Australian Mathematics Competition - 2015 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What does the digit 1 in 2015 represent?

(A) One
(B) Ten
(C) One hundred
(D) One thousand
(E) Ten thousand

Problem 2:

What is the value of 10 twenty-cent coins?
(A) $\$ 1$
(B) $\$ 2$
(C) $\$ 5$
(D) $\$ 20$
(E) $\$ 50$

Problem 3:

What temperature does this thermometer show?
(A) $25^{\circ}$
(B) $38^{\circ}$
(C) $27^{\circ}$
(D) $32^{\circ}$
(E) $28^{\circ}$

Problem 4:

Which number do you need in the box to make this number sentence true?

(A) 34
(B) 44
(C) 46
(D) 64
(E) 84

Problem 5:

Which number has the greatest value?
(A) 1.3
(B) 1.303
(C) 1.31
(D) 1.301
(E) 1.131

Problem 6:

The perimeter of a shape is the distance around the outside. Which of these shapes has the smallest perimeter?

Problem 7:

The class were shown this picture of many dinosaurs. They were asked to work out how many there were in half of the picture.

Simon wrote $6 \times 10$.
Carrie wrote $5 \times 12$.
Brian wrote $10 \times 12 \div 2$.
Rémy wrote $10 \div 2 \times 12$. Who was correct?
(A) All four were correct
(B) Only Simon
(C) Only Carrie
(D) Only Brian
(E) Only Rémy

Problem 8:

In the diagram, the numbers $1,3,5,7$ and 9 are placed in the squares so that the sum of the numbers in the row is the same as the sum of the numbers in the column.
The numbers 3 and 7 are placed as shown. What could be the sum of the row?
(A) 14
(B) 15
(C) 12
(D) 16
(E) 13

Problem 9:

To which square should I add a counter so that no two rows have the same number of counters, and no two columns have the same number of counters?
(A) A
(B) B
(C) C
(D) D
(E) E

Problem 10:

A half is one-third of a number. What is the number?
(A) three-quarters
(B) one-sixth
(C) one and a third
(D) five-sixths
(E) one and a half

Problem 11:

The triangle shown is folded in half three times without unfolding, making another triangle each time.

Which figure shows what the triangle looks like when unfolded?

Problem 12:

If $L=100$ and $M=0.1$, which of these is largest?
(A) $L+M$
(B) $L \times M$
(C) $L \div M$
(D) $M \div L$
(E) $L-M$

Problem 13:

You want to combine each of the shapes $A$ to $E$ shown below separately with the shaded shape on the right to make a rectangle.
You are only allowed to turn and slide the shapes, not flip them over. The finished pieces will not overlap and will form a rectangle with no holes.
For which of the shapes is this not possible?

Problem 14:

A plumber has 12 lengths of drain pipe to load on his ute. He knows that the pipes won't come loose if he bundles them so that the rope around them is as short as possible. How does he bundle them?

Problem 15:

The numbers 1 to 6 are placed in the circles so that each side of the triangle has a sum of 10 . If 1 is placed in the circle shown, which number is in the shaded circle?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 16:

Follow the instructions in this flow chart.

(A) 57
(B) 63
(C) 75
(D) 81
(E) 84

Problem 17:

A square piece of paper is folded along the dashed lines shown and then the top is cut off.

The paper is then unfolded. Which shape shows the unfolded piece?

(A)

(B)

(C)

(D)

(E)

Problem 18:

Sally, Li and Raheelah have birthdays on different days in the week beginning Sunday 2 August. No two birthdays are on following days and the gap between the first and second birthday is less than the gap between the second and third. Which day is definitely not one of their birthdays?
(A) Monday
(B) Tuesday
(C) Wednesday
(D) Thursday
(E) Friday

Problem 19:

A square of side length 3 cm is placed alongside a square of side 5 cm .

What is the area, in square centimetres, of the shaded part?
(A) 22.5
(B) 23
(C) 23.5
(D) 24
(E) 24.5

Problem 20:

A cube has the letters $A, C, M, T, H$ and $S$ on its six faces. Here are two views of this cube.

Which one of the following could be a third view of the same cube?

