 × 50= 20 000

5. So 368 × 52 is approximately 20 000.This is written as 368 × 52 ≈ 20 000

worked example 2

Estimate 67 483 ÷ 421.

Steps Solution1. Round off both numbers to the

first (i.e. leading) digit.67 483 ÷ 421

≈ 70 000 ÷ 400

2. Write the quotient as a fraction and cancel off the same number of zeros on top and bottom. =

3. Perform the division. 1 7 5

44. So 67 483 ÷ 421 is approximately 175.

This is written as 67 483 ÷ 421 ≈ 175

70 000400

------------------ 7004

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73020

worked example 3

When you complete a calculation, think: ‘Does this answer make sense?’


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1 Use rounding to the first digit to find the estimates for these multiplications.(a) 681 × 41 (b) 547 × 84 (c) 141 × 837(d) 104 × 8946 (e) 650 × 23 (f) 62 × 819(g) 38 944 × 771 (h) 7340 × 250 (i) 950 × 3489(j) 680 × 95 (k) 9 × 6511 (l) 8010 × 6(m) 65 000 × 70 (n) 56 439 × 9 (o) 95 × 75 000

2 In each case, choose the best estimate from the alternatives given. Don’t do the actual multiplication.(a) 321 × 73

A 210 B 643 C 2100 D 2163 E 21 000(b) 56 × 354

A 1200 B 2400 C 1500 D 15 000 E 24 000(c) 4570 × 429

A 2 000 000 B 1 600 000 C 180 000 D 160 000 E 200 000(d) 6500 × 78

A 480 000 B 420 000 C 350 000 D 42 000 E 560 000(e) 405 × 950

A 400 000 B 450 000 C 360 000 D 500 000 E 3 600 000

3 Write three different products for which the estimate would involve 20 000 × 60.

4 Use rounding to the first digit to find estimates for these quotients.(a) 2940 ÷ 41 (b) 3199 ÷ 62 (c) 8742 ÷ 31(d) 1955 ÷ 78 (e) 29 110 ÷ 59 (f) 52 511 ÷ 37(g) 5218 ÷ 8 (h) 7532 ÷ 4 (i) 44 895 ÷ 15(j) 75 342 ÷ 80 (k) 94 101 ÷ 60 (l) 10 803 ÷ 95(m) 95 000 ÷ 542 (n) 36 534 ÷ 35 (o) 3 082 817 ÷ 19

5 In each case, choose the best estimate from the alternatives given. Don’t do the actual division.(a) 7865 ÷ 24

A 768 B 900 C 1600 D 210 000 E 400(b) 7546 ÷ 84

A 100 B 1000 C 640 000 D 560 000 E 10(c) 5500 ÷ 29

A 350 B 200 C 700 D 50 E 3000(d) 99 160 ÷ 527

A 740 B 2000 C 330 D 660 E 200(e) 126 905 ÷ 9500

A 100 B 10 C 10 000 D 300 E 500

exercise 1.2 Estimating products and quotients

Preparation: Prep Zone Q2

Worksheet C1.1e

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Try keying this directly into your calculator: 20 − 10 × 2If you used a scientific calculator, you should get 0 as the answer. This

is because a scientific calculator does the multiplication first and then the subtraction. A simple calculator may give the answer 20. This is because a simple calculator does the subtraction before the multiplication.

Scientific calculators use order of operations automatically, which is not always built into simpler calculators.

If using a simple calculator you will have to know which parts to do first.

Order of operations rules

1. Brackets

2. Do × and ÷ in the order in which they appear.

3. Do + and − in the order in which they appear.

Simplify 24 + 6 ÷ 2 − 1 × 4.

Steps Solution1. Do multiplication and division first. 24 + 6 ÷ 2 − 1 × 4

= 24 + 3 − 4

2. Do addition and subtraction in the order in which they appear.

= 27 − 4= 23

worked example 4

Simplify 12 − 9 + 8 ÷ (2 + 2) × 3.

Steps Solution1. Do the brackets first. 12 − 9 + 8 ÷ (2 + 2) × 3

= 12 − 9 + 8 ÷ 4 × 3

2. Do multiplication and division in the order in which they appear.

= 12 − 9 + 2 × 3= 12 − 9 + 6

3. Do addition and subtraction in the order in which they appear.

= 3 + 6= 9

worked example 5



With brackets we follow these rules:• If there is more than one operation within a set of brackets, we follow the

ordinary order of operations rule.• If there are brackets inside brackets, we do the inside brackets first.

