Band 10 – Page 3

Measurement, NAP-D4-NC30 SA

When 1 mm of rain falls on 1 m² of a pool, 1 litre of water is collected.

A pool of what surface area is needed to collect 6000 litres from a rainfall of 15 mm?

m²

Show Answers Only

`400`

Show Worked Solution

`text(If 15 mm of rain falls on 1 m²)`

`=> 15\ text(litres collected)`

`:.\ text(Surface Area of pool needed)`

`= 6000/15`

`= 400\ text(m²)`

Geometry, NAP-D4-NC29 SA

A star is drawn on the inside of a regular pentagon, as shown below.

What is the size of the angle marked `x`?

degrees

Show Answers Only

`36^@`

Show Worked Solution

`text(Consider the triangle)\ \ ABC\ \ text(in)`

STRATEGY: The internal angle sum is 3 × 180 = 540 (since 3 triangles can be drawn internally from one point).

`text(the pentagon,)`

`text(Total degrees in a pentagon)`

`= 3 xx 180`

`= 540^@`

`:.\ text(Internal angle)`	`= 540/5`
	`= 108^@`

`DeltaABC\ text(is isosceles)`

`:. x + x + 108`	`= 180`
`2x`	`= 72`
`x`	`= 36^@`

Measurement, NAP-C4-NC32 SA

The total surface area of a cube is 150 cm².

How long is an edge of the cube?

Show Answers Only

`text(5 cm)`

Show Worked Solution

`text(Surface area of 1 face of the cube)`

`=150/6`

`=25\ text(cm²)`

`text(Let)\ \ x=\ text(length of 1 side)`

`x^2`	`=25`
`:.\ x`	`=5\ text(cm)`

Probability, NAP-C4-NC31 SA

Sharif has a lock on his bike.

To open his lock he moves the 4-digit dial to show 1221.

If each dial can be any number 0-9, how many different 4-digit codes, including Sharif's, are possible?

Show Answers Only

`10\ 000`

Show Worked Solution

`text(Each digit has 10 possibilities,)`

`:.\ text(Number of possible 4-digit codes)`

`=10 xx 10 xx 10 xx 10`

`=10\ 000`

Number, NAP-C4-NC30 SA

Terry picked some pears from his orchard.

Exactly 45% of the pears picked were not good to eat.

What is the smallest total number of pears that he could have collected?

pears

Show Answers Only

`20`

Show Worked Solution

`45text(%) = 9/20`

`text(The minimum number of pears that could be)`

`text(collected is the smallest number that can be)`

`text(be multiplied by)\ \ 9/20\ \ text(and equal a whole)`

`text(number.)`

`:.\ text(Smallest number of pears = 20)`

Algebra, NAP-C4-NC28 SA

Freya buys 2 of the same chocolate bars from the shop.

She breaks the first chocolate bar into 3 identical pieces.

She then breaks the second chocolate bar into 6 identical pieces.

One of the larger pieces of chocolate weighs 12 grams more than one of the smaller pieces.

What is the mass of one whole chocolate bar?

grams

Show Answers Only

`288`

Show Worked Solution

`text(Let)\ \ \ x=\ text(mass of one chocolate bar)`

`1/3x – 1/6x`	`=12`
`2/6x -1/6x`	`= 12`
`1/6 x`	`= 12`
`x`	`= 72\ text(grams)`

Geometry, NAP-C4-NC27

Eloise is taking a ride on a ferris wheel.

It takes her 5 minutes to complete a full lap.

What angle does the Ferris wheel turn through in 75 seconds?

Show Answers Only

`90^@`

Show Worked Solution

`text(5 minutes = 300 seconds)`

`:.\ text(Angle turned through)`	`=75/300 xx 360°`
	`=1/4 xx 360°`
	`=90°`

Measurement, NAP-D3-NC32 SA

Anthony creates a tile pattern using tiles in the shape of regular hexagons.

The perimeter of each tile is 24 cm.

He arranges them in the following way.

He keeps adding tiles in the same way until he has a row with a perimeter of 120 cm.

How many tiles are in Anthony's row?

Show Answers Only

`7`

Show Worked Solution

`text(Perimeter of:)`

STRATEGY: Each extra tile adds 24 cm but takes away 8 cm from perimeter (net add of 16 cm).