(A)

(B)

(C)

(D)

(E)

Problem 21:

A teacher gives each of three students Asha, Betty and Cheng a card with a 'secret' number on it. Each looks at her own number but does not know the other two numbers. Then the teacher gives them this information.
All three numbers are different whole numbers and their sum is 13 . The product of the numbers is odd. Betty and Cheng now know what the numbers are on the other two cards, but Asha does not have enough information. What number is on Asha's card?
(A) 9
(B) 7
(C) 5
(D) 3
(E) 1

Problem 22:

In this multiplication, $L, M$ and $N$ are different digits. What is the value of $L+M+N$ ?
(A) 13
(B) 15
(C) 16
(D) 17
(E) 20

Problem 23:

A scientist was testing a piece of metal which contains copper and zinc. He found the ratio of metals was 2 parts copper to 3 parts zinc. Then he melted this metal and added 120 g of copper and 40 g of zinc into it, forming a new piece of metal which weighs 660 g .
What is the ratio of copper and zinc in the new metal?
(A) 1 part copper to 3 parts zinc
(B) 2 parts copper to 3 parts zinc
(C) 16 parts copper to 17 parts zinc
(D) 8 parts copper to 17 parts zinc
(E) 8 parts copper to 33 parts zinc

Problem 24:

Jason had between 50 and 200 identical square cards. He tried to arrange them in rows of 4 but had one left over. He tried rows of 5 and then rows of 6 , but each time he had one card left over. Finally, he discovered that he could arrange them to form one large solid square. How many cards were on each side of this square?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Problem 25:

Eve has $\$ 400$ in Australian notes in her wallet, in a mixture of 5,10 , 20 and 50 dollar notes.
As a surprise, Viv opens Eve's wallet and replaces every note with the next larger note. So, each $\$ 5$ note is replaced by a $\$ 10$ note, each $\$ 10$ note is replaced by a $\$ 20$ note, each $\$ 20$ note is replaced by a $\$ 50$ note and each $\$ 50$ note is replaced by a $\$ 100$ note.
Eve discovers that she now has $\$ 900$. How much of this new total is in $\$ 50$ notes?
(A) $\$ 50$
(B) $\$ 100$
(C) $\$ 200$
(D) $\$ 300$
(E) $\$ 500$

Problem 26:

Alex is designing a square patio, paved by putting bricks on edge using the basketweave pattern shown.
She has 999 bricks she can use, and designs her patio to be as large a square as possible. How many bricks does she use?

Problem 27:

There are many ways that you can add three different positive whole numbers to get a total of 12 . For instance, $1+5+6=12$ is one way but $2+2+8=12$ is not, since 2,2 and 8 are not all different.
If you multiply these three numbers, you get a number called the product.
Of all the ways to do this, what is the largest possible product?

Problem 28:

I have 2 watches with a 12 hour cycle. One gains 2 minutes a day and the other loses 3 minutes a day. If I set them at the correct time, how many days will it be before they next together tell the correct time?

Problem 29:

A $3 \times 2$ flag is divided into six squares, as shown. Each square is to be coloured green or blue, so that every square shares at least one edge with another square of the same colour.
In how many different ways can this be done?

Problem 30:

The squares in a $25 \times 25$ grid are painted black or white in a spiral pattern, starting with black at the centre $\boldsymbol{*}$ and spiralling out.
The diagram shows how this starts. How many squares are painted black?

Australian Mathematics Competition - 2012 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1:

What is the value of $101-2+1+102$ ?
(A) 0
(B) 100
(C) 198
(D) 200
(E) 202

Problem 2:

Mary's soccer team wins a game by two goals. Between them the two teams scored 8 goals. How many goals did Mary's team score?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Problem 3:

Which of these spinners would be more likely to spin a rabbit?

Problem 4:

A small can of lemonade holds 250 mL . How many of these cans would fill a 1.5 L jug?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 10

Problem 5:

Which of the following numbers has a value between $\frac{1}{5}$ and $\frac{1}{4}$ ?
(A) 0.26
(B) 0.15
(C) 0.21
(D) 0.19
(E) 0.3

Problem 6:

The first time Bill looked at the clock it was 2.00 pm . Later that afternoon he saw it was 4.00 pm . Through how many degrees had the minute hand turned in this time?
(A) 90
(B) 180
(C) 360
(D) 270
(E) 720

Problem 7:

Mary colours in a honeycomb tessellation of hexagons. If hexagons share a common edge, she paints them in different colours.

What is the smallest number of colours she needs?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 8:

After half an hour Maya notices that she is one-third of the way through her homework questions. If she keeps working at a similar rate, how much longer, in minutes, can she expect her homework to take?
(A) 20
(B) 30
(C) 40
(D) 60
(E) 90

Problem 9:

Sentries marked $S$ guard the rows and columns they are on. Sentries marked $T$ guard diagonally.