1 Simplify the following.(a) 6 × 2 − 1 (b) 8 ÷ 4 − 2(c) 7 + 6 ÷ 2 (d) 1 + 8 × 3(e) 15 − 8 ÷ 4 (f) 8 − 5 ÷ 5(g) 8 + 3 × 10 (h) 25 − 2 × 11(i) 6 ÷ 3 + 3 × 5 (j) 8 × 5 − 4 × 10(k) 9 − 6 ÷ 2 + 7 (l) 8 − 24 ÷ 12 + 3(m) 9 × (10 − 7) ÷ 3 (n) 24 ÷ (7 + 5) × 6(o) 88 ÷ 8 − 6 × (5 − 4) (p) 12 × 5 + 4 × (10 − 4)(q) 18 − 7 × 2 + 13 − 4 ÷ 2 (r) 9 − 2 + 5 + 3 × 4 ÷ 6(s) 28 − 7 × 3 + (5 − 1) ÷ 2 + 3 (t) 23 − 5 + (17 − 2) × 3 + 5

2 Write and evaluate three different expressions using the numbers 2, 5 and 12, the operations + and × and one set of brackets.

3 Replace each with either �, � or = to make the statement true.(Remember: � means ‘is less than’ and � means ‘is greater than’.)(a) (1 + 4) × 20 ÷ 5 1 + (4 × 20) ÷ 5(b) 6 × (4 ÷ 2) × 3 (6 × 4) ÷ 2 × 3(c) 8 + (5 − 3) × 2 8 + 5 − (3 × 2)(d) 100 + 10 ÷ 10 (100 + 10) ÷ 10(e) 9 × 2 + 0 9 × (2 + 0)(f) 36 ÷ 6 × (3 − 3) 36 ÷ 6 × 3 − 3

(a) Simplify (5 + 3 × 2) × 4. (b) Simplify 6 × [(4 − 3) × 2].

Steps Solutions(a) 1. Do the brackets first. Within the bracket

follow the order of operations rule, so do the multiplication first, then the addition.

(a) (5 + 3 × 2) × 4= (5 + 6) × 4= 11 × 4

2. Do the remaining operation, multiplication. = 44

(b) 1. Do the brackets first. In this case this is [ ]. Within the [ ] brackets follow the order of operations rule, so do the inside brackets, ( ), then the multiplication.

(b) 6 × [(4 − 3) × 2]= 6 × [1 × 2]= 6 × 2

2. Do the remaining operation, multiplication. = 12

worked example 6

exercise 1.3 Order of operationsPreparation: Prep Zone Q3, Q4

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Worksheet C1.2eHinte

Homework 1.1e



DivisibilityWe often want to know if a smaller number will divide evenly into a larger number. If this occurs, we say the larger number is divisible by the smaller number. As an example, 10 is divisible by 5. On the other hand, 24 is not divisible by 9 because 24 ÷ 9 results in a remainder. We can use a calculator to check this but often we can find out more quickly by using some mental shortcuts.

1 (a) Which of these numbers are divisible by 5? 23, 65, 92, 10, 104, 234 625, 870, 88

(b) What sort of numbers are divisible by 5? Can you find a pattern?

2 (a) Which of these numbers are divisible by 10?70, 71, 5, 640, 235, 41 960, 500


3 (a) Which of these numbers are divisible by 2?19, 461, 2, 227, 56, 27 560, 24, 195, 768


Use a calculator for Questions 4 to 6.4 (a) Which of these numbers are divisible by 3?

247, 21, 64, 783, 6732, 9076, 34, 56 342, 798, 1223(b) What sort of numbers are divisible by 3?

Can you find a pattern?

5 (a) Which of these numbers are divisible by 9?81, 679, 2999, 82, 5634, 220 221, 87 984, 16 668, 562

(b) What sort of numbers are divisible by 9?Can you find a pattern?

6 (a) Which of these numbers are divisible by 4?516, 7612, 311, 608, 61, 64, 5364, 38 921, 500


Worksheet C1.3e

Try adding the digits.2 + 4 + 7 =

Look at the pattern you found for 3.

Look at the last two digits. 16, 12.



7 The shortcut to test if a number is divisible by 6 is to see if it is divisible by both 2 and 3.

(a) Use the test to see which of these numbers are divisible by 6. 436, 321, 132, 741, 8760, 4529, 3528, 705 630, 11 112

(b) The shortcut for 12 is similar to the shortcut for 6. What do you think the pattern is for seeing if a number is divisible by 12?

