9
Problem
solving I
9A Introduction to problem
solving — create a table
9B Draw a diagram
9C Look for a pattern
9D Work backwards from the answer
9E Elimination
9F Simplify the problem
9G Guess and check
9H Mixed problems I
9I Mixed problems II
9J Mixed problems III
Kitchen
Dining
Lounge
WC
Bedroom
Bathroom
Scale 1:200
opening question
When this house was going up for sale,
two real estate agents were asked to
quote. The first said that they could
sell the house for $180 000 and would
charge 2.5% commission on the sale.
The second said that they could sell the
house for $178 000 and would charge 2.3%
commission. If all other costs were equal,
which company was offering the better
deal?
9a
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More Maths
problems
Introduction to problem
solving — create a table
Introduction to problem solving
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When solving problems, the main processes that we can use are as follows:
1. Read the question at least twice and take note of all the important facts.
2. Identify the solution required.
3. Solve the problem using an appropriate strategy.
4. Communicate the 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? Could it have
been solved a better way?
The problem solving processes are interrelated. The importance of each process will depend
on the problem being solved. By practising the skills involved in using all processes, you
will learn to tackle new mathematics problems with confidence and arrive at the correct and
complete solution using the most appropriate methods.
READ THE QUESTION at least twice. Make sure you know what the question is asking
you to do. Do you have enough information to solve the problem?
IDENTIFY THE SOLUTION REQUIRED: What is the question asking you to do?
SOLVE THE PROBLEM using an appropriate strategy. Decide on a suitable strategy to
solve the problem. Examples of strategies that could be used are as follows:
• Create a table
• Draw a diagram
• Look for a pattern — using technology
• Work backwards from the answer
• Elimination
• Simplify the problem
• Guess and check
COMMUNICATE THE SOLUTION: Another person reading your work needs to be able
to follow your method or strategy. You need to present your data, explanation and solutions in
a clear and concise form, using correct mathematical terms and appropriate diagrams.
SUPPORT THE SOLUTION with mathematical reasoning. When you think you’ve solved
the problem, use mathematical reasoning to verify that your answer is correct and your
method is justified.
REFLECT ON THE SOLUTION: Have you answered the question? Think back over how
you solved the problem. Could it have been solved in a different or better way? Learn from
the experience, and use this knowledge to solve problems in the future.
Create a table
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A table is a way of organising or grouping numbers.
Think about the number of rows or columns that will be needed and label them appropriately.
A table can help you see patterns in the numbers you have organised.
A table can demonstrate to others how you arrived at your solution.
There are many different ways of presenting information in a table.
WorkeD exaMple 1
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
218
Maths quest 8 for the australian Curriculum
Think
Write/Display
1
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.
2
Identify the solution required.
The question asks to determine the total time,
expressed in hours and minutes, spent on the
phone.
3
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.
4
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).
5
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).
6
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.
7
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.
8
Answer the question.
■■
The total time spent on the telephone this week is
13 hours and 32 minutes.
Note this problem could be solved without a spreadsheet. The spreadsheet just made the
repetitive task faster.
Chapter 9 Problem solving I
219
remember
1. The five main processes used to solve problems are:
(a) questioning
(b)applying strategies
(c) communicating
(d)reasoning
(e) reflecting.
2. One strategy that can be used is to create a table. This strategy allows you to organise
or group numbers. It can help you see patterns in the numbers and can demonstrate to
others how you arrived at your solution.
Exercise
9A
Introduction to problem solving — create a table
Problem Solving
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 Peta is a great cook and buys her flour in 20 kg bags. Over the last few weeks she has had a
3
4
5
6
220
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.
My friend and I are trying to work out whether it is it is better to take a million dollars now
or to collect the money after 21 days by doing the following. Put in a dollar on the first day,
double that dollar the next day, then double the previous day’s dollars and continue this for
21 days. What would you advise? (Use mathematics to explain your answer.)
On a multiple choice test of 20 questions, each correct answer scores 5 points, each incorrect
answer scores -2 points, and each question left unanswered scores 0 points. If you answer all
questions on the test, what is the least number of questions you must answer correctly to still
get a positive score? Use algebra to explain your answer.
The Goods and Services Tax or GST rate is 10%. This means that when a business sells
1
something or provides a service it must charge an extra 10 more than what it is asking for.