How many squares are unguarded?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 8

Problem 10:

A rectangular rug is 3 times as long as it is wide. If it were 3 m shorter and 3 m wider it would be a square. How long, in metres, is the rug?
(A) 3
(B) 6
(C) 9
(D) 12
(E) 15

Problem 11:

Lee's mobile phone gives him a warning that only $20 \%$ of the battery charge remains. If it is 48 hours since he last charged his phone and he uses the phone in the same way, how much longer, in hours, can he expect to use the phone before it runs out of battery life?
(A) 12
(B) 20
(C) 24
(D) 80
(E) 192

Problem 12:

Michael threw 8 darts at the dartboard shown.

All eight darts hit the dartboard. Which of the following could have been his total score?
(A) 22
(B) 37
(C) 42
(D) 69
(E) 76

Problem 13:

A prime number is called a jillyprime when doubling it and adding 1 results in another prime. How many numbers less than 15 are jillyprimes? (Note that 1 is not a prime.)
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Problem 14:

The square $P Q R S$ is made up of 36 squares with side length one unit. The square $J K L M$ is drawn as shown.

What is the area, in square units, of $J K L M$ ?
(A) 18
(B) 20
(C) 24
(D) 25
(E) 30

Problem 15:

Following a recipe, Shane roasts a chicken for 20 minutes and then a further 30 minutes for each 500 g . How many minutes does it take Shane to cook a 1.75 kg chicken?
(A) 50
(B) 80
(C) 125
(D) 52.5
(E) 150

Problem 16:

Alex placed 9 number cards and 8 addition symbol cards on the table as shown.

Keeping the cards in the same order he decided to remove one of the addition cards to form a 2-digit number. If his new total was 99, which 2-digit number did he form?
(A) 32
(B) 43
(C) 54
(D) 65
(E) 76

Problem 17:

How many different isosceles triangles can be drawn with sides that can be only $2 \mathrm{~cm}, 3 \mathrm{~cm}, 7 \mathrm{~cm}$ or 11 cm in length? Note that equilateral triangles are isosceles triangles.
(A) 8
(B) 10
(C) 12
(D) 14
(E) 19

Problem 18:

There is a total of $\$ 25$ in $\$ 2, \$ 1$ and $ 50 c $ coins on a table. Peter notices that there are 20 coins altogether and that there are two more $\$ 2$ coins than $\$ 1$ coins. How many 50 c coins are there?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 14

Problem 19:

I can walk at $4 \mathrm{~km} / \mathrm{h}$ and ride my bike at $20 \mathrm{~km} / \mathrm{h}$. I take 24 minutes less when I ride my bike to the station than when I walk. How many kilometres do I live from the station?
(A) 1
(B) 1.5
(C) 2
(D) 2.5
(E) 4

Problem 20:

Jasdeep plays a game in which he has to write the numbers 1 to 6 on the faces of a cube. However, he loses a point if he puts two numbers which differ by 1 on faces which share a common edge. What is the least number of points he can lose?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Problem 21:

Twelve points are marked on a square grid as shown.

How many squares can be formed by joining 4 of these points?
(A) 5
(B) 6
(C) 9
(D) 11
(E) 13

Problem 22:

A rectangular tile has a perimeter of 24 cm . When Sally places four of these tiles in a row to create a larger rectangle, she finds the perimeter is double the perimeter of a single tile. What would be the perimeter of the rectangle formed by adding another 46 tiles to make a row of 50 tiles?
(A) 306
(B) 400
(C) 416
(D) 480
(E) 612

Problem 23:

The grid shown is part of a cross-number puzzle.

Clues
16 across is the reverse of 2 down
1 down is the sum of 16 across and 2 down
7 down is the sum of the digits in 16 across
What is 7 down?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15

Problem 24:

Damian makes a straight cut through a painted cube, dividing it into two parts. The unpainted face created by the cut could not be which of the following?
(A) an equilateral triangle
(B) a right-angled triangle
(C) a trapezium
(D) a pentagon
(E) a hexagon

Problem 25:

A $5 \times 5 \times 5$ cube has a $1 \times 1 \times 5$ hole cut through from one side to the opposite side, a $3 \times 1 \times 5$ hole through another and a $3 \times 1 \times 5$ hole through the third as shown in the diagram.

The number of $1 \times 1 \times 1$ cubes removed in this process is
(A) 25
(B) 29
(C) 36
(D) 48
(E) 92

Problem 26:

Traffic signals at each intersection on a main road all change on the same 2-minute cycle. A taxi driver knows that it is exactly 3.5 km from one intersection to the next. Without breaking the $50 \mathrm{~km} / \mathrm{h}$ speed limit, what is the highest average speed, in kilometres per hour, he can travel to get to each intersection as it just changes to green?

