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Term pack

Year 8 Maths

Name: ________________________

Skills in this pack

  • Properties of rational numbers
  • Playing with numbers
  • Squares and square roots
  • Cubes and cube roots
  • Exponents and powers
  • Discount and tax
  • Compound interest
  • Direct and inverse proportion
  • Linear equations with the unknown on both sides
  • Algebraic identities
  • Factorisation
  • Introduction to graphs
  • Mensuration
  • Understanding quadrilaterals
  • Grouped data, histograms and pie charts
  • Chance and probability

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Year 8 Maths · 1 of 16

Properties of rational numbers

Add, subtract, multiply and divide rational numbers, and use the properties that make the work shorter.

Before you start

Wander along this line in quarter steps. Nothing here is right or wrong — see how the numbers crowd together.

-2.00-1.000.001.002.00

The line only shows quarters, but between any two of them sits another rational number — an eighth, a sixteenth, and so on for ever. There is no 'next' rational number.

  1. 1.2/3 × 9/8 = ? Write your answer as a fraction in its simplest form.

  2. 2.Write the rational number that lies exactly halfway between 1/3 and 1/2.

  3. 3.4/9 × 7/5 + 4/9 × 3/5 = ? Write your answer as a fraction.

  4. 4.Sort each statement by whether it is true for every choice of rational numbers a, b and c.

    Groups: Always true · Not always true

    • a - b = b - a
    • a ÷ b = b ÷ a
    • a × b = b × a
    • a + b = b + a
    • (a + b) + c = a + (b + c)
    • (a - b) - c = a - (b - c)

  5. 5.Put these rational numbers in order, smallest first.

    • 0
    • -1/4
    • 3/8
    • 7/6
    • -5/3
    • -1/2

  6. 6.Tap where -7/4 sits.

    Mark the line with an X.

    -3.00-2.00-1.000.001.002.003.00

  7. 7.What is the reciprocal of -7/9? Write your answer as a fraction.

  8. 8.Match each rational number to its additive inverse — the number you add to it to get zero.

    • 2/7
    • -5/9
    • 3/4
    • -11/6
    • 8/3
    • 5/9
    • 11/6
    • -3/4
    • -2/7
    • -8/3

Properties of rational numbers · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 2 of 16

Playing with numbers

Use the divisibility tests and letter-for-digit puzzles to find missing digits.

  1. 1.5*7 is divisible by 3. What is the largest digit that can replace the star?

  2. 2.Put these numbers in order by how many of 2, 3, 5 and 9 divide into them exactly, fewest first.

    • 15
    • 7
    • 18
    • 90
    • 4

  3. 3.A two-digit number is 27 more than the number you get by reversing its digits, and its two digits add to 11. What is the number?

  4. 4.Match each divisibility test to the number it tests for.

    • The last digit is 0 or 5
    • The digits add to a multiple of 3
    • The last two digits make a multiple of 4
    • The digits add to a multiple of 9
    • The last digit is even
    • 3
    • 4
    • 2
    • 9
    • 5

  5. 5.In the sum A7 + 4A = 113, the letter A stands for one digit. What digit is A?

  6. 6.What is the smallest digit that can replace the star so that 3*5 is divisible by 3?

  7. 7.How can you tell that 4832 is divisible by 8 without dividing the whole number?

    • a) Its digits add to a multiple of 8
    • b) Its first digit is divisible by 8
    • c) It ends in an even digit
    • d) Its last three digits, 832, make a multiple of 8

  8. 8.What digit must replace the star so that 21*8 is divisible by 9?

Playing with numbers · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 3 of 16

Squares and square roots

Square whole numbers and find the square roots of perfect squares.

Before you start

Every whole number from zero to twenty-five sits on this line. Find 1, 4, 9, 16 and 25 on it, and look at the gaps between them.

0510152025

The gaps go 3, then 5, then 7, then 9 — the perfect squares spread out as you go, which is why so few numbers have a whole square root.

  1. 1.√144 = ?

  2. 2.17² = ?