`text(1 tile)`	`= 24\ text(cm)`
`text(2 tiles)`	`= 24 + 16 = 40\ text(cm)`
`text(3 tiles)`	`= 24 + 16 + 16 = 56\ text(cm)`
`vdots`
`text(7 tiles)`	`= 24 + (6 xx 16) = 24 + 96 = 120\ text(cm)`

Number, NAP-D3-NC31 SA

`3.85 ÷ 3.5 =`

Show Answers Only

`1.1`

Show Worked Solution

`text(One strategy:)`

`3.85 ÷ 3.5`	`=(3.5 + 0.35) ÷ 3.5`
	`=3.5/3.5 + 0.35/3.5`
	`=1 + 0.1`
	`=1.1`

Probability, NAP-D3-NC30 SA

In any standard six-sided dice, the sum of the opposite faces is 7.

Milo rolls 3 dice and the total of the top faces is 5.

What is the sum of the three opposite faces?

Show Answers Only

`16`

Show Worked Solution

`text(Sum of 3 top faces + 3 opposite)`

`= 3 xx 7`

`= 21`

`:.\ text(Sum of 3 opposite faces)`

`= 21 – 5`

`= 16`

Algebra, NAP-E3-CA29 SA

Bill creates a pattern of branches and nodes on a computer.

Bill continued the pattern until the shape had 3279 branches.

How many nodes did the shape have?

Show Answers Only

`1093\ text(nodes)`

Show Worked Solution

`text(Pattern shows that:)`

`text(Number of branches)/text(Number of nodes)`

`= 3/1 = 12/4 = 39/13 = 120/40 =3`

`:.\ text(If there are 3279 branches,)`

`text(Number of nodes)`	`= 3279/3`
	`= 1093\ text(nodes)`

Measurement, NAP-E3-CA28 SA

Guy builds a brick structure that is pictured below.

The structure is 7 bricks high, 7 bricks wide and 6 bricks deep.

The structure is solid brick but has a hole that goes from one side to the other which is 3 bricks high and two bricks wide, as shown in the diagram.

How many bricks are in the stack?

Show Answers Only

`258\ text(bricks)`

Show Worked Solution

`text(Bricks in the stack if no hole)`

`= 7 xx 7 xx 6`

`= 294`

`text(Bricks removed to make hole)`

`= 3 xx 2 xx 6`

`= 36`

`:.\ text(Bricks in stack)`	`= 294 – 36`
	`= 258`

Number, NAP-E3-CA32

Cassie has a money box with only 50 cent and 20 cent coins in it.

The total mass of the coins in her money box is 91.8 grams.

Cassie knows that:

the mass of a 50 cent coin is 10.7 grams.
the mass of a 20 cent coin is 7 grams

What is the smallest mass of exactly $1.20.

grams

What is the total value of the coins in Cassie's money box?

Show Answers Only

`text(Part 1:)\ 28.4`

`text(Part 2:)\ $3.40`

Show Worked Solution

`text(Part 1)`

`text(Smallest mass of $1.20)`

`= 2 xx text(mass of 50 cent) + 1 xx text(mass of 20 cent)`

`= 2 xx 10.7 + 7`

`= 21.4 + 7`

`= 28.4\ text(grams)`

`text(Part 2)`

`text(Total mass) = 91.8\ text(grams)`

`text(S) text(ince the total is a decimal and the mass of)`

`text(a 20 cent coin is a whole number,)`

`=>\ text(the total mass calculation of 50 cent coins)`

`text{must end in .8}`

`=>\ 4 xx 10.7 = 42.8`

`=>\ text(Check if 20 cent coins can)`

`text(have a mass of 49 grams?)`

`=>\ text(Yes)`

`:.\ text(Value of coins in the money box)`

`=4 xx 50\ text(cent) + 7 xx 20\ text(cent)`

`= $3.40`

Measurement, NAP-E3-NC31 SA

This triangle was made by cutting a square in half.

The perimeter of the triangle is 51.21 cm.

What is the area of the triangle?

square centimetres

Show Answers Only

`112.5\ text(cm²)`

Show Worked Solution

`text{Triangle is isosceles (half a square).}`

`=>\ text(Length of triangle side)`

`= 1/2 xx (51.21 – 21.21)`

`= 1/2 xx 30`

`= 15\ text(cm)`

`:.\ text(Area)`	`= 1/2 bh`
	`= 1/2 xx 15 xx 15`
	`= 112.5\ text(cm²)`

Geometry, NAP-E3-NC30 SA

The design below is created using squares and equilateral triangles only.

What is the length of the design?

Show Answers Only

`49 1/2`

Show Worked Solution

`text(Let)\ x = text(length of side of square)`

`2x`	`= 18`
`:. x`	`= 9\ text(cm)`

`text(Equilateral triangle must have)`

`text(side length = 9 cm)`

`:.\ text(Length)`	`= 5 1/2 xx 9`
	`= 49 1/2\ text(cm)`

Algebra, NAP-E3-NC28 SA

Damon bought 2 identical bags of salt for his swimming pool.

