## Long Multiplication and Division

Long Multiplication and Division The key to this topic is your working out which one of these they are asking for in the exam. If you think it’s a lon...
Author: Ira Wilkins
Long Multiplication and Division The key to this topic is your working out which one of these they are asking for in the exam. If you think it’s a long multiplication when really it’s a long division YOU WILL GET NO MARKS. (Even if your method is perfect). How do you avoid this? Well one way is to choose which one you think it is, do your working, THEN LOOK AT YOUR ANSWER. If it does not look realistic, then you have chosen the wrong one. Here is a good example: Mr Harrison is going to take all 650 kids in school on a trip to Blackpool (cos he’s the caring type). He wants to order some 53 seater coaches. He does 750 x 53 And gets 39750 He phones the coach company and orders nearly forty thousand coaches for the day. Can you imagine that! Obviously he should have divided. Get the picture? In all maths, not just this topic, look at your answer. Is it realistic? Long Multiplication These are multiplication questions where you are multiplying by a number of 2 digits or more. Eg 45 x 15 89 x 182 674 x 932 etc So what method should you use? The only method we teach. The GRIDS! Lets do each of the questions above 45 x 15

10 5

40 400 200

5 50 25

These are easy to do. Ignore the zeros, so for 10 x 40 do 1 x 4 = 4 then place the ignored zeros on the end of your answer. Then, being careful to line up the correct digits add all of the numbers in the grid using column addition. 4 0 0 5 0 2 0 0 + 2 5 6 7 5

89 x 182 100 8000 900

80 9

80 6400 720

2 160 18

70 63000 1400 210

4 3600 80 12

Again add the numbers inside your grid: 8 0 0 6 4 0 1 6 9 0 7 2 + 1

0 0 0 0 0 8

2

1 6 1 9 8 674 x 923 600 540000 12000 1800

900 20 3

5 4 0 0 6 3 0 3 6 1 2 0 1 4

0 0 0 0 0 8 1 8 0 2 1 1

+ 1

1

2

0 0 0 0 0 0 0 0 2

1

6 2 2 1 0 2 Long Division There are many fancy ways to do this, but we think that the old fashioned way is the best. Example 37908 ÷ 52 Set it out using the bus stop method.

52 3 7 9 0 8

space the digits out! So how many 52’s go into 3? None!

How many 52’s go into 37? Again, it won’t go. How many 52’s in 379? This is where the only way to answer it is by writing out the first 9 numbers in the 52 times table. You start with 52 and amazingly it goes up in 52s. You can do this by adding 50 then adding 2: 52

104

156

208

260

312

364

416

468

The closest number to 379 WITHOUT GOING PAST 379 is 364, which is the 7th number along. The remainder is found by doing 379 – 364 = 15 so put this in front of the 0 7 52 3 7 9 15 0 8 Now we do 52’s into 150, obviously twice (104 is closest WITHOUT GOING PAST 150), remainder is 46, which goes in front of the 8.

7 2 52 3 7 9 15 0 468 Now we do 52’s into 468. It is exactly the 9th number along So the final sum would look like this: 7 2 9 52 3 7 9 15 0 468

It looks difficult, but with most maths practice makes perfect.

1.

Work out 286 × 43

…………………….. (Total 3 marks)

2. Canal boat for hire £1785.00 for 14 days What is the cost per day of hiring the canal boat?

£ ................................. (Total 3 marks)

3.

‘Jet Tours’ has an aeroplane that will carry 27 passengers. Each of the 27 passengers pays £55 to fly from Liverpool to Prague. Work out the total amount that the passengers pay.

£ ………………………. (Total 2 marks)

4.

A school buys 34 books. Each book costs £5.21 Work out the total cost of the 34 books.

£ …………………. (Total 3 marks)

5.

The cost of a calculator is £6.79 Work out the cost of 28 of these calculators.

£……………………. (Total 3 marks)

6.

Fatima bought 48 teddy bears at £9.55 each. (a)

Work out the total amount she paid.

£ ............................. (3)

Fatima sold all the teddy bears for a total of £696. She sold each teddy bear for the same price. (b)

Work out the price at which Fatima sold each teddy bear.

£ ............................. (Total 6 marks)

7.

Nick takes 26 boxes out of his van. The weight of each box is 32.9 kg. (a)

Work out the total weight of the 26 boxes.