  3. 3.A square garden has an area of 169 square metres. How long is each side, in metres?

  4. 4.What is the smallest whole number, apart from zero, you can multiply 18 by to get a perfect square?

  5. 5.Work each one out, then put them in order, smallest first.

    • √144
    • √16
    • √81
    • √196
    • √49

  6. 6.√0.25 = ?

  7. 7.Which of these is a perfect square?

    • a) 250
    • b) 150
    • c) 200
    • d) 196

  8. 8.13² = ?

Squares and square roots · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 4 of 16

Cubes and cube roots

Cube a number, recognise a perfect cube, and find a cube root by prime factorisation.

  1. 1.Put these in order by value, smallest first.

    • The cube of 3
    • The cube root of 64
    • The cube root of 8000
    • The cube of 2
    • The cube root of 8

  2. 2.What happens when a negative number is cubed?

    • a) It becomes zero
    • b) It stays negative
    • c) A negative number cannot be cubed
    • d) It turns positive

  3. 3.What is the cube root of 1000?

  4. 4.Which of these is a perfect cube?

    • a) 600
    • b) 512
    • c) 500
    • d) 1200

  5. 5.What is the cube root of -125?

  6. 6.The prime factors of 5832 are 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3. What is the cube root of 5832?

  7. 7.What is 7 cubed?

  8. 8.Match each number to its cube.

    • 2
    • 3
    • 5
    • 9
    • 11
    • 27
    • 8
    • 729
    • 1331
    • 125

Cubes and cube roots · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 5 of 16

Exponents and powers

Work out powers and use the laws of exponents to simplify them.

  1. 1.Simplify (2³)².

    • a) 2¹
    • b) 2⁶
    • c) 2⁵
    • d) 2⁹

  2. 2.What is 10⁴?

  3. 3.What is 3⁰?

  4. 4.Simplify a⁵ × a³.

    • a) 2a⁸
    • b) a¹⁵
    • c) a⁸
    • d) a²

  5. 5.What is 5²?

  6. 6.64 can be written as 2 to some power. What is that power?

  7. 7.What is 2³ (two to the power 3)?

  8. 8.Work out 2⁵ ÷ 2³ as an ordinary number.

Exponents and powers · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 6 of 16

Discount and tax

Work out a discount, a sale price, and the price once tax has been added.

  1. 1.Which of these offers takes the most off an item marked $1000?

    • a) 10 per cent off, and then 10 per cent off again
    • b) A discount of 30 per cent
    • c) A discount of one fifth
    • d) $250 off

  2. 2.Match each marked price to its sale price after a discount of 20 per cent.

    • $150
    • $250
    • $400
    • $600
    • $900
    • $320
    • $200
    • $480
    • $120
    • $720

  3. 3.A discount of 20 per cent followed by a further 20 per cent off the new price is the same as one single discount of how much?

    • a) 36 per cent
    • b) 44 per cent
    • c) 40 per cent
    • d) 30 per cent

  4. 4.Each offer is on an item marked $2000. Put them in order by what you end up paying, cheapest first.

    • $600 off
    • A quarter off
    • Half price
    • A discount of 10 per cent
    • A discount of 35 per cent

  5. 5.Sort each change by whether it makes the price go up or down.

    Groups: Price goes up · Price goes down

    • A service charge of 5 per cent added
    • A third off
    • A quarter off
    • Tax of 12 per cent added
    • A discount of 10 per cent
    • Tax of 8 per cent added

  6. 6.After a discount of 20 per cent, a lamp sells for $960. What was its marked price? Give just the number.

  7. 7.A shirt is marked $800 and is sold at a discount of 15 per cent. How much is taken off the marked price? Give just the number.

  8. 8.A toy costs $600 before tax. With tax of 5 per cent added, what is the total to pay? Give just the number.

Discount and tax · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 7 of 16

Compound interest

Work out compound interest and the amount it grows to, and use the same idea for growth and depreciation.

  1. 1.A machine worth $20000 loses 10 per cent of its value every year. What is it worth after 2 years? Give just the number.