Problem 27:

Rani wrote down the numbers from 1 to 100 on a piece of paper and then correctly added up all the individual digits of the numbers. What sum did she obtain?

Problem 28:

This cube has a different whole number on each face, and has the property that whichever pair of opposite faces is chosen, the two numbers multiply to give the same result.

What is the smallest possible total of all 6 numbers on the cube?

Problem 29:

A rhombus-shaped tile is formed by joining two equilateral triangles together. Three of these tiles are combined edge to edge to form a variety of shapes as in the example given.

How many different shapes can be formed? (Shapes which are reflections or rotations of other shapes are not considered different.)

Problem 30:

Andrew has two children, David and Helen. The sum of their three ages is 49. David's age is three times that of Helen. In 5 years time, Andrew's age will be three times David's age. What is the product of their ages now?

Australian Mathematics Competition - 2008 - Upper Primary - Grade 5 & 6 - Questions and Solutions

Problem 1

What is the largest 4 -digit number which can be made from a selection of the following number cards if each card may be used only once?

(A) 4581
(B) 8542
(C) 8541
(D) 5480
(E) 8851

Problem 2

What is the number represented on the number line?

(A) 0.3
(B) 3
(C) 13
(D) 1.03
(E) 1.3

Problem 3

If 100 tickets are sold in a class raffle, how many tickets will Matthew have to buy to have a $\frac{1}{10}$ chance of winning?
(A) 100
(B) 1
(C) 20
(D) 10
(E) 5

Problem 4

A water tank holds 7500 litres when full. How many litres does it hold when it is exactly half full?
(A) 3275
(B) 3500
(C) 4000
(D) 3750
(E) 3725

Problem 5

If 1 bottle of water can fill 3 glasses, how many glasses of the same size would be filled with 7 bottles of water?
(A) 21
(B) 24
(C) 14
(D) 18
(E) 25

Problem 6

The cost of petrol is 149.9 cents per litre on a Tuesday and 153.5 cents per litre the next morning. What was the increase in cents per litre overnight?
(A) 6.4
(B) 4.3
(C) 16.4
(D) 3.5
(E) 3.6

Problem 7

Jill walked along a two-kilometre track. How far had she walked if she had walked $\frac{3}{4}$ of the track?
(A) $1 \frac{1}{2} \mathrm{~km}$
(B) 175 metres
(C) 300 metres
(D) 1750 metres
(E) 1.34 km

Problem 8

Which of the following shapes could not be formed from a number of equal-sized equilateral triangles?
(A) rhombus
(B) trapezium
(C) square
(D) hexagon
(E) parallelogram

Problem 9

Jenny has three boxes, each having the same number of toy cars inside. She finds two more cars down the back of the sofa. When she counts all her cars she finds that she has 17 . How many cars are there in each box?
(A) 3
(B) 5
(C) 6
(D) 8
(E) 15

Problem 10

In the number sentence,\[40 \times 40=20 \times 20 \times \square \]
what number do we put in the $\square$ to make the number sentence true?
(A) 2
(B) 4
(C) 8
(D) 16
(E) 40

Problem 11

A box holds socks which are all the same size. There are 6 white, 10 blue and 16 grey socks. What is the least number of socks I need to take out, without looking, so that I can be certain of getting a pair of matching socks?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 10

Problem 12

Kiesha has a fish tank which is the right size for 3 fish. A friend gave her 3 more fish and she needs to get a tank with double the volume. She could do this by buying a new tank which is:
(A) double the width and the length of the old tank
(B) double the width, length and depth of the old tank
(C) half the length and double the width and depth of the old tank
(D) half the length and double the depth of the old tank
(E) half the length and double the width of the old tank

Problem 13

If Joachim has 26 m of garden fencing, how many different ways can he make a rectangular garden, using all the fencing, if each side of his garden is a whole number of metres?
(A) 8
(B) 6
(C) 4
(D) 2
(E) 1

Problem 14

At half-time in a soccer match between Newcastle and Melbourne, the score was Newcastle 1, Melbourne 0 . Three goals were scored in the second half. Which of the following could not be the result of the match?

(A) The match was drawn
(B) Newcastle won by 2 goals
(C) Melbourne won by 2 goals
(D) Newcastle won by 1 goal
(E) Newcastle won by 4 goals

Problem 15

Two 1 litre jugs are each filled with a mixture of olive oil and vinegar. The first jug has three times as much oil as vinegar and the second has four times as much oil as vinegar. They are poured into a larger container. In the new mixture, the volume of oil will be
(A) 750 mL
(B) 800 mL
(C) 1250 mL
(D) 1450 mL
(E) 1550 mL

Problem 16

In a circle dance, everyone is evenly spaced around a circle and has a number in the order $1,2,3,4,5, \ldots$, and so on. The dancer with number 15 is directly opposite dancer number 3 . How many dancers are in the circle?
(A) 18
(B) 20
(C) 22
(D) 24
(E) 26

Problem 17

In the $5 \times 5$ square below, the numbers $1,2,3,4$ and 5 are to be placed in the squares so that each number appears exactly once in each row and once in each column. Some numbers have already been placed.