That extra money then must be sent to the tax office. For example, an item that would
otherwise be worth $100 now has GST of $10 added. So the price tag will show $110. The
business will then send to the tax office that $10 and all the other GST it has collected on
behalf of the government.
a Suppose a shopkeeper made sales totalling $15 400. How much GST must he put aside?
b Is there a number he can quickly divide by to figure out the GST?
1
Goldilocks and the three bears relate to each other with Baby bear being 2 the height of Papa
3
bear and Mama bear being 4 the height of Papa bear. Their possessions are all in proportion
3
too. This means that Mama bear’s bed is 4 the length of Papa bear’s bed. The items
belonging to Goldilocks do not fit this pattern. Some of her things are almost as big as
2
Papa bear’s and some are almost as small as baby bear’s. The height of the bowls is 3 the
diameter. Can you work out the owners of the items on the next page?
Maths Quest 8 for the Australian Curriculum
20 cm
10 cm
12 cm
1.2 m
10 cm
0.9 m
100 cm
0.8 m
80 cm
0.6 m
60 cm
40 cm
10 cm
7.5 cm
5 cm
4 cm
Chapter 9 Problem solving I
221
7 Cameron and William are running around a circular track. William can complete 1 circuit in
45 seconds. Cameron runs in the opposite direction and meets William every 20 seconds. How
long does it take Cameron to complete a circuit?
8 Polly and Neda had divided up some coins. Neda was upset as Polly had more. Polly said
‘Here’s one third of my coins’.
Neda was moved by Polly’s generosity and gave back one half of her total. Polly gave her
one quarter of her new total and an extra coin. ‘Now we both have 62 coins’. How many did
they start with?
9 Gwen likes soda. Her local store gives a free bottle for every 5 bottles recycled. If she has
collected 77 empty bottles, how many bottles of soda will she be able to drink free of charge?
(It might take her more than 1 day.)
1
10 Leonardo has a bank account that pays him 10 of the current balance as interest every month,
but only on amounts of up to $25 000. He already has $25 000 in the account and
decides to start a second account and deposit the interest from the $25 000 account into
it. Assume that the interest is calculated on the amount in the account at the time and
that Leonardo has time to move the interest from the first account to the second before
the interest is calculated there. How many months will it take for him to have at least
$25 000 in the second account.
9B
Draw a diagram
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When information is represented as a diagram, it can be easier to study all the information at
once.
There are many different types of diagram, so no single diagram is necessarily the best.
Worked Example 2
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
222
Write
1
Read the question at least twice and take note
of all the important facts.
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.
2
Identify the solution required.
The question asks to find the number
of guinea pigs after the second
litter.
3
Start a diagram by showing the original female
as F1, and original male as M1.
F1, M1
Maths Quest 8 for the Australian Curriculum
4
5
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.
F1, M1
F1, M1
F2, M2
F3, M3
6
7
The total number of guinea pigs must be counted
for a conclusion to be made. Remember to include
F1 and M1.
Answer the question.
F2
F3
M2
M3
F4
F5
M4
M5
F6
F7
M6
M7
F8
F9
M8
M9
There are 18 guinea pigs after two litters.
remember
Drawing a diagram can help to solve the problem.
Exercise
9B
Draw a diagram
Problem Solving
1 Nigel has one mating pair of rabbits. Assume that each litter of rabbits produces three males
2
3
4
5
6
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?
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?
Our Earth is just a mere speck in our solar system. The sun is our nearest star and is 150 000 000 km
away. Light from the sun travels to us (and the other planets) at 300 000 km/s. It takes about
12.5 minutes for light to reach Mars from the sun. Use this information to calculate the distance
between Earth and Mars when the sun, Earth and Mars are aligned.
Local time in Brisbane (there is no daylight saving) is 3 hours behind Auckland (NZ) and
one hour ahead of Tokyo (Japan). Auckland is 11 hours ahead of Paris (France) time. What is
the time difference between:
a Brisbane and Paris?
b Tokyo and Auckland?
At Miami High School there are 600 students. Three per cent of them wear one earring. Of the
other 97%, half wear 2 earrings and half do not wear any. What is the total number of earrings
being worn at Miami High School?
In a school of 600 students, 390 study Mathematics, 300 study Science and 185 study both.
Show this information on:
a a Venn diagram
b a Karnaugh map.
c What is the ratio of students who study Mathematics only to students who study Science
only?