  2. 2.$8000 is invested at 5 per cent per year, compounded yearly, for 2 years. What amount does it grow to? Give just the number.

  3. 3.$4000 is lent at 10 per cent per year, compounded half-yearly, for 1 year. What amount is owed at the end? Give just the number.

  4. 4.A town of 10000 people grows by 10 per cent every year. How many people live there after 2 years?

  5. 5.The simple interest on $4000 at 10 per cent for 2 years is $800. What is the compound interest on the same sum at the same rate and time? Give just the number.

  6. 6.$1000 is invested at 10 per cent per year, compounded yearly. Match each length of time to the amount it grows to.

    • 1 year
    • 2 years
    • 3 years
    • 4 years
    • 5 years
    • $1100
    • $1610.51
    • $1464.10
    • $1210
    • $1331

  7. 7.$2000 is invested at 10 per cent per year, compounded yearly, for 3 years. What amount does it grow to? Give just the number.

  8. 8.Each card is a rate at which $1000 is invested for 2 years, compounded yearly. Put them in order by the amount at the end, smallest first.

    • 8 per cent per year
    • 20 per cent per year
    • 15 per cent per year
    • 10 per cent per year
    • 5 per cent per year

Compound interest · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 8 of 16

Direct and inverse proportion

Tell direct proportion from inverse proportion, and use each to find a missing quantity.

  1. 1.A journey takes 4 hours at 60 km per hour. How many hours would it take at 80 km per hour?

  2. 2.If 5 pens cost $60, what do 8 pens cost? Give just the number.

  3. 3.y is in inverse proportion to x. When x is 4, y is 15. What is y when x is 10?

  4. 4.6 workers take 12 days to finish a job. Match each number of workers to the days they would take at the same rate.

    • 4 workers
    • 8 workers
    • 9 workers
    • 12 workers
    • 18 workers
    • 8 days
    • 4 days
    • 9 days
    • 6 days
    • 18 days

  5. 5.12 pipes fill a tank in 20 minutes. How many minutes would 15 pipes take?

  6. 6.If 4 kg of rice costs $200, what do 7 kg cost? Give just the number.

  7. 7.6 workers build a wall in 12 days. How many days would 8 workers take, working at the same rate?

  8. 8.A job takes 6 people 12 days. Put these teams in order by how long they would take, shortest first.

    • 12 people
    • 6 people
    • 9 people
    • 4 people
    • 18 people

Direct and inverse proportion · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 9 of 16

Linear equations with the unknown on both sides

Solve equations that have the unknown on both sides, and use them to answer word problems.

Before you start

The scale is level, and there is a mystery box on each side. Move the weights about and find out what keeps it level and what tips it over.

37

Lift one box off each side and the beam does not move at all. That is what “take the unknown off both sides” means, and it is why the boxes on the two sides can be gathered into one.

  1. 1.Solve x/5 + 2 = 5. x = ?

  2. 2.Solve 7 - 2x = x + 1. x = ?

  3. 3.Match each equation to its solution.

    • 2x + 3 = 11
    • 3x - 1 = 14
    • 5x + 4 = 4x + 10
    • 7x = 4x + 21
    • 2x - 9 = x - 1
    • 8
    • 4
    • 5
    • 6
    • 7

  4. 4.Sort each equation by whether its solution is positive or negative.

    Groups: x is positive · x is negative

    • x + 6 = 2x
    • 4x + 7 = 2x + 1
    • 6x + 5 = 4x - 1
    • 5x - 2 = 3x + 6
    • 3x = x + 8
    • 2x + 9 = x + 4

  5. 5.Which value of x makes 6x - 5 = 4x + 3 true?

    • a) 4
    • b) 1
    • c) 2
    • d) -4

  6. 6.The sum of two consecutive whole numbers is 45. Write the smaller number.

  7. 7.Solve 4(x - 3) = 2x + 6. x = ?

