Appendix

A p p e n d i x 697

Strategies forproblem solving

Introduction to problem solving ................................... 698

Strategies for investigation and problem solving ........ 699

Questioning skills............................................................ 699

Create a table, then look for a pattern or a result ......................................................................... 700

Draw a diagram, then look for a pattern or a result ......................................................................... 702

Use a pattern of numbers, making use of technology such as a computer spreadsheet................................ 703

Work backwards from the answer ........................... 704

Use a process of elimination ...................................... 706

Look at similar but simpler problems...................... 706

Use trial and error (guess and check), making use of technology such as a computer spreadsheet ............ 708

Communicating, reasoning and reflecting ................... 710

698 M a t h s Q u e s t 8 f o r V i c t o r i a

Introduction to problem solvingIn mathematics classes and in our everyday lives, we are presented with many problems

and situations that can be solved using mathematical processes. When we ‘solve

problems’, there are five main processes that we can use:

1. questioning

2. applying strategies

3. communicating

4. reasoning

5. reflecting.

To solve a simple problem, we may need to apply only two or three of the processes.

For example, imagine you are asked to write a simple formula to use on a scale map of

the world to convert from d (cm on a map) into k (km on Earth). You are told that the

map requires a scale of 1 : 2 000 000.

First you might question the purpose of the formula and whether to make the subject

of the formula d or k. You may think about the ratio 1 to 2 000 000 and make a sample

conversion using it to write down a possible conversion formula, say k = d × 20 or

d = . To test your formula you may generate a few conversions and then check with

the ratio 1 : 2 000 000 to see whether the results produced appear to be reasonable. Per-

haps you will question the advantages of using pencil and paper versus a spreadsheet

when generating a few conversions. Once you are happy with your first formula, you

may communicate your answer. You may then reflect on the two versions of the for-

mula with different subjects d or k. You may realise that one conversion formula

involves multiplying by the reciprocal of the number used in the other.

In this way, you would have focused on three processes:

1. questioning. What will be the subject of the formula? How best to generate and

display a set of conversions? Is my formula correct?

2. applying a strategy. In the process described above, a strategy was applied that

arrived at a formula and confirmed your findings.

3. reflecting. From sets of results produced by both conversion formulas, you observed

a connection between the two formulas.

The five processes in working mathematically are interrelated. By practising the

skills involved in using all five processes, you will learn to tackle new mathematics

problems with confidence and arrive at the correct and complete solution using the

most appropriate methods.

Questioning

Why do you think that teachers often say, ‘Read through the question a number of

times before you begin’? What does this achieve? The aim is to make you think

about the information that is needed and what will be the best approach to the

problem. You are questioning and looking ahead to what may be produced by certain

strategies. Be clear about the problem and the solution that is required. Consider

whether you have come across a similar problem before that could be useful. Is there

more than one method that you could use? Which method would be best to solve this

problem?

k

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S t r a t e g i e s f o r p r o b l e m s o l v i n g 699

Applying strategies

To apply a strategy when problem solving is to follow a plan or a method. The goal is

to complete an investigation, answer a question or find a solution to a problem. Exam-

ples of strategies that can be used to achieve this include:

1. drawing up tables and diagrams

2. finding patterns of numbers

3. making use of technology, such as a computer spreadsheet

4. working backwards from the answer

5. using a process of elimination

6. looking at similar but simpler problems

7. using trial and error (guess and check).

Communicating

Another person who reads your work should be able to follow your method or strategy.

It is important to learn to present data, explanations and solutions in a clear and concise

form, using correct mathematical terms and appropriate diagrams.

Reasoning

When you reach a solution to a particular problem, make sure that you check it and can

prove with mathematical reasoning that it is correct and that the method was appro-

priate.

Reflecting

Whenever you use the processes of problem solving, there is an opportunity to reflect

on what you did and why, and to improve your understanding of the strategies and

concepts. Consider whether the solution could have been obtained in a different or

better way. Learn from the experience and look forward with confidence to using the

knowledge gained when you are faced with the next problem or situation.