After he used `2/3` of one bag in the pool, he had a total of 20 kilograms of salt left.

How many kilograms of salt were in one full bag?

kilograms

Show Answers Only

`15`

Show Worked Solution

`text(Let)\ \ \ S =\ text(weight of 1 bag of salt)`

`1/3S + S`	`= 20`
`4/3 S`	`= 20`
`:. S`	`= (3 xx 20)/4`
	`= 15\ text(kg)`

Measurement, NAP-F3-CA32 SA

A square has an area of 169 square centimetres.

What is the perimeter?

centimetres

Show Answers Only

`52`

Show Worked Solution

`text(Let)\ \ x = text(length of 1 side)`

`text(Area) = x xx x`	`= 169`
`x^2`	`= 169`
`x`	`= sqrt 169`
	`= 13\ text(cm)`

`:.\ text(Perimeter)`	`= 4 xx 13`
	`= 52\ text(cm)`

Number, NAP-F3-CA31 SA

Gary used 4 litres of paint to paint a wall.

The wall was a rectangle 2 metres high and 3 metres wide.

How many litres of paint would he need to paint a rectangular wall which is 3 metres high and 5 metres wide?

litres

Show Answers Only

`10`

Show Worked Solution

`text(Area of smaller wall)`

`= 2 xx 3`

`= 6\ text(m²)`

`text(S)text(ince 4 litres of paint are needed to)`

`text(paint the small wall:)`

`=>\ text(Paint needed for 1 m²)`

`= 4/6`

`= 2/3\ text(litre)`

`=>\ text(Area of larger wall)`

`= 3 xx 5`

`= 15\ text(m²)`

`:.\ text(Paint needed)`	`= 15 xx 2/3`
	`= 10\ text(litres)`

Number, NAP-F3-CA30 SA

A pet store sells its high growth dog food in four different sizes.

What is the lowest price a customer can pay for 10 kg of high growth dog food?

Show Answers Only

`$44.65`

Show Worked Solution

`text(Calculate the cheapest way to purchase 1 kg:)`

`text(Option 1: C)text(ost)\ = $4.60`

`text(Option 2: C)text(ost)\ =9.10/2 = $4.55`

`text(Option 3: C)text(ost)\ = 13.35/3 = $4.45`

`text(Option 4: C)text(ost)\ =22.50/4 = $4.50`

`=>\ text(Cheapest option will be to buy three 3 kg bags)`

`text(and one 1 kg bag.)`

`:.\ text(The lowest price for 10 kg)`

`= 3 xx $13.35 + 1 xx $4.60`

`= $44.65`

Geometry, NAP-F3-CA29 SA

Towns `A, B and C` are marked on the scale diagram below.

The distance from Town `A` to Town `B` is 9 km.

What is the distance from Town `B` to Town `C?`

kilometres

Show Answers Only

`3.6`

Show Worked Solution

`text(Let)\ \ d = text(distance of Town)\ B\ text(to Town)\ \ C`

`text(S)text(ince the diagram is a scale diagram,)`

`d/9`	`= 2/5`
`:. d`	`= (9 xx 2)/5`
	`= 3.6\ text(km)`

Number, NAP-F3-CA28 SA

Richard bought a second hand snowboard at a cost of $390.

Richard had to spend 65% of his savings to buy the snowboard.

How much did Richard have in his savings after buying the snowboard?

Show Answers Only

`$210`

Show Worked Solution

`65text(%) xx text(Savings)`	`= $390`
`:.\ text(Savings)`	`= 390/0.65`
	`= $600`

`:.\ text(Savings left)`	`= 600 – 390`
	`= $210`

Number, NAP-E3-NC27 SA

Nona makes maple syrup on her farm.

The mass of 400 millilitres of her maple syrup is 600 grams.

How many litres is 2.4 kilogram of Nona's maple syrup?

litres

Show Answers Only

`1.6\ text(litres)`

Show Worked Solution

`text(Multiples of 600 grams in 2.4 kilograms:)`

`=> 2400/600 = 4`

`:.\ text(Volume of maple syrup)`

`= 4 xx 400\ text(mL)`

`= 1600\ text(mL)`

`= 1.6\ text(litres)`

Number, NAP-C3-CA32 SA

John's old tractor used 8.3 litres of fuel per 100km.

His new tractor uses 5.9 litres of fuel per 100 km.

John pays $1.35 per litre and drives 20,000 km each year.

How much money will John save on fuel each year with his new tractor?