  8. 8.A rectangle is 3 cm longer than it is wide, and its perimeter is 26 cm. How wide is it, in cm? Give just the number.

Linear equations with the unknown on both sides · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 10 of 16

Algebraic identities

Use (a + b)², (a - b)² and a² - b² to expand brackets and to multiply quickly.

  1. 1.If a + b = 10 and ab = 21, what is a² + b²?

  2. 2.Expand (x + 3)².

    • a) x² + 3x + 9
    • b) x² + 9
    • c) x² + 6x + 6
    • d) x² + 6x + 9

  3. 3.Factorise a² - b².

    • a) (a + b)(a - b)
    • b) a(a - b)
    • c) (a - b)(a - b)
    • d) (a + b)(a + b)

  4. 4.Expand (a + b)².

    • a) 2a + 2b
    • b) a² + b²
    • c) a² - 2ab + b²
    • d) a² + 2ab + b²

  5. 5.Expand (a - b)².

    • a) a² - b²
    • b) a² + 2ab + b²
    • c) a² + b²
    • d) a² - 2ab + b²

  6. 6.Use a² - b² = (a + b)(a - b) to work out 105 × 95.

  7. 7.Expand (x - 5)².

    • a) x² - 10x - 25
    • b) x² - 25
    • c) x² - 10x + 25
    • d) x² + 10x + 25

  8. 8.Use (100 - 2)² to work out 98².

Algebraic identities · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 11 of 16

Factorisation

Break an algebraic expression into factors, and use those factors to divide one expression by another.

  1. 1.Which of these is x² + 10x + 25 written in factorised form?

    • a) (x + 5)²
    • b) (x + 10)²
    • c) (x + 25)(x + 1)
    • d) (x + 5)(x - 5)

  2. 2.Divide x² + 5x + 6 by x + 2. Write the answer.

  3. 3.Take a out of a² + ab as a common factor. What is left inside the bracket?

  4. 4.Put these expressions in order by the highest common factor of their terms, smallest first.

    • 14x + 21
    • 4x + 8
    • 10x + 25
    • 6x + 18
    • 3x + 9

  5. 5.Which of these is 4x² - 9 written in factorised form?

    • a) (2x - 3)(2x - 3)
    • b) (2x - 9)(2x + 1)
    • c) (2x - 3)(2x + 3)
    • d) (4x - 3)(x + 3)

  6. 6.Take 5 out of 5y - 20 as a common factor. What is left inside the bracket?

  7. 7.Divide 24y³ - 16y² by 8y². Write the answer.

  8. 8.Match each expression to its factorised form.

    • x² - 16
    • x² + 8x + 16
    • 3x + 12
    • x² - 4x
    • 2x² - 32
    • 2(x - 4)(x + 4)
    • (x - 4)(x + 4)
    • x(x - 4)
    • (x + 4)²
    • 3(x + 4)

Factorisation · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 12 of 16

Introduction to graphs

Plot and read points on a grid, read a line graph, and work out values from a simple linear relation.

Before you start

This graph follows one journey from start to finish, with time along the bottom and distance up the side. Find out what each stretch of the line says about it.

1234
  1. 1. Rising — moving away
  2. 2. Flat — standing still
  3. 3. Steeper — faster
  4. 4. The origin

Flat does not mean time has stopped — the clock still runs along the bottom. It means the distance stopped changing, and standing still is exactly what that looks like on a graph.

  1. 1.A line graph shows a plant 4 cm tall on day 1 and 16 cm tall on day 7. How much did it grow over those six days, in cm? Give just the number.

  2. 2.On a distance-time graph, what does a flat horizontal part of the line mean?

    • a) The object is going backwards
    • b) The graph has been drawn wrongly
    • c) The object is speeding up
    • d) The object is not moving

  3. 3.Sort each point by where it sits on the grid.

    Groups: On the x-axis · On the y-axis · On neither axis

    • (0, -1)
    • (2, 7)
    • (-4, 0)
    • (0, 5)
    • (6, 6)
    • (3, 0)

  4. 4.A car travels at a steady 50 km per hour. On a distance-time graph, how far has it gone after 3 hours, in km? Give just the number.