Throughout this textbook there are many questions and problems that can be

answered using the five processes. On the following pages, some examples and prac-

tice questions are supplied. These will help you to focus on improving your skills

relating to each process. The examples will also help you to become aware of the

need to use all the processes to arrive at the correct solution in the most appropriate

and efficient way.

Strategies for investigation and problem solving

Questioning skillsBefore beginning work on a problem, devise some questions that ensure you set off on

an effective path to solving it. Questions can help us to organise the information, test an

idea, add another key piece of information and/or decide what to do next. There may be

more than one question that could be asked.

The first important question is, ‘What is this problem asking me to do or find?’ In

order to answer this question, it is good practice to read the question at least twice.


Create a table, then look for a pattern or a resultA table is a way of organising or grouping numbers. You should consider the number of

rows and columns that will be needed and label them appropriately. A well-designed

table helps you to see any patterns or results in the numbers you have organised and

also demonstrates to others how you were able to arrive at your solution. There are

many different ways of presenting information in a table.

Sallie makes long telephone calls to her relatives overseas. The duration of each call made

last week is listed below. Use a table to calculate the total time spent on the telephone,

expressed in hours and minutes.

2 h 23 min, 1 h 57 min, 3 h 16 min, 59 min, 3 h 21 min, 44 min, 52 min.

THINK WRITE/DISPLAY

Read the question at least twice and take

note of all the important facts.

The duration of each of the seven individual

calls is listed.

Identify the solution required. The question asks to determine the total time,

expressed in hours and minutes, spent on the

phone.

Decide how the ‘amounts’ of time

can be arranged in a table.

Set up a spreadsheet and enter each

‘amount’ of time into separate columns.

Enter the hours in cells A2, A3, A4, A5,

A6, A7, A8; enter the minutes in cells

C2, C3, C4, C5, C6, C7, C8. Place the

heading ‘Hours’ in cell A1 and ‘Minutes’

in cell C1.

We need to total the hours column and

the minutes column.

In the cell at the bottom of the A column,

cell A9, enter =sum(A2:A8). This will total

the hours. Similarly, for the minutes

column in cell C9, enter =sum(C2:C8).

For the total of the minutes column, in

cell C9, we need to calculate the number

of whole hours (integer value of hours).

Alongside the total of the minutes, in cell

E9, enter =int(C9/60).

We need to know what’s left from the

amount in cell C9 when the whole hours,

expressed as minutes, are subtracted

from the current total of the minutes.

In cell G9 enter =C9-60*int(C9/60), and

then in cells D9, F9, H9 enter =, hours,

minutes.

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1WORKEDExample


Try these

Construct an active spreadsheet to solve each of the following problems.

1 A small drapery store had a stocktake. The lengths of cloth in the store were recorded

in feet and inches as shown below. We are advised that 12 inches equals 1 foot. Use a

table to calculate the total length of material in the store. Express your answer in feet

and inches.

14 feet 56 inches 9 feet 64 inches 12 feet 36 inches

24 feet 22 feet 56 inches 14 feet 53 inches

5 feet 8 inches 16 feet 23 inches 4 feet 54 inches

98 inches 5 feet 20 inches 5 feet 62 inches

2 A sports reporter was researching some famous football stars from the 1950s. The

football players were weighed and records kept. The weights were recorded in stones

and pounds. We are advised that 14 pounds equals 1 stone. From the following

records, construct a table to find the total weight of the 12 players. Express your answer

in stones and pounds.

18 stone 8 pounds 19 stone 4 pounds 13 stone 12 pounds




3 Your school is conducting a ‘Get fit’ campaign. Students are organised into groups of

12. Each student wears a pedometer and records the distance walked in a week. The

results for one group are recorded as follows. Find the total distance this group walked,

expressing your answer in kilometres and metres.

6.8 km 10.3 km 12.4 km

5.9 km 8.8 km 16.5 km

8.4 km 11.9 km 9.3 km

13.7 km 15.7 km 10.2 km

4 In a retirement village there are 9 residents who are over the age of 90. Their ages are:

90 years 6 months 92 years 1 month 94 years 3 months

91 years 2 months 96 years 4 months 90 years 5 months

90 years 11 months 95 years 10 months 97 years 7 months

What is the total age of these residents? Express your answer in years and months.