Chapter 9 Problem solving I
223
7 The perimeters of two squares are in the ratio 4 to 9. What is the ratio of their areas?
8 ABCD is a square with NM bisecting both AD and BC and PN = NQ = NM. Determine the
magnitude of ±PNM.
A
Q B
M
N
D
P C
9 Sally lives 35 km from her work. On her way to work this morning she averaged 60 km/h.
Her average speed for her trip to and from work today was 56 km/h. What was Sally’s average
speed on her way home from work?
10 The ratio of black to white balls in a bag is 3 : 4. There are 48 white balls. How many white
balls must be removed to make the ratio of black to white balls 4 : 3?
9C
Look for a pattern
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Repetitive tasks are suited to a spreadsheet.
A spreadsheet can list patterns of numbers from which a result can be found.
Worked Example 3
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
1
Read the question at least twice and take note
of all the important facts.
2
Identify the solution required.
3
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.
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5
6
7
8
224
We need a heading for this cell.
In cell C30 enter Multiplication 5.462 ì 29.
Answer the question.
Maths Quest 8 for the Australian Curriculum
Write/display
Using a spreadsheet, calculate 5.462 ì 29 by
repeated addition; that is, add 29 amounts of
5.462.
The question asks to obtain a spreadsheet
solution to 5.462 ì 29 using repeated addition.
5.462 ì 29 = 158.398
remember
1. Repetitive tasks are suited to a spreadsheet.
2. If a pattern of numbers is listed in a spreadsheet, the result can be found.
Exercise
9C
Look for a pattern
Problem Solving
1 Verity wants to use a spreadsheet to show that $783.57 ì 43 can be calculated by repeated
addition using a spreadsheet. Use a spreadsheet to calculate the answer.
3
ì 43 can be calculated by repeated addition
57
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.) Use a spreadsheet to
calculate the answer.
3 Amelia wants to use a spreadsheet to show that 144 ó 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.) Use a spreadsheet to calculate the answer.
8
4 Sonja wants to use a spreadsheet to show that  1  can be calculated using repeated
2 Phillip wants to use a spreadsheet to show that
2
5
6
7
8
9
10
multiplication by 1 . (Hint: Enter 1/2 in cell A1; then enter =A1*(1/2) in cell A2. Drag this
2
formula down from A2 to A8.) Use a spreadsheet to calculate the answer.
It is common to use shortcuts when performing calculations. Consider the following case:
6.293 ì 9 = 6.293 ì 10 - 6.293 ì 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.
If this pattern continues, how many cubes will it take to make 10 layers?
If today is Monday, what day will it be 50 days from now?
In the song The Twelve Days of Christmas the person receives
gifts on each of the 12 days leading up to Christmas — 1 the
first day, 2 the second day, 3 the third day and so on. How many
gifts did the person receive altogether?
a Calculate the values of the following pairs of expressions:
1 1 1
1
i i +
+ ;1 −
2 4 8
8
1
1
1
1
1
ii +
+ +
;1 −
2 4 8 16
16
1
1
1
1
1
1
iii +
+ +
+
;1 −
2 4 8 16 32
32
1 1 1
1
1
1
1
b Consider this sum: +
+ +
+
+
+
+ . . . It goes on forever. To
2 4 8 16 32 64 128
express it we could include as many terms as we like as long as we add the dashes to
indicate a never ending series.
i i What are the next two terms?
ii This sum cannot be done in the same fashion as ordinary fraction sums but there are
various ways to determine its value. Can you explain why it equals 1?
In Mathville, there are no house numbers that are either a multiple of 3 or a multiple of 4. If the
highest number is a street is 67, how many houses are there?
Chapter 9 Problem solving I
225
9D
Work backwards from the answer
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If there is a sequence of steps for which we know the final answer, then a useful strategy is to
work backwards from this final result or answer.
Worked Example 4
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
1
Read the question at least twice and take note of
all the important facts.
2
Identify the solution required.
3
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.
Overnight from Monday to Tuesday, the clerical
staff added 7 reports. Before that there must have
been 7 fewer reports in the in-tray.
Milo removed, processed and filed 25 reports on
Monday.
Answer the question.
4
5
6
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.
The question asks to determine how many reports
were originally in the in-tray Monday morning.