  5. 5.A shop sells notebooks at $20 each. On a graph of cost against number bought, what is the cost of 7 notebooks? Give just the number.

  6. 6.This distance-time graph shows one journey in three parts. Tap the part where the car was not moving.

    Write the number of the part.

    123

  7. 7.A point sits 4 units to the right of the origin and 7 units up. Write its y-coordinate.

  8. 8.For the relation y = 3x, what is y when x is 6?

Introduction to graphs · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 13 of 16

Mensuration

Find areas of trapeziums and circles, and volumes and surface areas of cuboids and cylinders.

  1. 1.A cube has edges of 5 cm. What is its total surface area, in square centimetres?

  2. 2.A circle has a radius of 14 cm. Take π = 22/7. What is its area, in square centimetres?

  3. 3.A cylinder has a radius of 7 cm and a height of 10 cm. Take π = 22/7. What is its curved surface area, in square centimetres?

  4. 4.A cube has edges of 5 cm. What is its volume, in cubic centimetres?

  5. 5.A trapezium has parallel sides of 10 cm and 6 cm, and a height of 4 cm. What is its area, in square centimetres?

  6. 6.A cylinder has a radius of 7 cm and a height of 10 cm. Take π = 22/7. What is its total surface area, in square centimetres?

  7. 7.Sort each formula by what it measures.

    Groups: Measures area · Measures volume

    • π × r × r
    • length × width × height
    • edge × edge × edge
    • π × r × r × h
    • length × width
    • 2 × π × r × h

  8. 8.Which of these gives the volume of a cylinder?

    • a) π × r × r
    • b) 2 × π × r × h
    • c) 2 × π × r
    • d) π × r × r × h

Mensuration · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 14 of 16

Understanding quadrilaterals

Work out the angles in a polygon and use the properties of parallelograms, rhombuses, rectangles and squares.

Before you start

The four corners of this parallelogram hold four angles. Find out how big each one is — nothing here is right or wrong.

1234
  1. 1. Angle A — 45°
  2. 2. Angle B — 135°
  3. 3. Angle C — 45°
  4. 4. Angle D — 135°

Opposite corners match. Any two corners next to each other add up to 180°, which is why one known angle gives you all four.

  1. 1.Three angles of a quadrilateral are 80, 100 and 65 degrees. What is the fourth, in degrees? Give just the number.

  2. 2.What do the interior angles of a hexagon add up to, in degrees? Give just the number.

  3. 3.Which of these is true of every rhombus but not of every rectangle?

    • a) Opposite sides are parallel
    • b) Opposite angles are equal
    • c) The angles add up to 360 degrees
    • d) All four sides are the same length

  4. 4.What do the interior angles of a pentagon add up to, in degrees? Give just the number.

  5. 5.Sort each property by whether it is true of every parallelogram.

    Groups: True of every parallelogram · Not true of every parallelogram

    • Every angle is a right angle
    • The diagonals cut each other in half
    • All four sides are equal
    • Opposite angles are equal
    • The diagonals are equal in length
    • Opposite sides are equal

  6. 6.Each exterior angle of a regular polygon is 45 degrees. How many sides has it?

  7. 7.In which quadrilateral do the diagonals always cross at right angles and always come out the same length?

    • a) A parallelogram
    • b) A square
    • c) A rectangle
    • d) A rhombus

  8. 8.Put these polygons in order by what their interior angles add up to, smallest first.

    • Octagon
    • Hexagon
    • Quadrilateral
    • Pentagon
    • Triangle

Understanding quadrilaterals · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 15 of 16

Grouped data, histograms and pie charts

Group data into class intervals, read a histogram, and work out the angles in a pie chart.

Before you start

This pie chart shows what a class of 40 children picked for their club. Find out what each slice is and how wide its angle is — nothing here is right or wrong.

