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Term pack

Class 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
  • Linear equations with the unknown on both sides
  • Algebraic identities
  • Factorisation
  • Discount and tax
  • Compound interest
  • Direct and inverse proportion
  • Understanding quadrilaterals
  • Mensuration
  • Grouped data, histograms and pie charts
  • Chance and probability
  • Introduction to graphs

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Class 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.Tap where -7/4 sits.

    Mark the line with an X.

    -3.00-2.00-1.000.001.002.003.00

  2. 2.-3/5 + 3/5 = ?

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

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

  4. 4.4/9 × 7/5 + 4/9 × 3/5 = ? Write your answer as a fraction.

  5. 5.1/2 ÷ -2/3 = ? Write your answer as a fraction.

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

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

  8. 8.Rational numbers are closed under addition, subtraction and multiplication. Which operation is the exception, because one case has no answer at all?

    • a) Division, when you divide by zero
    • b) Addition, when both numbers are negative
    • c) Subtraction, when the answer comes out negative
    • d) Multiplication, when one number is a fraction

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

Class 8 Maths · 2 of 16

Playing with numbers

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

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

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

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

    • 15
    • 4
    • 7
    • 18
    • 90

  4. 4.A two-digit number is written as 10a + b, where a is its tens digit and b is its units digit. If a is 4 and b is 7, what is the number?

  5. 5.Take any two-digit number and subtract the number made by reversing its digits. The difference is always divisible by which number?

  6. 6.5*7 is divisible by 3. What is the largest digit that can replace the star?

  7. 7.Take any two-digit number and add the number made by reversing its digits. The total is always divisible by which number?

  8. 8.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
    • 5
    • 4
    • 9
    • 2

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

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

  2. 2.√0.25 = ?

  3. 3.13² = ?

  4. 4.√576 = ?

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

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

  6. 6.17² = ?

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

  8. 8.√225 = ?

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

Class 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.What is the cube root of 3375?

  2. 2.What is 7 cubed?

  3. 3.What is the cube root of 216?

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

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

  5. 5.Sort each number by whether it is a perfect cube.

    Groups: A perfect cube · Not a perfect cube

    • 300
    • 343
    • 100
    • 500
    • 64
    • 125
    • 216

  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 the cube root of 1000?

  8. 8.392 is 2 × 2 × 2 × 7 × 7. What is the smallest number, apart from zero, it must be multiplied by to make a perfect cube?

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

Class 8 Maths · 5 of 16

Exponents and powers

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

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

  2. 2.What is 10⁴?

  3. 3.Match each power to its value.

    • 2⁴
    • 3³
    • 5²
    • 10³
    • 7²
    • 49
    • 27
    • 1000
    • 25
    • 16

  4. 4.Simplify (2³)².

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

  5. 5.Work out each one, then sort it by its value.

    Groups: Equals 8 · Equals 16

    • 16¹
    • 8¹
    • 2 × 4
    • 4²
    • 2⁴
    • 2³

  6. 6.What is 3⁰?

  7. 7.What is 5²?

  8. 8.Write 2⁵ ÷ 2³ as a power of 2.

    • a) 2
    • b) 2²
    • c) 2⁸
    • d) 2¹⁵

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

Class 8 Maths · 6 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 7 - 2x = x + 1. x = ?

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

    Groups: x is positive · x is negative

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

  3. 3.Solve (x + 4)/3 = 2. x = ?

  4. 4.Why may the same number be taken away from both sides of an equation?

    • a) Because taking away always makes an equation simpler
    • b) Because the two sides stay equal if you do the same thing to each
    • c) Because the unknown must end up on the left
    • d) Because the left side is always the more important one

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

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

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

  7. 7.Match each equation to its solution.

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

  8. 8.Solve 3x + 5 = 2x + 11. x = ?

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

Class 8 Maths · 7 of 16

Algebraic identities

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

  1. 1.Expand (a + b)².

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

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

  3. 3.Factorise a² - b².