56 + 19 = 75 at the start of Tuesday
75 - 7 = 68 at the end of the day on Monday
68 + 25 = 93 at the start of Monday
There were 93 reports in the in-tray at the start
of Monday.
remember
If there is a sequence of steps for which we know the final answer, then a useful strategy is
to work backwards from this final result or answer.
Exercise
9D
Work backwards from the answer
Problem Solving
1 Mike works on a cattle station. At the start of the year 2010 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 2010, there were
245 animals. How many were there at the start of the year 2010?
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?
226
Maths Quest 8 for the Australian Curriculum
3 Grace sells tropical fish. Last week Grace counted the fish and then she added 3 dozen new
4
5
6
7
8
9
10
9E
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?
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 purchased another dozen. A
harsh winter claimed the life of another
5 roses. Her six children 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?
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 personal
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. If Dan removes emails from
his inbox once he’s answered them, how many emails were in Dan’s inbox on Monday
morning?
Craig has bought shares in CSL which have a value $34.56 per share one week after he bought
them. Each day the change in share price was given as -0.45, +1.23, -2.56, +0.07 and -0.75.
What did Craig pay for the shares?
Give an example of a set of 5 numbers that have a mean of 7, a median of 8 and a range
of 5.
Sean has passed 20 out of 25 tasks. He has 15 tasks left to complete. He needs to pass a
minimum of 75% of the tasks. What is the minimum number of the remaining tasks he needs
to pass?
Charles took a maths exam with 20 questions. For every correct question, Charles received
10 marks. For every incorrect or unanswered question, he lost 5 marks. If his final score was
140 marks, how many questions were answered correctly?
James had 18 litres of water shared unequally between three buckets. Then he:
a Poured three quarters of the water in bucket 1 into bucket 2.
b Poured half the water that was now in bucket 2 into bucket 3.
c Poured a third of the water that was now in bucket 3 into bucket 1.
After the pouring, all the buckets contained equal amounts of water.
How much water did each bucket start with?
Elimination
■■
■■
When using a process of elimination we remove or eliminate possible solutions that do not
match the given information.
We first write down all the possible combinations or solutions in a grid or table. From the
information supplied, we cross out (eliminate) those combinations that do not match.
Chapter 9 Problem solving I
227
Worked Example 5
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
1
Read the question at least twice and take note of all
the important facts.
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.
2
Identify the solution required.
The question asks to find the required
number.
3
List the numbers from between 20 and 30.
21, 22, 23, 24, 25, 26, 27, 28, 29
4
Eliminate (strike through) the odd number(s)
that are odd.
21, 22, 23, 24, 25, 26, 27, 28, 29
5
Eliminate the number(s) that are multiples of 4;
that is, 24, 28.
21, 22, 23, 24, 25, 26, 27, 28, 29
6
Eliminate the number(s) that are multiples of 13;
that is, 26.
21, 22, 23, 24, 25, 26, 27, 28, 29
7
The remaining number is the answer.
The answer is 22.
8
Answer the question.
remember
1. When using a process of elimination we remove or eliminate possible solutions that do
not match the given information.
2. We first write down all the possible combinations or solutions in a grid or table.
From the information supplied, we cross out (eliminate) those combinations that do
not match.
Exercise
9E
Elimination
Problem Solving
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?
228
Maths Quest 8 for the Australian Curriculum
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.
6 The number 15 can be represented as the sum of two or more consecutive positive integers in
three different ways. One of them begins with 1, that is, 1 + 2 + 3 + 4 + 5. What do the other
sequences begin with?
7 There is a number.
If it is not a multiple of 4, it is between 60 and 69.
If it is a multiple of 3, it is between 50 and 59.
If it is not a multiple of 6, it is between 70 and 79.
What is the number?
8 Identify the following shape:
Four of the sides are equal.
The angles add to 360è.
It is not a square.
9 Place the following numbers in the grid below. (There may be more than one way to do this.)
 6
 7
 9
10
11
12
15
16
18
20
21
23
24
25
30
35
36
45
55
60
Multiple of 5
 5
Multiple of 3
 4
Square
 3
Triangular
 2
Prime
 1
Odd
Even
Less than 20
Greater than 20
Factor of 60
10 Emily bought a 5-scoop ice-cream. Her brother wanted to know what she’d bought. Can you
figure it out from the clues she gave him?
The flavours are Bubblebum, Chocolate, Pistachio, Strawberry and Vanilla.