THINK WRITE/DISPLAY

We need to add all the hours and then

make a conclusion about time spent on

the telephone this week.

In cell A12 enter

=A9+E9&‘HOURS’&G9&‘MINUTES’ spent on

the telephone this week.

Your spreadsheet should appear as

shown. Notice that your table is an

active spreadsheet; if you change an

amount in columns A or C, the final

values will adjust accordingly.

Answer the question.The total time spent on the telephone this week

is 13 hours and 32 minutes.

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5 Peta is a great cook and buys her flour in 20 kg bags. Over the last few weeks she has

had a baking spree and recorded the quantities of flour used in her dishes.

1.2 kg, 750 g, 1.25 kg, 275 g, 125 g, 1 kg, 800 g, 2.2 kg, 950 g, 1.3 kg, 950 g, 1.8 kg

How much flour is left in her 20 kg bag? Express your answer in kilograms and grams.

Draw a diagram, then look for a pattern or a resultWhen information is represented in the form of a diagram, it can be easier to study all

of the information at once. There are many types of diagrams, so no single diagram is

necessarily the best.

The Rowe family currently have one male and one female guinea pig. Assume each litter of

guinea pigs will produce two females and two males. Also assume that a mating pair of

guinea pigs will have three litters per year and that new guinea pigs will be mature enough

to have their own litter when they are just 3 months old. Use a drawing to represent

growing numbers of guinea pigs and calculate how many guinea pigs there will be after

the second litter.

THINK WRITE



A pair of male (M1) and female (F1) guinea

pigs produce 2 female and 2 male (F2, F3, M2,

M3) guinea pigs in each litter. Each mating pair

produces 3 litters per year. New guinea pigs are

mature enough to mate at 3 months of age.

Identify the solution required. The question asks to find the number of guinea

pigs after the second litter.

Start a diagram by showing the original

female as F1, and original male as M1.

F1, M1

Connect F1, M1 to each member of the

litter, F2, F3, M2, M3.

Start a new diagram because now there

are three mating pairs, each producing

another litter of four.

The total number of guinea pigs must

be counted for a conclusion to be made.

Remember to include F1 and M1.

There are 18 guinea pigs after two litters.

Answer the question.

1

2

3

4

F1, M1

F2

F3

M2

M3

5

F1, M1

F4

F5

M4

M5

F3, M3

F8

F9

M8

M9

F2, M2

F6

F7

M6

M7

6

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2WORKEDExample


Try these

1 Nigel has one mating pair of rabbits. Assume that each litter of rabbits produces three

males and three females. Rabbits are mature enough to have their own litter by the time

their parents have another litter. How many rabbits will Nigel have after the second litter?

2 If the rabbits in question 1 produce only two males and two females per litter, how

many rabbits would Nigel have after the third litter?

3 There are six faces on a normal die, numbered 1 to 6. If such a die is rolled twice, how

many different combinations of numbers can result?

4 Jocelyn and Bernard have three children. They are all married and each has a daughter

and a son. The daughters are all married, each with one child. The sons do not have

children as yet. How many people are in Jocelyn and Bernard’s extended family?

5 A medical laboratory is studying the growth of a bacterium. A single bacterium splits

into two bacteria in 30 seconds. These two cells then each split in two in 30 seconds. If

this pattern continues, how many bacteria will there be after 5 minutes?

Use a pattern of numbers, making use of technology such as a computer spreadsheetRepetitive tasks are well suited to spreadsheets because once a spreadsheet is set up,

the repetitive tasks are achieved in an instant.

A spreadsheet can list patterns of numbers from which a result can be found.

Paul has been asked to demonstrate that the decimal number 5.462 multiplied by 29 can

be calculated by repeated addition using a spreadsheet.

THINK WRITE/DISPLAY

Read the question at least twice and take note

of all the important facts.

Using a spreadsheet, calculate 5.462 × 29

by repeated addition; that is, add 29

amounts of 5.462.