Show Answers Only

`$648`

Show Worked Solution

`text(Fuel cost of old tractor)`

`=8.3 xx $1.35 xx (20\ 000)/100`

`=8.3 xx $1.35 xx 200`

`=$2241`

`text(Fuel cost of new tractor)`

`=5.9 xx $1.35 xx (20\ 000)/100`

`=5.9 xx $1.35 xx 200`

`=$1593`

`:.\ text(John’s fuel savings each year)`

`=2241 – 1593`

`=$648`

Algebra, NAP-C3-CA31 SA

A construction company uses the following rule to calculate the cost of concrete.

cost in dollars = ($59.50 × volume in cubic metres) + ($8.90 × delivery distance in km)

Jim paid the company $713 for 6 cubic metres of concrete.

What delivery distance was Jim charged for?

Show Answers Only

`40`

Show Worked Solution

`text(Total cost of concrete)`

`= 6 xx 59.50`

`= $357`

`=>\ text(Delivery charges)`	`=713 -357`
	`=$356`

`:.\ text(Delivery distance)`	`=356/8.90`
	`=40\ text(km)`

Number, NAP-C3-CA29 SA

Natalie is building a brick wall 9 metres high.

The height of a column of 15 bricks is 1.8 metres as shown.

How many bricks will Natalie need for a wall 9 metres high?

Show Answers Only

`75\ text(bricks)`

Show Worked Solution

`text(Let)\ \ h=\ text(height of 1 brick)`

`15 xx h`	`=1.8`
`h`	`= 1.8/15`
	`=0.12\ text(m)`

`text(If the wall is 9 metres high,)`

`:.\ text(Number of bricks needed)`

`=9/0.12`

`=75\ text(bricks)`

Measurement, NAP-F3-NC32 SA

A farmer uses an existing stone wall and fencing to create a large grazing paddock for his sheep, as shown in the diagram below.

The fencing has 3-strand wiring which is also shown below.

How many kilometres of fencing will be required?

kilometres

Show Answers Only

`5.7`

Show Worked Solution

`text(Fencing required)`

`= 3xx (450 + 200 + 150 + 500 + 600)`

`= 3 xx 1900`

`= 5.7\ text(km)`

Number, NAP-F3-NC31 SA

Adam is laying tiles in the pattern shown below.

He uses three white tiles for every four gray tiles as shown.

He uses a total of 133 tiles.

How many gray tiles does he use?

Show Answers Only

`76`

Show Worked Solution

`text(In every 7 tiles, 3 are white and 4 are black.)`

`=> 133/7 = 19\ text(groups of 7 tiles)`

`:.\ text(Number of gray tiles)`

`= 19 xx 4`

`= 76`

Number, NAP-F3-NC29

The graph below can be used to calculate the width of any photograph from its height.

Using the graph, a photo with a height of 5 centimetres should be approximately what width?

	`text(5.8 centimetres)`
	`text(6.0 centimetres)`
	`text(6.4 centimetres)`
	`text(7.0 centimetres)`

Show Answers Only

`text(7.0 centimetres)`

Show Worked Solution

`text(Width increase for every 1 cm increase in height)`

`= 4.2/3`

`=1.4\ text(cm)`

`:.\ text(Width when height is 5 cm)`

`= 5 xx 1.4`

`=7.0\ text(cm)`

Statistics, NAP-F3-NC28 SA

Tom collected data on the number of big game animals in an African wildlife park.

This graph shows how many big game animals lived in the park.

There were 16 more elephants than rhinos in the wildlife park.

How many big game animals were in the park in total?

Show Answers Only

`80`

Show Worked Solution

`text(Let)\ \ n = text(total big games animal in the park)`

`text(Elephants − Rhinos)`	`= 16`
`7/20 n – 3/20 n`	`= 16`
`4/20 n`	`= 16`
`n`	`= (16 xx 20)/4`
	`= 80`

Number, NAP-G3-NC32 SA

The fraction `3/8` equals 0.375 as a decimal.

What does `3/16` equal as a decimal?

Show Answers Only

`0.1875`

Show Worked Solution

`3/16`	`= 3/8 xx 1/2`
	`= 0.375 ÷ 2`
	`= 0.1875`

Number, NAP-G3-NC27

Which fraction is halfway between `1/5` and `1/2` on the number line?

`1/4`	`2/7`	`1/3`	`7/20`	`7/10`

Show Answers Only

`7/20`

Show Worked Solution

`text(Halfway between)\ \ 1/5 and 1/2`

`=1/2 xx (1/5 +1/2)`

`=1/2 xx (2/10 + 5/10)`

`=1/2 xx 7/10`

`=7/20`

Number, NAP-G3-CA32

Ali had a bag of different coloured marbles.