1234
  1. 1. Reading — 90°, so 10 children
  2. 2. Sport — 90°, so 10 children
  3. 3. Music — 108°, so 12 children
  4. 4. Art — 72°, so 8 children

The four angles add up to 360°, one whole turn. Music is the biggest slice because 12 out of 40 is the biggest share — and 12/40 of 360° really is 108°.

  1. 1.A pie chart shows 200 people. One sector measures 72 degrees. How many people does that sector stand for?

  2. 2.This histogram shows the marks of a class. Tap the interval that holds the most students.

    Write the number of the part.

    1234

  3. 3.These marks are grouped in intervals of 10 starting at 0: 5, 12, 18, 23, 27, 31, 34, 38, 45. How many marks fall in the interval 30 to 40?

  4. 4.What is the main difference between a bar graph and a histogram?

    • a) A histogram has to be drawn in colour
    • b) A histogram has no gaps between its bars, because its classes run on from each other
    • c) A histogram always has more bars than a bar graph
    • d) A bar graph cannot show how many there are of anything

  5. 5.A pie chart is split between 4 subjects with an equal share each. What angle does each sector take, in degrees? Give just the number.

  6. 6.Match each share of the whole to the angle of its sector in a pie chart, in degrees.

    • Half
    • A quarter
    • A third
    • A fifth
    • A tenth
    • 90
    • 120
    • 72
    • 36
    • 180

  7. 7.Put these pie-chart sectors in order by the size of their angle, smallest first.

    • 1/4 of the whole
    • 1/6 of the whole
    • 1/3 of the whole
    • 1/12 of the whole
    • 1/8 of the whole

  8. 8.Sort each mark into the class interval it belongs to.

    Groups: 0 to 20 · 20 to 40 · 40 to 60

    • 41
    • 27
    • 19
    • 8
    • 52
    • 35

Grouped data, histograms and pie charts · Year 8 Maths · www.arenapublications.com/learn

Year 8 Maths · 16 of 16

Chance and probability

Say how likely an event is, and write its probability as a fraction.

Before you start

Every chance there is sits somewhere on this line, from impossible at one end to certain at the other. Move along it.

0.00.51.0

0.5 in the middle is an even chance — as likely to happen as not. To the left of it things are unlikely and to the right they are likely, and nothing at all lives past the two ends.

  1. 1.A box holds 4 green, 6 yellow and 10 white counters. One is drawn without looking. What is the probability that it is yellow? Write your answer as a fraction.

  2. 2.An event that is certain to happen has which probability?

    • a) 1/2
    • b) 1
    • c) 0
    • d) 100

  3. 3.A bag holds 5 red balls and 3 blue balls. One is drawn without looking. What is the probability that it is red? Write your answer as a fraction.

  4. 4.A fair die numbered 1 to 6 is rolled. What is the probability of getting an even number? Write your answer as a fraction.

  5. 5.A fair die numbered 1 to 6 is rolled. Which of these is impossible?

    • a) Getting an odd number
    • b) Getting a number less than 3
    • c) Getting a 6
    • d) Getting a 7

  6. 6.A fair coin is tossed. What is the probability of getting a head? Write your answer as a fraction.

  7. 7.Put these events in order, least likely first.

    • Rolling a 5 on a die numbered 1 to 6
    • Rolling a number above 4 on a die numbered 1 to 6
    • Rolling a number below 7 on a die numbered 1 to 6
    • Rolling a 7 on a die numbered 1 to 6
    • Tossing a fair coin and getting a head

  8. 8.A fair die numbered 1 to 6 is rolled. Match each event to its probability.

    • Rolling a 3
    • Rolling an odd number
    • Rolling a number above 4
    • Rolling a number below 7
    • Rolling a 9
    • 1/2
    • 1
    • 1/3
    • 0
    • 1/6

Chance and probability · Year 8 Maths · www.arenapublications.com/learn

Answer keys — Year 8 Maths

In the same order as the worksheets.