The flavour on the bottom does not have 9 letters in the name.
Strawberry touches both Chocolate and Vanilla.
Bubblegum is not on top.
9F
Simplify the problem
■■
■■
If 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.
Chapter 9 Problem solving I
229
Worked Example 6
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
1
Read the question at least twice and take note of all
the important facts.
It takes 57 days to install 153.6 km of cable;
a total of 170 km must be installed.
2
Identify the solution required.
The question asks to find the time taken to
reach the target length of 170 km.
3
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.
4
Calculate the time taken to install 1 km of cable in the
simplified problem.
Time to install 1 km of cable
= 62 day
1
= 3 day
5
Calculate the remaining length of cable still to be
installed.
Remaining length of cable to install
=8-6
= 2 km
6
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 2 km of cable
1
=3ì2
7
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.
8
Calculate the time taken to install 1 km of cable.
Time to install 1 km of cable
57
= 153.6 day
=
2
3
day
570
= 1536 day
=
9
10
11
95
256
(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
95
= 256 ì 16.4
Answer the question.
To reach the target of installing 170 km of
cable, approximately 6 more days are needed.
= 6.085 937 5 days
remember
1. If 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.
2. After finding the answer to the simpler question, the same method can be used to solve
the original problem.
230
Maths Quest 8 for the Australian Curriculum
exerCise
9F
simplify the problem
probleM solving
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 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?
3 If you calculated the following sum 9 + 99 + 999 + 9 999 + 99 999 + . . ., where the last number
consists of ten digits of 9, how many times would the number 1 appear in your answer?
(Hint: 9 = 10 - 1, 99 = 100 - 1)
4 A chess board is made up of 64 small squares.
There are many more than 64 squares in total on the board.
a Use this board to investigate the total number of squares on an 8 ì 8 chess board.
b Explain how you could deduce a formula to determine the total number of squares on a board
with 12 rows and 12 columns, and hence find the number of squares on a board with 12 rows
and 12 columns.
5 A motorist was driving in
the rain at a steady speed of
72 km/h. As the car passed
under a bridge, the rain
stopped falling on the car for
2.7 seconds. How wide was
the bridge in metres?
6 You are in a hallway with
100 closed lockers. The
first time you walk through the hallway, you open every locker. The second time you walk
through, you close every second locker. The third time through, you change every third locker,
this means that you open it (if it’s closed) or close it (if it’s open). The fourth time through, you
change every fourth locker. This pattern continues until the 100th walk through the hallway,
where the 100th locker is changed. How many lockers are left open?
7 Mrs Mather won $25 and decided to share her winnings with her 5 children.
The first child received $1 plus 1 of the money remaining.
6
The second child received $2 plus 1 of the money remaining.
6
The third child received $3 plus 1 of the money remaining and so on.
6
a How much money did each child receive?
When Mrs Smith won some money, she divided the money in a similar manner (but she only
had 4 children):
The first child received $1 plus 1 of the money remaining.
5
Chapter 9 problem solving I
231
The second child received $2 plus 1 of the money remaining and so on.
5
b If each child received the same amount of money, how much did Mrs Smith win?
c Mrs Brown has 8 children. If she wants to share money using a method similar to
Mrs Mather and Mrs Smith, how much money does she need and what fraction would
she use?
8 Is the number 433 + 344 divisible by 5?
9 A motorist, travelling at 100 km/h,
overtakes an 4WD towing a caravan. The
4WD and caravan together are 13 metres
long and have a speed of 64 km/h. The car
is 5 metres long. How many seconds will it
take from the time the front of the car is
level with the back of the caravan to the
time the back of the car is level with the
front of the 4WD?
10 If all of the odd numbers from 1–199 inclusive are added together, what is the total?
9g
guess and check
■
■
Sometimes is may not be easy to solve a problem directly, in this case we can use a strategy in
which we guess at the solution. We test this guess by using the available information supplied
in the problem to check whether it is the solution. We continue to guess and check until the
solution is found.
Technology, such as a spreadsheet, can be used to give instant feedback about the guess.
WorkeD exaMple 7
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
232
Write/Display
1
Read the question at least
twice and take note of all the
important facts.
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.
2
Identify the solution
required.
The question asks to find the number of chocolates group 1, group 2
and group 3 have received.
3
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.