Identify the solution required. The question asks to obtain a spreadsheet

solution to 5.462 × 29 using repeated

addition.

Set up a spreadsheet and write a heading in

the first cell.

In cell A1 enter repeated addition.

We need to enter 5.462 in 29 cells from A2 to

A30 and add them up.

In cells A2, A3, A4 enter 5.462 then select

these cells and drag the mouse down all the

way to A30.

Add the numbers in the cells automatically.

In cell A31, enter =sum(A2:A30).

We need to multiply 5.462 and 29

automatically on the spreadsheet.

In cell C31 enter =5.462*29.

We need a heading for this cell.

In cell C30 enter Multiplication 5.462 x 29.

Answer the question. 5.462 × 29 = 158.398

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3WORKEDExample


Try these

1 Verity wants to use a spreadsheet to show that $783.57 × 43 can be calculated by

repeated addition using a spreadsheet.

2 Phillip wants to use a spreadsheet to show that × 43 can be calculated by repeated

addition using a spreadsheet. (Hint: First format the cells to accept fractions with three

digits in numerator and denominator, and enter the fraction by typing 3/57.)

3 Amelia wants to use a spreadsheet to show that 144 ÷ 3 can be calculated by repeated

subtraction of 3 from 144 using a spreadsheet. (Hint: Enter 144 in a cell A1, then enter

=A1–3 in cell A2, and enter =A2–3 in cell A3. Carefully select just cells A2 and A3 and

then drag downwards.)

4 Sonja wants to use a spreadsheet to show that 8

can be calculated using repeated

multiplication by . (Hint: Enter 1/2 in cell A1; then enter =A1*(1/2) in cell A2. Drag

this formula down from A2 to A8.)

5 It is common to use shortcuts when performing calculations. Consider the following

case: 6.293 × 9 = 6.293 × 10 – 6.293 × 1.

Use a spreadsheet to calculate the first term of the right-hand side by repeated

addition; then subtract the second term. Confirm that your answer is the same as the

left-hand side value. Set out your spreadsheet so that it is logical and clear.

Work backwards from the answer

If there is a sequence of steps for which we know the final result, then a useful strategy

may be to work backwards from this final result or answer. We start with the last step

of the sequence.

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Milo arrived at his desk on Monday morning to find that his in-tray was full of reports. He

spent all day processing and filing 25 of these reports. Overnight the clerical staff added

an extra 7 reports. On Tuesday Milo processed and filed 19 reports, but there were still 56

reports left in his in-tray. How many reports were in Milo’s in-tray on Monday morning?

THINK WRITE

Read the question at least twice and

take note of all the important facts.

Milo filed 25 reports on Monday. Seven reports

were then added to the in-tray that evening.

Nineteen reports were filed Tuesday, leaving 56

reports in the in-tray that evening.

Identify the solution required. The question asks to determine how many

reports were originally in the in-tray Monday

morning.

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4WORKEDExample


Try these

1 Mike works on a cattle station. At the start of the year 2005 there were a certain number

of cattle. Due to the drought, 15 cattle perished during summer. Fortunately there were

18 healthy calves born that year. Just before the cattle were rounded up to be counted,

3 strayed on to a major highway and were killed. When the cattle were finally counted

later in 2005, there were 245 animals. How many were there at the start of the year 2005?

2 Rae opened a savings account with a certain amount of money earning interest

monthly. During October, $2.65 interest was added. In November, Rae withdrew $450

for Christmas gifts and other expenses. The November interest added was $2.44. Rae

was paid $1048 for a week’s work during December and this money was deposited into

the account. By Christmas, Rae had $1393 in the account. How much was there when

the account was opened?

3 Grace sells tropical fish. Last week Grace counted the fish and then she added 3 dozen

new fish. She sold 17 fish the next day. Over the following week fin rot claimed 13 fish,

which had to be thrown away. The next week Grace added another 10 fish. When she

then counted the fish in the tank there were 223. How many were in the tank when she

first counted them?