Properties of rational numbers

  1. 1. 3/4
  2. 2. 5/12
  3. 3. 8/9
  4. 4. a + b = b + a → Always true; a - b = b - a → Not always true; a × b = b × a → Always true; a ÷ b = b ÷ a → Not always true; (a + b) + c = a + (b + c) → Always true; (a - b) - c = a - (b - c) → Not always true
  5. 5. 1. -5/3 2. -1/2 3. -1/4 4. 0 5. 3/8 6. 7/6
  6. 6. -1.75
  7. 7. -9/7
  8. 8. 2/7 → -2/7; -5/9 → 5/9; 3/4 → -3/4; -11/6 → 11/6; 8/3 → -8/3

Playing with numbers

  1. 1. 9
  2. 2. 1. 7 2. 4 3. 15 4. 18 5. 90
  3. 3. 74
  4. 4. The last digit is 0 or 5 → 5; The digits add to a multiple of 3 → 3; The last two digits make a multiple of 4 → 4; The digits add to a multiple of 9 → 9; The last digit is even → 2
  5. 5. 6
  6. 6. 1
  7. 7. d) Its last three digits, 832, make a multiple of 8
  8. 8. 7

Squares and square roots

  1. 1. 12
  2. 2. 289
  3. 3. 13
  4. 4. 2
  5. 5. 1. √16 2. √49 3. √81 4. √144 5. √196
  6. 6. 0.5
  7. 7. d) 196
  8. 8. 169

Cubes and cube roots

  1. 1. 1. The cube root of 8 2. The cube root of 64 3. The cube of 2 4. The cube root of 8000 5. The cube of 3
  2. 2. b) It stays negative
  3. 3. 10
  4. 4. b) 512
  5. 5. -5
  6. 6. 18
  7. 7. 343
  8. 8. 2 → 8; 3 → 27; 5 → 125; 9 → 729; 11 → 1331

Exponents and powers

  1. 1. b) 2⁶
  2. 2. 10000
  3. 3. 1
  4. 4. c) a⁸
  5. 5. 25
  6. 6. 6
  7. 7. 8
  8. 8. 4

Discount and tax

  1. 1. b) A discount of 30 per cent
  2. 2. $150 → $120; $250 → $200; $400 → $320; $600 → $480; $900 → $720
  3. 3. a) 36 per cent
  4. 4. 1. Half price 2. A discount of 35 per cent 3. $600 off 4. A quarter off 5. A discount of 10 per cent
  5. 5. A discount of 10 per cent → Price goes down; Tax of 8 per cent added → Price goes up; A quarter off → Price goes down; A service charge of 5 per cent added → Price goes up; A third off → Price goes down; Tax of 12 per cent added → Price goes up
  6. 6. 1200
  7. 7. 120
  8. 8. 630

Compound interest

  1. 1. 16200
  2. 2. 8820
  3. 3. 4410
  4. 4. 12100
  5. 5. 840
  6. 6. 1 year → $1100; 2 years → $1210; 3 years → $1331; 4 years → $1464.10; 5 years → $1610.51
  7. 7. 2662
  8. 8. 1. 5 per cent per year 2. 8 per cent per year 3. 10 per cent per year 4. 15 per cent per year 5. 20 per cent per year

Direct and inverse proportion

  1. 1. 3
  2. 2. 96
  3. 3. 6
  4. 4. 4 workers → 18 days; 8 workers → 9 days; 9 workers → 8 days; 12 workers → 6 days; 18 workers → 4 days
  5. 5. 16
  6. 6. 350
  7. 7. 9
  8. 8. 1. 18 people 2. 12 people 3. 9 people 4. 6 people 5. 4 people

Linear equations with the unknown on both sides

  1. 1. 15
  2. 2. 2
  3. 3. 2x + 3 = 11 → 4; 3x - 1 = 14 → 5; 5x + 4 = 4x + 10 → 6; 7x = 4x + 21 → 7; 2x - 9 = x - 1 → 8
  4. 4. 3x = x + 8 → x is positive; 2x + 9 = x + 4 → x is negative; 5x - 2 = 3x + 6 → x is positive; 4x + 7 = 2x + 1 → x is negative; x + 6 = 2x → x is positive; 6x + 5 = 4x - 1 → x is negative
  5. 5. a) 4
  6. 6. 22
  7. 7. 9
  8. 8. 5