A
1
B
C
D
First group chocolates Second group chocolates Third group chocolates
Sum of chocolates
= A2*4
= B2 + 10
= A2 + B2 + C2
3
20
80
90
190
4
10
40
50
100
5
11
44
54
109
2
Maths quest 8 for the australian Curriculum
remember
1. Sometimes it may not be easy to solve a problem directly, in this case we can use a
strategy in which we guess at the solution. We test this guess by using the available
information supplied in the problem to check whether it is the solution. We continue to
guess and check until the solution is found.
2. Technology, such as a spreadsheet, can be used to give instant feedback about the guess.
Exercise
9G
Guess and check
Problem Solving
1 Chocolates were distributed among three groups of children. The second group received
2
3
4
5
6
7
8
9
10
9H
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 distributed altogether.
How many chocolates did the first, the second and the third group receive?
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,
2
there were 13 more white cars than red cars, and the number of blue cars was 3 the number of
white cars. What was the most popular colour, and how many did Toula see of that colour?
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?
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 integers?
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?
The Puregold Jewellery company makes bracelets and necklaces. A bracelet has 6 links of
gold and no gem stone. A necklace has 12 links of gold and 3 gem stones. A marketing review
found that they use 5 times as many links as gemstones and 60 links were used. What is the
ratio of bracelets to necklaces made by the Puregold Jewellery company. Explain your answer.
Using the numbers 1, 2, 3, 4, in that order, generate the numbers from −1 to −10. You may use
any of the operations (+, - , ì, ó, ø).
There are two integers whose square root is the last 2 digits of the integer. Find the integers.
A two-digit number is such that if the digits are added together and divided by 5, the result is
the same as placing a decimal point between the digits of the original number. Find the
two-digit number.
Fill in the blanks in the following addition sum using all the digits 1 to 9 only once each.
−−−
+ −−−
−−−
Mixed problems I
Communicating, reasoning and reflecting
■■
It is important to understand that solving problems involves much more than just writing
numbers on a page. Words should accompany the mathematics and these words should be in
the form of appropriate English and use correct mathematical terms.
Chapter 9 Problem solving I
233
■■
■■
■■
Care should be taken that when the equals (=) sign is used, the mathematics following is
indeed equal.
After providing a solution to a question, it is good practice to review your solution 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 understanding
of the problem and the strategies you used to solve it. You may be able to connect this to
previous experiences as well as to future problems you will have to tackle.
Worked Example 8
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
1
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 followed by a
further 10% GST.
2
Identify the solution required.
The question asks to find the percentage equivalent to these two
percentages.
3
Consider an appropriate strategy. In Let x = original price of a stationery item.
this case, define a variable to be
used to represent the original price.
4
Perform the first of the two
separate calculations.
Adding a 20% profit margin:
x + 20% of x
or
120% of x =
20
ì x 10 0
= x + 0.2x = 1.2x
=x+
5
Perform the second of the two
separate calculations.
= 1.2 ì x
= 1.2x
Adding a further 10% GST:
1.2x + 10% of 1.2x
or
110% of 1.2x =
10
ì 1.2x 10 0
= 1.2x + 0.1 ì 1.2x
= 1.2x + 0.12x
= 1.32x
= 1.2x +
234
12 0
ìx
10 0
6
Devise a question that will lead
Abdul to the required equivalent
single calculation.
How can I relate 1.32x as a percentage increase?
7
First express 1.32 as an improper
fraction and then as an equivalent
percentage. Hence, find the
required percentage increase.
132
100
= 132%
so 1.32x = 132% of x.
This means that x has been increased by 32%.
Maths Quest 8 for the Australian Curriculum
1.32 =
11 0
ì 1.2x
10 0
= 1.1 ì 1.2x
= 1.32x
8
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 multiplying the
price by 1.32.
We can check this by considering an item that costs $5.00.
Applying a 20% increase gives 1.2 ì $5.00 = $6.00; then a further
increase of 10% gives 1.1 ì $6.00 = $6.60. A single calculation of
applying a 32% increase gives 1.32 ì $5.00 = $6.60.
9
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 variable to
represent the original price lets us work back to the single
calculation shortcut.
remember
1. It is important to communicate your solution clearly.
2. Take care with how equals (=) signs are used.
3. After you have written your solution, review your work.
4. Reflect on your work.
Exercise
9H
Mixed problems I
Problem Solving
1 A sports reporter was researching the physical characteristics
of football players. As she was writing for an English
newspaper, she reported their weights in the English system
of stones and pounds. (In that system, 14 pounds equal one
stone.) From the following records, construct a table to find
the total weight of the 12 players. Express your answer in
stones and pounds.