4 Diana had always wanted to have a large rose garden. When the local garden nursery

had a special on roses, she purchased as many as she could fit in her car. Unfortunately,

4 of the plants died during the first week. In anticipation of further losses, Diana pur-

chased another dozen. A harsh winter claimed the life of another 5 roses. Her six chil-

dren came to the rescue by each giving her a rosebush for her birthday. The number of

roses in the garden now totalled 57. How many plants did Diana buy initially?

5 Dan constantly struggled to deal with all his email at work, trying to answer his client’s

queries as quickly as possible. Monday morning he arrived at work to find his inbox

needing urgent attention. He started by deleting all the spam email — 57 in all. By the

end of the day he had answered 35 genuine work enquiries. He also answered 10 per-

sonal emails. During the day he received another 15 emails, 5 of which were spam

emails, which he immediately deleted. Tuesday morning he arrived at work to find

another 18 emails had been received overnight. Dan now had 38 emails in his inbox.

How many emails were in Dan’s inbox on Monday morning?

THINK WRITE

The last facts we know are that on

Tuesday, Milo took 19 reports from his

in-tray and there were 56 reports left.

This means that 56 + 19 gives the

number at the start of Tuesday.

56 + 19 = 75 at the start of Tuesday

Overnight from Monday to Tuesday,

the clerical staff added 7 reports.

Before that there must have been 7

fewer reports in the in-tray.

75 − 7 = 68 at the end of the day on Monday

Milo removed, processed and filed 25

reports on Monday.

68 + 25 = 93 at the start of Monday

Answer the question. There were 93 reports in the in-tray at the start

of Monday.

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Use a process of eliminationWhen using a process of elimination we remove or eliminate possible solutions that do

not match the given information. We first write all the possible combinations or solu-

tions in a grid or table. From the information supplied, we cross out (eliminate) those

combinations that do not match.

Try these

1 Mr Bateaux will give away a diamond ring to the first person that correctly guesses its

value. Here are the clues. The amount is a multiple of $50 and is more than $500 but

less than $2000. It is not a multiple of $200, and it is not a multiple of $350. The first

digit is a prime number. One of the digits is repeated.

2 Find the terminating decimal that has three digits after the decimal point and lies

between 0.5 and 0.6. No digits are repeated. The first two digits are prime. The digits

are increasing in value from left to right. The third digit is a multiple of 3.

3 Find an integer between 50 and 80 that is both a perfect square and a perfect cube.

4 What two numbers multiply to give –12 and add to give –11?

5 Triangular numbers are those whose dots form the pattern of a triangle, for example,

1, 3, 6, 10, … Find a triangular number below 50 that is also a square number.

Look at similar but simpler problemsIf you are overwhelmed by the size of the numbers involved in a question, try to solve

a similar but simpler question. This can be achieved by changing the numbers in the

original question to smaller numbers. After finding the answer to the simpler question,

the same method can be used to solve the original problem.

Harvey is solving a riddle. He needs to find a number between 20 and 30 that is not odd,

is not a multiple of 4, and is not a multiple of 13.

THINK WRITE



The clues concerning the required number

are: it is between 20 and 30; it is not odd; it

is neither a multiple of 4 nor a multiple of

13.

Identify the solution required. The question asks to find the required

number.

List the numbers from between 20 and 30. 21, 22, 23, 24, 25, 26, 27, 28, 29

Eliminate (strike through) the odd

number(s) that are odd.

21, 22, 23, 24, 25, 26, 27, 28, 29

Eliminate the number(s) that are multiples

of 4; that is, 24, 28.

21, 22, 23, 24, 25, 26, 27, 28, 29

Eliminate the number(s) that are multiples

of 13; that is, 26.

21, 22, 23, 24, 25, 26, 27, 28, 29

The remaining number is the answer. The answer is 22.


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5WORKEDExample


Try these

1 After 73 days, a sailing boat had travelled 1264 nautical miles. How much longer

would it take to reach the target of 2000 nautical miles?

2 In a random sample of 64 823 cars, exactly 1741 had defective brake lights. How many

cars would you expect to have defective brake lights in a sample of 68 cars?