Algebraic identities

  1. 1. 58
  2. 2. d) x² + 6x + 9
  3. 3. a) (a + b)(a - b)
  4. 4. d) a² + 2ab + b²
  5. 5. d) a² - 2ab + b²
  6. 6. 9975
  7. 7. c) x² - 10x + 25
  8. 8. 9604

Factorisation

  1. 1. a) (x + 5)²
  2. 2. x + 3
  3. 3. a + b
  4. 4. 1. 3x + 9 2. 4x + 8 3. 10x + 25 4. 6x + 18 5. 14x + 21
  5. 5. c) (2x - 3)(2x + 3)
  6. 6. y - 4
  7. 7. 3y - 2
  8. 8. x² - 16 → (x - 4)(x + 4); x² + 8x + 16 → (x + 4)²; 3x + 12 → 3(x + 4); x² - 4x → x(x - 4); 2x² - 32 → 2(x - 4)(x + 4)

Introduction to graphs

  1. 1. 12
  2. 2. d) The object is not moving
  3. 3. (3, 0) → On the x-axis; (0, 5) → On the y-axis; (2, 7) → On neither axis; (-4, 0) → On the x-axis; (0, -1) → On the y-axis; (6, 6) → On neither axis
  4. 4. 150
  5. 5. 140
  6. 6. 2 — Stopped by the roadside
  7. 7. 7
  8. 8. 18

Mensuration

  1. 1. 150
  2. 2. 616
  3. 3. 440
  4. 4. 125
  5. 5. 32
  6. 6. 748
  7. 7. π × r × r → Measures area; length × width → Measures area; 2 × π × r × h → Measures area; length × width × height → Measures volume; π × r × r × h → Measures volume; edge × edge × edge → Measures volume
  8. 8. d) π × r × r × h

Understanding quadrilaterals

  1. 1. 115
  2. 2. 720
  3. 3. d) All four sides are the same length
  4. 4. 540
  5. 5. Opposite sides are equal → True of every parallelogram; Opposite angles are equal → True of every parallelogram; The diagonals cut each other in half → True of every parallelogram; All four sides are equal → Not true of every parallelogram; The diagonals are equal in length → Not true of every parallelogram; Every angle is a right angle → Not true of every parallelogram
  6. 6. 8
  7. 7. b) A square
  8. 8. 1. Triangle 2. Quadrilateral 3. Pentagon 4. Hexagon 5. Octagon

Grouped data, histograms and pie charts

  1. 1. 40
  2. 2. 2 — 10 to 20 marks — 9 students
  3. 3. 3
  4. 4. b) A histogram has no gaps between its bars, because its classes run on from each other
  5. 5. 90
  6. 6. Half → 180; A quarter → 90; A third → 120; A fifth → 72; A tenth → 36
  7. 7. 1. 1/12 of the whole 2. 1/8 of the whole 3. 1/6 of the whole 4. 1/4 of the whole 5. 1/3 of the whole
  8. 8. 8 → 0 to 20; 35 → 20 to 40; 52 → 40 to 60; 19 → 0 to 20; 41 → 40 to 60; 27 → 20 to 40

Chance and probability

  1. 1. 3/10
  2. 2. b) 1
  3. 3. 5/8
  4. 4. 1/2
  5. 5. d) Getting a 7
  6. 6. 1/2
  7. 7. 1. Rolling a 7 on a die numbered 1 to 6 2. Rolling a 5 on a die numbered 1 to 6 3. Rolling a number above 4 on a die numbered 1 to 6 4. Tossing a fair coin and getting a head 5. Rolling a number below 7 on a die numbered 1 to 6
  8. 8. Rolling a 3 → 1/6; Rolling an odd number → 1/2; Rolling a number above 4 → 1/3; Rolling a number below 7 → 1; Rolling a 9 → 0

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