2
3
4
5
6
18 stone 8 pounds
19 stone 4 pounds
13 stone 12 pounds
18 stone 13 pounds
17 stone 8 pounds
17 stone 4 pounds
18 stone 5 pounds
14 stone 11 pounds
18 stone 10 pounds
18 stone 2 pounds
14 stone 11 pounds
17 stone 13 pounds
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?
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?
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.
The canteen sells sandwiches on white, brown or grain bread. The filling can be either egg,
cheese, chicken or ham. These can be served with tomato sauce, BBQ sauce or no sauce. How
many different types of sandwiches are available at the canteen?
Use a spreadsheet to show that 48 can be calculated by repeated multiplication.
Chapter 9 Problem solving I
235
7 At the beginning of the year, Sue noticed that her cupboard was
overflowing with shoes, many of which she hadn’t worn for
some time. This prompted her to clear out eight pairs of
unwanted summer shoes and buy two pairs in the latest style. As
winter approached, Sue purchased two new pairs of boots and
discarded four pairs of old winter shoes. With the Christmas
holidays approaching, Sue treated herself to three new pairs of
sandals. She now had eleven pairs of shoes. How many shoes
were in Sue’s cupboard at the beginning of the year?
8 Ken’s house number is a number between 50 and 100. It is a
multiple of 3 and has a prime number as one of its factors. The sum
of its digits is 15 and the first digit is larger than the second digit.
Use a spreadsheet to determine Ken’s house number.
9 John’s pedometer showed that he had taken 65 423 paces and travelled a distance of 42.5 km.
At this rate, how far would John travel every 100 paces?
10 Three numbers have a product of 400. The second number is the cube of the first and the
third number is a square number between 20 and 30. Use a spreadsheet to determine the three
numbers.
9I
Exercise
9I
Mixed problems II
Mixed problems II
Problem Solving
1 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?
2 Jocelyn and Bernard have three children. They are all married and each has a daughter and a
3
4
5
6
7
236
son. The daughters are all married, each with one child. The sons are not married and do not
have children as yet. How many people are in Jocelyn and Bernard’s extended family?
Your school is taking part in a community ‘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
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?
Gemma was asked to use a single calculation that will decrease the pre-GST selling price of a
car by 10% and then add GST of 10%. Find this single calculation.
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?
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?
Maths Quest 8 for the Australian Curriculum
8 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.
9 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 approximately 750 megabytes). Assume
1 page uses approximately 40 kilobytes of memory.
10 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.
9J
Exercise
9J
Mixed problems III
Mixed problems III
Problem Solving
1 In a retirement village there are 9 residents who are over the age of 90. Their ages are:
2
3
4
5
6
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.
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?
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?
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).
The sides of a triangles are in the ratio 5 : 12 : 15. If the longest side
measures 60 cm, what is the perimeter of the triangle?
A garden sunshade viewed from above looks like the diagram below.
Four metal arms hold up the material of the sunshade. Each arm is 1 m long. Supporting
wires are also wrapped around the arms to support the material. What length of wire is
required to make the outer perimeter? Give your answer in exact form. How much material
is required to make the sunshade (in square metres)? If you had 6 m2 of material, how long
would the arms need to be to support this sunshade if its shape is similar?
Chapter 9 Problem solving I
237
7 A household of five used 84 kilolitres of water during
the quarter October–December.
a Find the average daily water use for the family.
b Find the average daily water use per person.
c If there is a target of 155 L per person per day,
how much water should this family have used in
this quarter?
d If the first 14 kL is charged at a rate of $1.05/kL
and the rest is at $1.35/kL, what is the household
water charge?
1
8 The square base of a tent has an area of 6 square metres.
4
a How long is one side of the tent?
b What is the perimeter?
9 You are using a natural spray to feed the flowers in
your garden. The instructions are to mix 15 mL with
every 2 litres of water. Your bottle is 200 mL. How
many litres of water will you mix up if you if you use
the entire bottle?
3
10 A car uses 4 of a tank of petrol to travel 504 km. The
tank holds 52 litres. How far can the car
go on one litre?
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238
Maths quest 8 for the australian Curriculum