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

  4. 4.Match each product to its expansion.

    • (x + 1)²
    • (x - 1)²
    • (x + 2)(x - 2)
    • (x + 4)²
    • x² - 4
    • x² + 8x + 16
    • x² - 2x + 1
    • x² + 2x + 1

  5. 5.Expand (x - 5)².

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

  6. 6.Expand (a - b)².

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

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

  8. 8.Use (50 + 1)² to work out 51².

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

Class 8 Maths · 8 of 16

Factorisation

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

  1. 1.x² - 49 factorises as (x - 7)(x + ?). What number belongs in place of the question mark?

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

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

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

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

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

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

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

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

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

  8. 8.What is the highest common factor of the two terms in 12x + 18?

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

Class 8 Maths · 9 of 16

Discount and tax

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

  1. 1.A bag is marked ₹1200 and is sold at a discount of 25 per cent. What is the sale price? Give just the number.

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

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

    Groups: Price goes up · Price goes down

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

  4. 4.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 one fifth
    • c) ₹250 off
    • d) A discount of 30 per cent

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

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

  6. 6.A book costs ₹250 and tax of 8 per cent is added. How much is the tax? Give just the number.

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

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

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

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

Class 8 Maths · 10 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.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.

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

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

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

  4. 4.₹8000 is invested at 5 per cent per year, compounded yearly, for 2 years. What amount does it grow to? Give just the number.

  5. 5.Find the compound interest on ₹1000 at 20 per cent per year for 2 years. Give just the number.

  6. 6.Sort each description by which kind of interest it belongs to.

    Groups: Simple interest · Compound interest

    • The amount grows faster and faster as the years go by
    • Interest is always worked out on the original sum only
    • Doubling the time doubles the interest exactly
    • A population rising by the same percentage every year
    • The interest is the same amount every year
    • Each year's interest is added on before the next year is worked out

  7. 7.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.

  8. 8.₹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
    • ₹1464.10
    • ₹1210
    • ₹1100
    • ₹1610.51
    • ₹1331

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

Class 8 Maths · 11 of 16

Direct and inverse proportion

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

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

  2. 2.A car travels 180 km in 3 hours at a steady speed. How far does it go in 5 hours, in km? Give just the number.

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

  4. 4.Two quantities are in inverse proportion. Which of these stays the same as they change?

    • a) Their product
    • b) Their sum
    • c) Their difference
    • d) Their ratio

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

  6. 6.12 pipes fill a tank in 20 minutes. How many minutes would 15 pipes take?

  7. 7.6 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
    • 6 days
    • 18 days
    • 9 days
    • 8 days
    • 4 days

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

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

Class 8 Maths · 12 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 pentagon add up to, in degrees? Give just the number.

  3. 3.One angle of a parallelogram is 65 degrees. What is the angle next to it, in degrees? Give just the number.

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

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

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

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

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

  6. 6.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

  7. 7.What do the four angles of any quadrilateral add up to, in degrees? Give just the number.

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

    • Hexagon
    • Octagon
    • Pentagon
    • Quadrilateral
    • Triangle

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

Class 8 Maths · 13 of 16

Mensuration

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

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

    Groups: Measures area · Measures volume

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

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

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

  4. 4.A cuboid is 12 cm long, 8 cm wide and 5 cm tall. What is its total surface area, in square centimetres?

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

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

  7. 7.A cuboid is 12 cm long, 8 cm wide and 5 cm tall. What is its volume, in cubic centimetres?

  8. 8.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?

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

Class 8 Maths · 14 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 class interval runs from 20 to 30. What is its class size?

  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.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.

  4. 4.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
    • 180
    • 36
    • 120
    • 90
    • 72

  5. 5.Data is grouped as 10-20, 20-30 and 30-40. Which interval does a value of exactly 20 belong to?

    • a) 10-20
    • b) Both of them
    • c) 20-30
    • d) Neither of them

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

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

  7. 7.In a pie chart of 60 students, 15 of them chose reading. What angle does that sector take, in degrees? Give just the number.