`1/6` of the marbles were blue and `1/4` of the marbles were yellow.

Her bag contained 4 more yelllow marbles than blue.

How many marbles were in Ali's bag?

`16`	`20`	`24`	`48`

Show Answers Only

`N = 48`

Show Worked Solution

`text(Let)\ \ N = text(number of marbles in Ali’s bag)`

`1/4 xx N – 1/6 xx N`	`= 4`
`3/12 xx N – 2/12 xx N`	`= 4`
`1/12 xx N`	`= 4`
`:. N`	`= 48`

Number, NAP-G3-CA31 SA

One litre of softdrink contains 90 grams of sugar.

How many millilitres of softdrink contain 4.5 grams of sugar?

millilitres

Show Answers Only

`50`

Show Worked Solution

`text(Millilitres)`	`= 4.5/90 xx 1000\ text(mL)`
	`= 50\text(mL)`

Measurement, NAP-G3-CA30 SA

Sam moves into a new rectangular office that measures 4.5 m × 3 m.

He decides to cover the floor with square carpet tiles.

How many boxes of carpet tiles does Sam need to order?

Show Answers Only

`24\ text(boxes)`

Show Worked Solution

`text(Area of office in the number of tiles)`

`= 4.5/0.25 xx 3/0.25`

`= 18\ text(tiles) xx 12\ text(tiles)`

`= 216\ text(tiles)`

`:.\ text(Number of boxes to order)`

`= 216/9`

`= 24\ text(boxes)`

Number, NAP-H3-NC32 SA

Charlie is buying sweets for a party.

He buys twice as many candy canes as chocolate bars, and six times as many candy canes as sherbert sticks.

He has 24 more chocolate bars than sherbert sticks.

How many sherbert sticks does Charlie have?

Show Answers Only

`12`

Show Worked Solution

`text(Expressing the information as equations:)`

`text(Let)\ \ X = text(candy canes)`

`text(Let)\ \ Y = text(chocolate bars)`

`text(Let)\ \ Z = text(sherbert sticks)`

`X = 2Y`	`\ …\ (1)`
`X = 6 Z`	`\ …\ (2)`
`Y = Z + 24`	`\ …\ (3)`

`text(Substitute)\ \ X = 2 xx Y\ \ text{into (2)}`

`2Y = 6 Z\ …\ (2)′`

`text(Multiply)\ \ (3) xx 2`

`2Y = 2Z + 48\ …\ (3)′`

`text(Subtract)\ \ (2)′ – (3)′`

`0`	`= 4 Z – 48`
`4Z`	`= 48`
`Z`	`= 12`

`:.\ text(Charlie has 12 sherbert sticks.)`

Number, NAP-H3-NC30

Which of these is not equal to `42 xx 30`?

`42 xx 10 xx 3`	`420 xx 3`	`40 xx 30 + 2 xx 30`	`40 + 2 xx 30`

Show Answers Only

`40 + 2 xx 30`

Show Worked Solution

`40 + 2 xx 30`

`= 40 + 60`

`= 100`

`!= 42 xx 30`

Number, NAP-H3-NC28

Which two odd numbers, greater than 1, have a product of 91?

and

Show Answers Only

`text(7 and 13)`

Show Worked Solution

`text(One Strategy)`

`text(Test if different odd numbers are factors:)`

`text(Is 3 a factor? ⇒ No)`

`text(Is 5 a factor? ⇒ No)`

`text(Is 7 a factor? ⇒ Yes)`

`:.\ text(7 and 13 are the factors.)`

Measurement, NAP-I3-CA32 SA

Ky designs an outdoor table that is in the shape of a trapezium.

The dimensions of the table top are shown in the picture below.

What is the area of Ky's table top?

cm²

Show Answers Only

`2600\ text(cm²)`

Show Worked Solution

`text(Area)`	`= 1/2 h (a + b)`
	`= 1/2 xx 40 xx (50 + 80)`
	`= 20 xx (130)`
	`= 2600\ text(cm²)`

Number, NAP-I3-CA30 SA

Kurt is travelling from Newcastle to Sydney. The journey is 165 kilometres.

His car uses 8.35 litres of fuel per 100 kilometres.

How much fuel will Kurt need to make the journey?

Round your answer to the nearest litre.

litres

Show Answers Only

`14\ text{litres (nearest whole)}`

Show Worked Solution

`text(Fuel needed)`	`= 165/100 xx 8.35`
	`= 13.77…`
	`= 14\ text{litres (nearest whole)}`

Algebra, NAP-I3-CA27 SA

Tim is going on a golfing trip.