After 57 days, a team installed 153.6 km of fibre-optic cable. How much longer would it

take to reach the target of 170 km?

THINK WRITE



It takes 57 days to install 153.6 km of cable; a

total of 170 km must be installed.

Identify the solution required. The question asks to find the time taken to

reach the target length of 170 km.

Consider a similar but much easier

question.

Consider a team that takes 2 days to install

6 km of cable with a target of completing 8 km.

Calculate the time taken to install 1 km

of cable in the simplified problem.

Time to install 1 km of cable

= day

= day

Calculate the remaining length of cable

still to be installed.

Remaining length of cable to install

= 8 − 6

= 2 km

Use the time taken to install 1 km of

cable to calculate the time needed to

install the remaining length of cable.


= × 2

= day

Repeat the same method to work out

the answer for the original question.

First list the given information.

In 57 days, 153.6 km of cable is installed. Need

to install a total of 170 km of cable.

Calculate the time taken to install 1 km

of cable.


= day

= day

= (or approximately 0.37 day)

Calculate the remaining length of cable

still to be installed.

Remaining length of cable to install

= 170 − 153.6

= 16.4 km

Use the time taken to install 1 km of

cable to calculate the time needed to

install the remaining length of cable.

Time to install 16.4 km of cable

= × 16.4

= 6.0859375 days

Answer the question. To reach the target of installing 170 km of

cable, approximately 6 more days are needed.

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1

3---

5

6

1

3---

2

3---

7

8

57

153.6-------------

570

1536------------

95

256---------

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10

95

256---------

11

6WORKEDExample


3 A popular website has been visited a total of 15 670 234 times in 28 hours. How many

times might the website be visited every 5 seconds?

4 On a charity walk, Gordon covered the 150 km distance in 22 hours 12 minutes. At this

rate, how far did Gordon walk every 15 minutes?

5 The school’s Building Fund hopes to raise $2 000 000 for a new swimming pool and

sporting complex. During one month $24 892 was raised. How long would it take to

raise the $2 000 000 at this contribution rate?

Use trial and error (guess and check), making use of technology such as a computer spreadsheetSometimes it may not be easy to solve a problem directly; in this case we can use a

strategy by which we guess at the solution. We test this value (guess), using the avail-

able information supplied in the problem, to check whether it is the solution. Even if it

is not the solution to the problem, this process provides us with further information that

we can use to try another, better-informed guess.

We can continue to guess and check until we reach the solution. Since this can be a

lengthy process, we can use technology such as a spreadsheet to provide instant feed-

back on our checking.

Chocolates were distributed among three groups of children. The second group received 4

times the number of chocolates of the first group. The third group received 10 more

chocolates than the second group. One hundred and nine chocolates were distributed

altogether. How many chocolates did the first, the second and the third group receive?

THINK WRITE/DISPLAY



Three groups of children receive chocolates.

Group 2 receives 4 times as many chocolates as

group 1.

Group 3 receives 10 more chocolates than

group 2.

In total 109 chocolates are distributed.

Identify the solution required. The question asks to find the number of

chocolates group 1, group 2 and group 3 have

received.

Since the number of

chocolates received by

the first group is an

unknown value, guess

any number (say 20). In

a spreadsheet, type the

heading 1st group

chocolates in cell A1.

Enter 20 in cell A2.

1

2

3 A

First group chocolates1 Second group chocolates Third group chocolates Sum of chocolates

2 = A2*4 = B2 + 10 = A2 + B2 + C2

203 80 90 190

104 40 50 100

115 44 54 109

B C D

7WORKEDExample


Try these

1 Chocolates were distributed among three groups of children. The second group

received 4 more chocolates than the first group. The third group received 3 times the

number of chocolates of the second group. One hundred and one chocolates were dis-

tributed altogether. How many chocolates did the first, the second and the third group

receive?

2 Toula observed the number of times she saw a red car, white car or blue car pass

through an intersection. She kept a tally of these colours. She ignored all other colours.

Out of 219 cars, there were 13 more white cars than red cars, and the number of blue

cars was the number of white cars. What was the most popular colour, and how

many did Toula see of that colour?