When the square is completed, the number which is in the square marked with an X is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Problem 18

The perimeter of this rectangular paddock is 700 m . It is subdivided into six identical paddocks as shown.

The perimeter, in metres, of each of the six smaller paddocks is
(A) $116 \frac{1}{3}$
(B) 300
(C) 200
(D) 150
(E) 600

Problem 19

You roll three regular 6-sided dice. The product of the numbers rolled is 24 . What is the maximum possible sum of three such numbers?
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

Problem 20

Which of the following methods would divide seven identical loaves of bread equally between 12 hungry people?
(A) cut 5 into quarters and 2 into thirds
(B) cut 3 into thirds and 4 into quarters
(C) cut 5 into thirds and 2 into quarters
(D) cut 3 into quarters and 4 into thirds
(E) cut 5 into halves and 2 into thirds

Problem 21

Three water pipes can be used to fill a water tank. The first pipe by itself takes 8 hours to fill the tank, the second pipe by itself takes 12 hours to fill the tank and the third pipe by itself takes 24 hours to fill the tank. How long would it take to fill the tank from empty if all pipes were used at the same time?
(A) 2 hrs
(B) 3 hrs
(C) 6 hrs
(D) 5 hrs
(E) 4 hrs

Problem 22

The average age of the 11 players in the Australian soccer team is 22.

One player got a red card and had to leave the field. Then, the average age of the remaining 10 players on the field was 21 . How old was the player with the red card?
(A) 21
(B) 31
(C) 22
(D) 32
(E) 24

Problem 23

Anne designs the dart board shown, where she scores $P$ points in the centre circle, $Q$ points in the next ring and $R$ points in the outer ring. She throws three darts in each turn. In her first turn, she gets two darts in ring $Q$ and one in ring $R$ and scores 10 points. In her second turn, she gets two in circle $P$ and one in ring $R$ and scores 22 points.
In her next turn, she gets one dart in each of the regions. How many points does she score?
(A) 12
(B) 13
(C) 15
(D) 16
(E) 18

Problem 24

Paradise is a circular island. There are 4 villages on the island at the $\mathrm{N}, \mathrm{E}, \mathrm{S}$ and SE points. Each village has fishing rights for a single continuous strip of coastline, which must include the village plus 3 km either side. If the fishing rights are distributed as evenly as possible between the four villages under the above rules, and the difference between the length of coastline fished by the village at N and the village at SE is as small as possible, then this difference, in kilometres, is
(A) 6
(B) 9
(C) 12
(D) 15
(E) 36

Problem 25

A town centre has a series of roads which form a two by two square as shown. On any particular journey from $P$ to $Q$, I may not drive down the same section of road twice, though I may cross any intersection more than once. How many different journeys are there from $P$ to $Q$ ?

(A) 6
(B) 10
(C) 12
(D) 14
(E) 16

Problem 26

What is the largest number less than 1000 , which is odd, leaves a remainder of 2 when divided by 3 , and a remainder of 4 when divided by $5 ?$

Problem 27

A cube with edge length 1 m is cut up into cubes each with edge length 5 cm . If all these cubes were stacked one on top of the other to form a tower, what would be the height, in metres, of this tower?

Problem 28

A beetle sets out on a journey. On the first day it crawls 1 m in a straight line. On the second day it makes a right-angled turn (in either direction) and crawls 2 m in a straight line. On the third day it makes a right-angled turn (in either direction) and crawls 3 m in a straight line. This continues each day with the bug making a rightangled turn (in either direction) and crawling 1 m further than it did the day before. What is the least number of days before the beetle could find itself stopped at its starting point?

Problem 29

Two identical coins are placed one inside and one outside a thin circular wire, which has twice the diameter of each coin. Each coin is rolled, without slipping, around the wire until it returns to its starting point. If the direction of rotation does not matter, what is the difference, in degrees, between the angle through which the outer coin rotates and the angle through which the inner coin rotates?

Problem 30

I have a number of bricks which are each 3 units long, 1 unit deep and 1 unit wide. I want to stack them in a tower 3 units wide, 1 unit deep and 10 units high. In how many ways can I do this? (One such way is shown.)