He wants to purchase balls before the trip, which are sold in boxes of a dozen.

The more boxes of balls that Tim buys, the less it costs per box, as shown in the table below.

If this price pattern continues, how much will it cost for 7 boxes of balls?

Show Answers Only

`$133`

Show Worked Solution

`text(The boxes are reducing in cost)`

`text(by 50 cents for each extra box.)`

`=>\ text(6th box costs $18.00)`

`=>\ text(7th box costs $17.50)`

`:. 7\ text(boxes cost)`	`= 97.50 + 18.00 + 17.50`
	`= $133`

Geometry, NAP-H3-CA32 SA

In the diagram `AB` is a straight line.

What is the size of the angle marked `x°`?

degrees

Show Answers Only

`30^@`

Show Worked Solution

`x^@`	`= 90-60`
	`= 30^@`

Number, NAP-H3-CA31

Maria has a square photo with an area of 16 cm².

She enlarges the photo to have an area of 64 cm².

The enlargement multiplied the side lengths of the original photo by a factor of

`text(two.)`	`text(three.)`	`text(four.)`	`text(five.)`	`text(six.)`

Show Answers Only

`text(two)`

Show Worked Solution

`text(Let)\ \ \ s`	`=\ text(side length of original photo)`
`s^2`	`= 16`
`s`	`= 4`

`text(Let)\ \ \ S`	`=\ text(side length of enlarged photo)`
`S^2`	`= 64`
`S`	`= 8`

`:.\ text(The side lengths have increased from 4 to 8,)`

`text(or by a factor of 2.)`

Number, NAP-H3-CA29 SA

Charlie is five years younger than Murray.

Murray is four years younger than half of Barry's age.

The sum of all three ages is 59.

How old is Barry?

Show Answers Only

`36`

Show Worked Solution

`text(Strategy one:)`

`text(Try some educated guesses,)`

`text(If Charlie is 7,)`

`text(Total of ages)\ = 7 + 12 + 32 = 51`

`text(If Charlie is 9,)`

`text(Total of ages)\ = 9 + 14 + 36 = 59`

`text{Strategy two (using algebra):}`

`text(Express the information into 3 equations:)`

`C`	`=M-5\ …\ (1)`
`M`	`=1/2 B – 4, or`
`2M – B`	`= -8\ …\ (2)`
`C+M+B`	`=59\ …\ (3)`

`text(Substitute)\ \ C=M-5\ \ text{into (3):}`

`M-5+M+B`	`=59`
`2M+B`	`=64\ …\ (3′)`

`text{Subtract (3′) – (2):}`

`2B`	`=72`
`B`	`=36`

`:.\ text(Barry is 36 years old.)`

Number, NAP-H3-CA28 SA

Harrison was saving his money to buy a cricket bat.

After 1 month, he had saved 45% of the bat's cost.

After 2 months, he had saved another $90 and now had 75% of the bat's cost.

How much does the bat cost?

Show Answers Only

`$300`

Show Worked Solution

`text(75%)\ -\ text(45%)`	`= $90`
`text(30%)`	`= $90`
`:. 1text(%)`	`= $3`

`:.\ text(The bat cost)`	`= 100 xx 3`
	`= $300`

Geometry, NAP-I3-NC32 SA

The objects below can be used as dice in a board game.

How many edges does one of these objects have?

Show Answers Only

`20`

Show Worked Solution

`text(Horizontal edges)\ = 4+4+4 =12`

`text(Vertical edges)\ = 4+4=8`

`:.\ text(Total edges)\ = 20`

Number, NAP-I3-NC31 SA

Jack is at school and his age this year is a prime number and a factor of 65.

In two years' time, Jack's age will again be a prime number.

What is Jack's age this year?

Show Answers Only

`5`

Show Worked Solution

`text(Consider the factors of 65:)`

IMPORTANT: The factors 65 × 1 also work for this question but do not fit with the school age requirement.

`5 xx 13 = 65`

`text(In two years’ time,)`

`5 -> 7\ (text(which is prime))`

`13 -> 15\ (text(which is not prime))`

`:.\ text(Jack is 5 this year.)`

Number, NAP-I3-NC27 SA

A box of golf balls normally cost $19.50.

During a sale, the golf shop took 10% off the regular price of the balls.

What is the new sale price of the golf balls?