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

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

Class 8 Maths · 15 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.Sort each event by how likely it is.

    Groups: Impossible · Possible but not certain · Certain

    • Rolling a number below 7 on a die numbered 1 to 6
    • Drawing a red ball from a bag of only red balls
    • Drawing a blue ball from a bag of only red balls
    • Rolling a 7 on a die numbered 1 to 6
    • Tossing a fair coin and getting a tail
    • Rolling an even number on a die numbered 1 to 6

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

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

  3. 3.A spinner has 8 equal sections numbered 1 to 8. What is the probability of landing on a number greater than 5? 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. 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
    • 1/6
    • 1
    • 1/2
    • 0
    • 1/3

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

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

  7. 7.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.

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

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

Class 8 Maths · 16 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 point sits 4 units to the right of the origin and 7 units up. Write its y-coordinate.

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

  3. 3.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.

  4. 4.For the relation y = 4x - 1, match each value of x to its value of y.

    • x = 1
    • x = 2
    • x = 3
    • x = 5
    • x = 6
    • 19
    • 23
    • 7
    • 11
    • 3

  5. 5.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

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

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

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

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

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

  8. 8.For the relation y = 2x + 1, what is y when x is 5?

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

Answer keys — Class 8 Maths

In the same order as the worksheets.

Properties of rational numbers

  1. 1. -1.75
  2. 2. 0
  3. 3. 1. -5/3 2. -1/2 3. -1/4 4. 0 5. 3/8 6. 7/6
  4. 4. 8/9
  5. 5. -3/4
  6. 6. 2/7 → -2/7; -5/9 → 5/9; 3/4 → -3/4; -11/6 → 11/6; 8/3 → -8/3
  7. 7. 5/12
  8. 8. a) Division, when you divide by zero

Playing with numbers

  1. 1. 7
  2. 2. 6
  3. 3. 1. 7 2. 4 3. 15 4. 18 5. 90
  4. 4. 47
  5. 5. 9
  6. 6. 9
  7. 7. 11
  8. 8. 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

Squares and square roots

  1. 1. 13
  2. 2. 0.5
  3. 3. 169
  4. 4. 24
  5. 5. 1. √16 2. √49 3. √81 4. √144 5. √196
  6. 6. 289
  7. 7. 2
  8. 8. 15

Cubes and cube roots

  1. 1. 15
  2. 2. 343
  3. 3. 6
  4. 4. d) 512
  5. 5. 64 → A perfect cube; 100 → Not a perfect cube; 125 → A perfect cube; 216 → A perfect cube; 300 → Not a perfect cube; 343 → A perfect cube; 500 → Not a perfect cube
  6. 6. 18
  7. 7. 10
  8. 8. 7

Exponents and powers

  1. 1. 8
  2. 2. 10000
  3. 3. 2⁴ → 16; 3³ → 27; 5² → 25; 10³ → 1000; 7² → 49
  4. 4. d) 2⁶
  5. 5. 2³ → Equals 8; 2⁴ → Equals 16; 8¹ → Equals 8; 4² → Equals 16; 2 × 4 → Equals 8; 16¹ → Equals 16
  6. 6. 1
  7. 7. 25
  8. 8. b) 2²

Linear equations with the unknown on both sides

  1. 1. 2
  2. 2. 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
  3. 3. 2
  4. 4. b) Because the two sides stay equal if you do the same thing to each
  5. 5. b) 4
  6. 6. 22
  7. 7. 2x + 3 = 11 → 4; 3x - 1 = 14 → 5; 5x + 4 = 4x + 10 → 6; 7x = 4x + 21 → 7; 2x - 9 = x - 1 → 8
  8. 8. 6