8 Copy and complete the following tests for divisibility.

State TRUE or FALSE for each of the following.(a) 346 is divisible by 3. (b) 6872 is divisible by 6.(c) 548 348 is divisible by 2. (d) 552 is divisible by 4.(e) 18 342 is divisible by 9. (f) 5 633 902 is divisible by 3.(g) 4 332 112 is divisible by 5. (h) 56 432 is divisible by 2.(i) 67 432 is divisible by 6. (j) 3935 is divisible by 5.

Number Divisibility test

2 Look at the digit only. If it is or zero, then the number is divisible by 2.

3 up all the digits and see if the is divisible by 3. If it is, then the original number is divisible by 3.

4 Look at the number formed by the last digits only. If this number is divisible by 4, then the number is divisible by 4.

5 Look at the digit only. If it is or , then the number is by 5.

6 Do two tests. See if the number is divisible by and .

9 up all the and see if the is divisible by 9. If it is, original number is by .

10 Look the only. If it is , then number is .

One of the numbers involved is 3.



Number sequences A number sequence is a list of numbers that form a pattern. The numbers that make up the sequence are called terms.

MultiplesMultiples of a number can be thought of as the answers to the ‘times tables’ for that number.

For example, the times table for 7 starts:1 × 7 = 72 × 7 = 143 × 7 = 21 7, 14, 21, 28, 35 are some of the multiples of 7.4 × 7 = 285 × 7 = 35

Another way of finding multiples of 7 is to count by 7s:7, 14, 21, 28, 35, . . .

9 Copy the table and do the divisibility tests. Circle the number if the original number is divisible by it. The first one has been done for you.

100 000202 008

12 121 212300 300 300

7 500900 090

123 456 7892 5643 429

2 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 102 3 4 5 6 9 10

eTutoriale

The list of multiples for any number will continue forever.{

The first multiple of any number is the number itself. eTutoriale



Lowest common multiplesThe lowest common multiple, or LCM, of two or more numbers is the lowest number that is in the multiples list of all of the given numbers.

Factors A factor is a number that divides into another number exactly, with no remainder. For example, 2 is a factor of 12 since 2 ‘goes into’ 12 (six times), with no remainder. One of the simplest ways to find all the factors of a number is to write all the pairs of numbers that multiply to give the original number.

Find the lowest common multiple of 4 and 6.

Steps Solution1. List several multiples of 4. 4, 8, 12, 16, 20, 24, ...

2. List several multiples of 6. 6, 12, 18, 24, 30, 36, ...

3. Write down the numbers that are in both lists. These are the common multiples.

12, 24, ...

4. Write the smallest one of these. It is the lowest common multiple.

LCM = 12

worked example 7

eTutorialeeQuestionse

It is not possible to find the highest common multiple. Why not?

The largest factor of any number is the number itself.

Find all the factors of 12.

Steps Solution1. Find all pairs of numbers that multiply to

give the original number.1 × 12 = 122 × 6 = 123 × 4 = 12

2. List the factors in numerical order. The factors of 12 are 1, 2, 3, 4, 6, 12.

worked example 8

eTutoriale



We can tell we have listed all the pairs of numbers if we follow a pattern in finding the pairs of numbers. In Worked Example 8 we start on the left with 1, dividing each number into 12 to complete the pair. When we get to 3 we find that the number paired with it is 4, so there is no point in trying 4 or larger numbers as factors.

Common factors and HCF A common factor is one that appears in the factor lists of two or more given numbers. The highest common factor (HCF) is the largest of the common factors. Every pair of numbers has the same lowest common factor. It is 1, as 1 is a factor of every number.

1 Copy these sequences and write down the next three terms in each.(a) 1, 3, 5, . . . (b) 24, 22, 20, . . . (c) 10, 15, 20, . . .(d) 1, 2, 4, 8, . . . (e) 64, 32, 16, . . . (f) 8, 1, 7, 2, 6, . . .(g) 54, 8, 56, 6, 58, 4, . . . (h) 1, 2, 4, 7, . . . (i) 34, 33, 31, 28, . . .(j) 12, 22, 33, 45, . . . (k) 1, 1, 2, 3, 5, . . . (l) 1, 2, 5, 14, . . .

2 Write down the first five terms of the sequences that follow these rules.(a) Start with 5 and add 3 each time.(b) Start with 29 and take away 4 each time.(c) Start with 1 and multiply by 3 each time.(d) Start with 80 and divide by 2 each time.(e) Start with 3 and multiply by 2, then add 1 each time.(f) Start with 1 and multiply by 4, then subtract 2 each time.

3 Find the first five multiples of each of these.(a) 2 (b) 3 (c) 4 (d) 5 (e) 6 (f) 8(g) 9 (h) 11 (i) 14 (j) 15 (k) 16 (l) 19

By writing the pairs of factors underneath each other we can easily list them in numerical order.