Show Answers Only

`$17.55`

Show Worked Solution

`text(10% discount)`	`= 19.50/10`
	`= $1.95`

`:.\ text(Sale price)`	`= $19.50 – $1.95`
	`= $17.55`

Measurement, NAP-E4-CA32 SA

Two views of a trapezoidal prism are shown below.

Each square on this grid has an area of one square centimetre.

The vertical edges of the prism are 4 centimetres.

What is the volume of the prism in cubic centimetres?

cubic centimetres

Show Answers Only

`72\ text(cm³)`

Show Worked Solution

`text(The area of the base)`

`= 12\ text(squares) + 9\ text(triangles)\ (1 xx 1) + 1\ text(triangle)\ (1 xx 3)`

`= 12 + (9 xx 1/2) + 1 1/2`

`= 18\ text(cm²)`

`:.\ text(Volume)`	`=18 xx 4`
	`=72\ text(cm³)`

Measurement, NAP-E4-CA31 SA

A net of a cube is pictured below.

The cube has a volume of 512 cubic centimetres.

What is the height of the net in centimetres?

centimetres

Show Answers Only

`24\ text(cm)`

Show Worked Solution

`text(Let)\ \ s`	`=\ text(length of 1 side of the cube)`
`s^3`	`= 512`
`:. s`	`= 8\ text(cm)`

`:.\ text(Height of net)`	`= 3 xx 8`
	`= 24\ text(cm)`

Algebra, NAP-E4-CA30 SA

In a ping pong competition with 4 players, each person plays all other players once.

In a competition with 4 players, there are 6 games.

The formula below gives the total number of games `G` for a competition with `N` players.

`G = 0.5 N (N - 1)`

What is the value of `N` if the total games played `G` is 120?

Show Answers Only

`16`

Show Worked Solution

`text(Strategy 1)`

`text(By trial and error:)`

`text(If)\ \ G=10, C=0.5xx10xx9=45`

`text(If)\ \ G=14, C=0.5xx14xx13=91`

`text(If)\ \ G=16, C=0.5xx16xx15=120`

`text{Strategy 2 (advanced)}`

`G`	`= 0.5 N (N – 1)`
`120`	`= 0.5 N (N – 1)`
`240`	`= N^2 – N`

`N^2 – N – 240`	`= 0`
`(N – 16) (N + 15)`	`= 0`

`:. N = 16\ \ \ text{(}N\ text{must be positive)}`

Geometry, NAP-E4-CA29

A company logo is 14 cm by 9 cm.

Pablo wants to project it onto a screen that is 4.2 m by 3 m.

Pablo enlarges the logo as big as possible without distorting its proportions.

What are the dimensions of the enlarged logo, once projected, in cm?

cm by

Show Answers Only

`420\ text(cm) xx 270\ text(cm)`

Show Worked Solution

`4.2\ text(m) = 420\ text(cm)`

`text(Magnification) = 420/14 = 30\ text(times).`

`text(Check if height fits on a 30 times enlargement:)`

`30 xx 9\ text(cm) = 270\ text{cm (which fits)}`

`:.\ text(Dimensions are:)\ \ 420\ text(cm) xx 270\ text(cm)`

Geometry, NAP-E4-CA28 SA

Gabby put 5 points on a grid and labelled them `A` to `E`, as shown on the diagram below.

Point `A` is 35 millimetres from point `D.`

Gabby adds a sixth point, `F` so that the arrangement of points has one line of symmetry.

How far is point `F` from point `B?`

millimetres

Show Answers Only

`42\ text(mm)`

Show Worked Solution

`A\ text(to)\ D\ text(is 35 mm)`

`=> 1\ text(grid is 7 mm × 7 mm)`

`text(Point)\ F\ text(is added below,)`

`:.\ text(Distance)\ B\ text(to)\ F`

`= 6 xx 7`

`= 42\ text(mm)`

Number, NAP-E4-CA27 SA

A marine conservationist records the number of eggs laid by 3 turtles and the number of those eggs that hatched.

Her results are recorded in the table below:

What percentage of the eggs from all 3 turtles hatched?

Show Answers Only

`text(41%)`

Show Worked Solution

`text(Total turtle eggs hatched)`

`= (25text{%} xx 300) + (75text{%} xx 120) + (50text{%} xx 80)`

`= 75 + 90 + 40`

`=205`

`text(Total eggs laid)`	`= 300 +120 + 80`
	`= 500`

`:.\ text(Percentage of eggs that hatched)`

`= 205/500 xx 100`

`=\ text(41%)`

Measurement, NAP-E4-CA26

A tiler is tiling a wall that is 90 cm wide and 160 cm high.

He uses triangular tiles that are drawn below.

The tiler uses all of his tiles and has no gaps between them.