Algebraic identities

  1. 1. d) a² + 2ab + b²
  2. 2. 9975
  3. 3. a) (a + b)(a - b)
  4. 4. (x + 1)² → x² + 2x + 1; (x - 1)² → x² - 2x + 1; (x + 2)(x - 2) → x² - 4; (x + 4)² → x² + 8x + 16
  5. 5. d) x² - 10x + 25
  6. 6. c) a² - 2ab + b²
  7. 7. 9604
  8. 8. 2601

Factorisation

  1. 1. 7
  2. 2. c) (2x - 3)(2x + 3)
  3. 3. c) (x + 5)²
  4. 4. y - 4
  5. 5. a + b
  6. 6. x + 3
  7. 7. 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)
  8. 8. 6

Discount and tax

  1. 1. 900
  2. 2. 630
  3. 3. 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
  4. 4. d) A discount of 30 per cent
  5. 5. 1. Half price 2. A discount of 35 per cent 3. ₹600 off 4. A quarter off 5. A discount of 10 per cent
  6. 6. 20
  7. 7. ₹150 → ₹120; ₹250 → ₹200; ₹400 → ₹320; ₹600 → ₹480; ₹900 → ₹720
  8. 8. 1200

Compound interest

  1. 1. 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
  2. 2. 12100
  3. 3. 16200
  4. 4. 8820
  5. 5. 440
  6. 6. The interest is the same amount every year → Simple interest; Each year's interest is added on before the next year is worked out → Compound interest; Interest is always worked out on the original sum only → Simple interest; The amount grows faster and faster as the years go by → Compound interest; Doubling the time doubles the interest exactly → Simple interest; A population rising by the same percentage every year → Compound interest
  7. 7. 840
  8. 8. 1 year → ₹1100; 2 years → ₹1210; 3 years → ₹1331; 4 years → ₹1464.10; 5 years → ₹1610.51

Direct and inverse proportion

  1. 1. 6
  2. 2. 300
  3. 3. 350
  4. 4. a) Their product
  5. 5. 96
  6. 6. 16
  7. 7. 4 workers → 18 days; 8 workers → 9 days; 9 workers → 8 days; 12 workers → 6 days; 18 workers → 4 days
  8. 8. 3

Understanding quadrilaterals

  1. 1. 115
  2. 2. 540
  3. 3. 115
  4. 4. 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
  5. 5. d) A square
  6. 6. d) All four sides are the same length
  7. 7. 360
  8. 8. 1. Triangle 2. Quadrilateral 3. Pentagon 4. Hexagon 5. Octagon

Mensuration

  1. 1. 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
  2. 2. 32
  3. 3. 616
  4. 4. 392
  5. 5. 88
  6. 6. 125
  7. 7. 480
  8. 8. 748

Grouped data, histograms and pie charts

  1. 1. 10
  2. 2. 2 — 10 to 20 marks — 9 students
  3. 3. 90
  4. 4. Half → 180; A quarter → 90; A third → 120; A fifth → 72; A tenth → 36
  5. 5. c) 20-30
  6. 6. c) A histogram has no gaps between its bars, because its classes run on from each other
  7. 7. 90
  8. 8. 40

Chance and probability

  1. 1. Rolling a 7 on a die numbered 1 to 6 → Impossible; Tossing a fair coin and getting a tail → Possible but not certain; Rolling a number below 7 on a die numbered 1 to 6 → Certain; Drawing a red ball from a bag of only red balls → Certain; Drawing a blue ball from a bag of only red balls → Impossible; Rolling an even number on a die numbered 1 to 6 → Possible but not certain
  2. 2. 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
  3. 3. 3/8
  4. 4. 1/2
  5. 5. 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
  6. 6. c) Getting a 7
  7. 7. 3/10
  8. 8. 1/6

Introduction to graphs

  1. 1. 7
  2. 2. 18
  3. 3. 12
  4. 4. x = 1 → 3; x = 2 → 7; x = 3 → 11; x = 5 → 19; x = 6 → 23
  5. 5. 2 — Stopped by the roadside
  6. 6. d) The object is not moving
  7. 7. (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
  8. 8. 11

Class 8 Maths · www.arenapublications.com/learn