Find the common factors of 12 and 18, and state the highest common factor (HCF).

Steps Solution1. List all the factors of 12. 1, 2, 3, 4, 6, 12

2. List all the factors of 18. 1, 2, 3, 6, 9, 18

3. Pick out the factors appearing in both lists. The common factors are 1, 2, 3, 6.

4. Pick out the largest of these. This is the highest common factor.

HCF = 6

worked example 9

eTutorialeeQuestionse

exercise 1.4 Sequences, multiples and factors

Preparation: Prep Zone Q5, Q6


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4 Find the LCM of:(a) 5 and 6 (b) 2 and 5 (c) 8 and 12 (d) 7 and 9(e) 10 and 12 (f) 3 and 9 (g) 6 and 11 (h) 4 and 7(i) 5 and 25 (j) 16 and 24 (k) 21 and 28 (l) 20 and 50(m) 50 and 60 (n) 35 and 55 (o) 3, 4 and 5 (p) 2, 5 and 7

5 Choose the correct answer.(a) The lowest common multiple of 3 and 7 is:

A 3 B 6 C 21 D 14 E 7(b) The lowest common multiple of 5 and 10 is:

A 15 B 10 C 5 D 50 E 1(c) The lowest common multiple of 8 and 1 is:

A 8 B 4 C 10 D 80 E 1(d) The lowest common multiple of 5, 3 and 2 is:

A 1 B 30 C 10 D 6 E 15

6 Find all the factors of each of the following numbers.(a) 4 (b) 5 (c) 7 (d) 8 (e) 10(f) 9 (g) 13 (h) 11 (i) 16 (j) 18(k) 19 (l) 23 (m) 20 (n) 24 (o) 32(p) 36 (q) 30 (r) 60 (s) 77 (t) 55

7 Choose the correct answer.(a) Which of the following is a factor of 17?

A 7 B 14 C 34 D 17 E 2(b) Which of the following is a factor of 25?

A 3 B 2 C 5 D 50 E 250(c) Which of the following is a factor of 34?

A 4 B 12 C 17 D 24 E 8(d) Which of the following is a factor of 47?

A 17 B 1 C 9 D 94 E 3

8 Choose the correct answer.(a) Which of the following is not a factor of 15?

A 3 B 1 C 15 D 30 E 5(b) Which of the following is not a factor of 22?

A 22 B 4 C 1 D 11 E 2(c) Which of the following is not a factor of 14?

A 4 B 7 C 2 D 14 E 1(d) Which of the following is not a factor of 21?

A 1 B 21 C 7 D 3 E 14(e) Which of the following is not a factor of 33?

A 11 B 33 C 1 D 22 E 3(f) Which of the following is not a factor of 42?

A 14 B 8 C 7 D 6 E 3(g) Which of the following is not a factor of 50?

A 10 B 25 C 15 D 50 E 2(h) Which of the following is not a factor of 63?

A 1 B 63 C 21 D 3 E 13

eTestereWorksheet C1.5eHinte


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9 Find the highest common factor (HCF) for each pair of numbers.(a) 10 and 15 (b) 8 and 24 (c) 5 and 12 (d) 26 and 36(e) 11 and 33 (f) 28 and 70 (g) 44 and 22 (h) 10 and 30(i) 40 and 70 (j) 32 and 60 (k) 35 and 70 (l) 42 and 48

10 Find three different pairs of numbers whose HCF is 6.

11 Find the HCF for each group of numbers.(a) 8, 40 and 60 (b) 10, 44 and 99(c) 14, 77 and 90 (d) 32, 56 and 80

Worksheet C1.7e10

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The number gameYou will need: A calculator.Melinda and Craig were playing a game with their calculators. They would first agree on a target number and then see who could get closer to the target number by multiplying together two 2-digit numbers. They could not see what numbers the other person entered.

An example of the game is as follows:Target number: 8202 Melinda: 92 × 90 = 8280Craig: 96 × 87 = 8352

In this case Melinda would win because her guess was closer.

1 If the target number was 6151 and Melinda used 90 × 59 and Craig used 79 × 77, how far was each from the target number, and who won the round?

2 The target number was 2521 and Craig had used 50 × 50. If Melinda had entered 61, which number or numbers could she have finished with to win the round?

3 When the target was 4880 Melinda used 60 × 80. Find three different pairs of numbers Craig could have used to win the round.

4 The target number is 2649. Find two numbers that get you as close as possible to the target. Show the pairs of numbers you try in this exercise in search of the best pair.


exercise1.1 whole number problem solving - hi.com.au · 1 whole numbers 3 until now you would no...

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