3 Four people, Max, Kim, Lilla and Harvey, decided to pool their money and hire a taxi.

They each contributed $1 or $2 coins. Together they had $66. Kim had $5 less than

Max. Lilla had $3 more than Max and Harvey had half as much as Lilla. How much

did Harvey contribute?

4 Three integers have a sum of 50. The second integer is four times as large as the first

integer and the third integer is 4 less than the second integer. What are the three inte-

gers?

5 Each digit in a four-digit number is a prime number. The first digit is the smallest digit.

The third digit is two smaller than the last digit and the number is a multiple of 5. What

is the four-digit number?

THINK WRITE/DISPLAY

The second group’s number of

chocolates is 4 times the first group’s

number of chocolates; so type the

heading second group chocolates in cell

B1 and enter =A2*4 in cell B2.

The third group’s number of chocolates

is 10 more than the second group’s

number of chocolates so type the

heading third group chocolates in cell C1

and enter =B2+10 in cell C2.

As we know that the sum of the

chocolates is 109, type the heading sum

of chocolates in cell D1 and enter

=A2+B2+C2 in cell D2.

Highlight the cells A2 to D2, and drag

down to copy. Enter the first guess —

20 — and check cell D3. This answer is

not 109; so, in the next cell down, enter

a smaller number and repeat until 109

appears in column D.


The first group has 11 chocolates, the second

group has 44 chocolates and the third group has

54 chocolates.

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Communicating, reasoning and reflecting

Once you have applied a strategy, there remains the task of communicating your find-

ings. (You are usually expected to support your findings with some reasoning.) It is

important to understand that solving problems involves much more than just writing

numbers on a page. When you solve a problem, you reason, or think in a rational way,

and the words and numbers that you use should accurately show the reasoning you fol-

lowed to solve the problem. Your words should be in an appropriate form of English

and use correct mathematical terms. Remember, however, that communicating well is

important in general, not only when solving mathematical problems! After providing

your solution, it is good practice to review it to see whether another person could

understand your work without first reading the question.

Finally, if you take the time to reflect on your work, you may increase your under-

standing of the problem and the strategies you used to solve it. You may be able to con-

nect this to previous experiences as well as to future problems you will have to tackle.

Let us review the processes suggested last year. To solve problems successfully, the

following steps should be followed.

1. Read the question at least twice and take note of all the important facts.

2. Identify the solution required.

3. Decide on and apply an appropriate strategy (for example, create a table, draw a dia-

gram, use technology, work backwards, use a process of elimination, look at similar

but simpler problems, use trial and error).

4. Communicate the whole solution using appropriate language and mathematical

terms.

5. Support the solution with mathematical reasoning.

6. Reflect on the solution — does it answer the question and does it make sense?

Abdul is updating prices on all the stationery items in the shop. He adds a 20% profit

margin and a further 10% GST. How can he easily update the price in one calculation?

THINK WRITE

Read the question at least

twice and take note of all the

important facts.

A 20% profit margin is to be added on all items fol-

lowed by a further 10% GST.

Identify the solution required. The question asks to find the percentage equivalent to

these two percentages.

Consider an appropriate

strategy. In this case, define a

pronumeral to be used to

represent the original price.

Let x = original price of a stationery item.

1

2

3

8WORKEDExample


Try these

1 Ten children were using buckets to fill a drum with water from a creek. On their first

trip they brought the following quantities.

6.8 L, 8.5 L, 7.7 L, 8.9 L, 9.5 L, 7.6 L, 8.4 L, 9.3 L, 8.1 L, 7.9 L

They tipped their buckets of water into the drum. Use a spreadsheet to determine the

volume of water in the drum at this stage. Express your answer in litres and millilitres.

2 The canteen sells sandwiches on white, brown or grain bread. The filling can be eitheregg, cheese, chicken or ham. These can be served with tomato sauce, BBQ sauce orno sauce. How many different types of sandwiches are available at the canteen?

THINK WRITE

Perform the first of the two

separate calculations.