How many tiles does he need?

`200`	`300`	`450`	`600`

Show Answers Only

`600\ text(tiles)`

Show Worked Solution

`text(Area of wall)`	`= 90 xx 160`
	`= 14\ 400\ text(cm²)`

`text(Area of 1 tile)`	`= 1/2 xx 6 xx 8`
	`= 24\ text(cm²)`

`:.\ text(Tiles needed)`	`= (14\ 400)/24`
	`= 600`

Geometry, NAP-C3-NC32 SA

Two squares are drawn on the sides of an equilateral triangle as shown.

A straight line was drawn through the point where the 3 shapes touch.

What is the size of the shaded angle?

Show Answers Only

`15`

Show Worked Solution

`text(S)text(ince there is 180° in a straight line:)`

`x`	`=180 – (60 + 45)`
	`=75`
`:. y`	`=90-x`
	`=90-75`
	`=15°`

Number, NAP-C3-NC31 SA

Billy has some bananas to give to his friends.

If he gives each friend 3 bananas. he will have 5 left over.

He cannot give each friend 4 bananas because he would need 7 more bananas.

How many bananas does Billy have?

Show Answers Only

`41`

Show Worked Solution

`text(Let)\ \ n=\ text(the number of Billy’s friends)`

`text(The information given tells us that:)`

`3n + 5`	`=\ text(# of bananas)`
`4n-7`	`=\ text(# of bananas)`
`:. 4n-7`	`=3n+5`
`n`	`=12`

`:.\ text(Number of bananas Billy has)`

`=(3 xx 12) + 5`

`=41`

Number, NAP-C3-NC29 SA

Freya buys 2 of the same chocolate bars from the shop.

She breaks the first chocolate bar into 3 identical pieces.

She then breaks the second chocolate bar into 6 identical pieces.

One of the larger pieces of chocolate weighs 12 grams more than one of the smaller pieces.

What is the mass of one whole chocolate bar?

grams

Show Answers Only

`72`

Show Worked Solution

`text(Let)\ \ \ x=\ text(mass of one chocolate bar)`

`1/3x – 1/6x`	`=12`
`2/6x -1/6x`	`= 12`
`1/6 x`	`= 12`
`x`	`= 72\ text(grams)`

Statistics, NAP-C3-NC28

Each bar on this graph shows the population of a country and the population of its capital city.

The white section is the population that lives in the capital city.

The black section is the population that lives outside the capital city.

Which of the following countries has the highest percentage of its population living in its capital city?

	`text(Australia)`
	`text(Belgium)`
	`text(France)`
	`text(England)`

Show Answers Only

`text(Belgium)`

Show Worked Solution

`text(Considering the length of the white section of each)`

`text(bar and comparing it to the length of the total bar,)`

`text(Belgium easily has the highest percentage living in)`

`text(its capital city.)`

Number, NAP-E4-CA24 SA

John and Fiona each bought a quantity of the same prawns from the fish markets.

John bought a quarter of the quantity that Fiona bought.

The total cost of John and Fiona's prawns was $30.60.

What was the cost of Fiona's prawns?

Show Answers Only

`$24.48`

Show Worked Solution

`text(Strategy 1)`

`text(Ratio of prawns)`

`text(Fiona : John = 4 : 1)`

`$30.60 -: 5 = $6.12`

`:.\ text(C)text(ost of Fiona’s prawns = 4 × $6.12 = $24.48`

`text(Strategy 2)`

`text(Let)\ \ \ p`	`=\ text(cost of John’s prawns)`
`:. 4p`	`=\ text(cost of Fiona’s prawns)`

`p + 4p`	`= 30.60`
`5p`	`= 30.60`
`p`	`= $6.12`

`:.\ text(Fiona’s prawns cost)`

`= 4 xx 6.12`

`= $24.48`

Geometry, NAP-E4-NC32 SA

Two circles have the same centre, `O`, as shown in the diagram below.

The radius of the small circle is `2/3` the radius of the large circle.

Arc `CD` is 21 cm and the angle between the lines `AC` and `BD` is 45°.

What is the length of the arc `RS` in centimetres?

Show Answers Only

`text(42 cm)`

Show Worked Solution

`text(Arc)\ ST`	`= 2/3 xx text(arc)\ CD`
	`= 2/3 xx 21`
	`= 14\ text(cm)`

`angleSOR`	`= 180-45`
	`= 135^@`
	`= 3 xx angleSOT`

`:. text(Arc)\ RS`	`= 3 xx text(arc)\ ST`
	`= 3 xx 14`
	`= 42\ text(cm)`