Adding a 20% profit margin:

x + 20% of x or 120% of x = × x

= x + × x = 1.2 × x

= x + 0.2x = 1.2x

= 1.2x

Perform the second of the two

separate calculations.

Adding a further 10% GST:

1.2x + 10% of 1.2x or 110% of 1.2x = × 1.2x

= 1.2x + × 1.2x = 1.1 × 1.2x

= 1.2x + 0.1 × 1.2x = 1.32x

= 1.2x + 0.12x

= 1.32x

Devise a question that will lead

Abdul to the required

equivalent single calculation.

How can I relate 1.32x as a percentage increase?

First express 1.32 as an

improper fraction and then as

an equivalent percentage.

Hence, find the required

percentage increase.

so x1.32 =

so x1.32 = 132%

so 1.32x = 132% of x.

This means that x has been increased by 32%.

Communicate the answer with

reasoning.

Abdul can easily update the price of any stationery item

by increasing the price by 32%. This he can do by multi-

plying the price by 1.32.

We can check this by considering an item that costs

$5.00. Applying a 20% increase gives

1.2 × $5.00 = $6.00; then a further increase of 10%

gives 1.1 × $6.00 = $6.60. A single calculation of

applying a 32% increase gives 1.32 × $5.00 = $6.60.

Reflect on the solution. It would be tempting to think that successive increases

of 20% and then 10% would be the same as increasing

by 30%. Performing each calculation separately with a

pronumeral to represent the original price lets us work

back to the single calculation shortcut.

4120

100---------

20

100---------

5110

100---------

10

100---------

6

7 132

100---------

8

9


3 Use a spreadsheet to show that 48 can be calculated by repeated multiplication.

4 At the beginning of the year, Sue noticed that her cupboard was overflowing withshoes, many of which she hadn’t worn for some time. This prompted her to clear outeight pairs of unwanted summer shoes and buy two pairs in the latest style. As winterapproached, Sue purchased two new pairs of boots and discarded four pairs of oldwinter shoes. With the Christmas holidays approaching, Sue treated herself to threenew pairs of sandals. She now had eleven pairs of shoes. How many shoes were inSue’s cupboard at the beginning of the year?

5 Ken’s house number is a number between 50 and 100. It is a multiple of 3 and has aprime number as one of its factors. The sum of its digits is 15 and the first digit islarger than the second digit. Use a spreadsheet to determine Ken’s house number.

6 John’s pedometer showed that he had taken 65 423 paces and travelled a distance of42.5 km. At this rate, how far would John travel every 100 paces?

7 Three numbers have a product of 400. The second number is the cube of the first andthe third number is a square number between 20 and 30. Use a spreadsheet todetermine the three numbers.

8 Julius was asked to use a single calculation that will increase the selling price by 5%and then add GST of 10%. How can he easily update the price in one calculation?

9 Gemma was asked to use a single calculation that will decrease the pre-GST sellingprice of a car by 10% and then add GST of 10%. Find this single calculation.

10 Alfie had $x in the bank. He was given 5% interest and then charged $1. How can a

single calculation be made to obtain the final balance?

11 Jeff left a sum of money in a bank for exactly 3 years. The interest (8% p.a.) was

added to the account at the end of each 12 months. How can a single calculation be

made to obtain the final balance?

12 Write a simple formula to use on a house plan for converting millimetres on the plan

(d) into metres on the land (m). You are told that the plan has a scale of 1 : 2000.

13 Write a simple formula to calculate the number of pages (p) of writing that can be

stored on a writable CD with memory (d) (assume that one writable CD can store

approx 750 megabytes). Assume 1 page uses approx 40 kilobytes of memory.

14 Write a simple formula to calculate the number of millilitres of oil (m) to add to the

number of litres of petrol (p) when mixing lawnmower petrol. You are told that the

ratio of oil to petrol is 1 to 250.

15 Write a formula to calculate, V (the value in dollars and cents of a quantity of coins)

if we know the number of A (of 5 cent coins) and B (of 10 cent coins) and C (of

20 cent coins) and D (of 50 cent coins).

Documents

Appendix