....................... kg (3)

Then Nick fills the van with large wooden crates. The weight of each crate is 69 kg. The greatest weight the van can hold is 990 kg. (b)

Work out the greatest number of crates that the van can hold.

.......................... (4)

(Total 7 marks)

8.

Enzo makes pizzas. One day he makes 36 pizzas. He charges £2.45 for each pizza. (a)

Work out the total amount he charges for 36 pizzas.

£ ................................ (3)

Mario delivers pizzas. He is paid 65p for each pizza he delivers. One day he was paid £27.30 for delivering pizzas. (b)

How many pizzas did Mario deliver?

........................ pizzas (3) (Total 6 marks)

Converting fractions into decimals One of the benefits of being able to do this is so that we can order fractions. Example Write the following fractions in ascending order of size 2/

3

3/

4

3/

5

Note: Ascending means smallest to largest. Well there are various methods that can be used to do this, but we in the maths department think that the best method is to change them all into decimals. Method Put the denominator on the outside of the division sum.

3

2

Now add 2 decimal points and a nought (you may need more noughts later on).

3

. 2.0

Now do the sum

3

0. 6 6 2 . 20 20

It’s a recurring decimal. Now for the other 2 fractions:

4

0. 7 5 3. 30 20

5

0. 6 3. 30

So 2/

3

3/

4

3/

5

Are 0.66, 0.75,

0.6

Now we can order the fractions 3/ 2/ 3/ 5 3 4

1.

5 8

1 2

3 4

Write these fractions in order of size. Start with the smallest fraction.

.............................................................................. (Total 2 marks)

2.

Write these fractions in order of size. Start with the smallest fraction.

9 16

3 4

1 2

5 8 …………………………… (Total 2 marks)

3.

Write these numbers in order of size. Start with the smallest number. (i)

0.56, 0.067, 0.6, 0.65, 0.605 ...........................................................................................

(ii)

5, – 6, – 10, 2, – 4 ...........................................................................................

(iii)

1 2 2 3 , , , 2 3 5 4

........................................................................................... (Total 4 marks)

4.

Here are six numbers 75%

8 10

9 12

Two of the numbers are not equal to

0.75

66

2 % 3

6 8

3 4

Draw a circle around each of the two numbers. (Total 2 marks)

5.

Write these numbers in order of size.

76, 103, 13,

130, 67 ……………………………………………………… (1)

(b)

–3,

5, 0, –7, –1 ……………………………………………………… (1)

(c)

70%,

3 , 4

0.6,

2 3

……………………………………………………… (2) (Total 4 marks)

6.

Write these numbers in order of size. Start with the smallest number. (i)

75, 56, 37, 9, 59

........................................................................................... (ii)

0.56, 0.067, 0.6, 0.65, 0.605

........................................................................................... (iii)

5, – 6, – 10, 2, – 4

........................................................................................... (iv)

1 2 2 3 , , , 2 3 5 4

........................................................................................... (Total 5 marks)

7.

Write these numbers in order of size.

0.56, 0.067, 0.6, 0.65, 0.605 ..................................................................................................

(ii)

5, – 6, – 10, 2, – 4 ..................................................................................................

(iii)

1 2 2 3 , , , 2 3 5 4

.................................................................................................. (Total 4 marks)

8.

(a)

Write these five fractions in order of size. Start with the smallest fraction. 3 4

1 2

3 8

2 3

1 6

................................................................ (2)

(b)

Write these numbers in order of size. Start with the smallest number. 65%

3 4

0.72

2 3

3 5

................................................................ (2) (Total 4 marks)

Percentages of amounts These questions can turn up on the calculator or non-calculator section of the exam. Calculator method 24% of £60 Change the percentage to a fraction (easy, because it’s always out of 100) and change the ‘of’ to a x. 24 x 60 100 Now just press it in to your calculator = £14.40 Non calculator method I call this the 10% method eg 1)

30% of 80 10% is 8

So 30% is 8 x 3 = 24

eg 2) Find the VAT on £120 Now VAT is always 17.5% So 10% is £12 5% is £6 2.5% is £3 Therefore 17.5% is 12 + 6 + 3 = £21

eg 3)

Find 21% of 140 10% is 14 20% is 28 1% is 1.4 So 21% is 29.4

In some questions you will need to do an addition or subtraction at the end. For the following examples we will use the calculator method.

eg 1)

A gas bill of £60 is reduced by 20%. What is the final bill?

20% of £60 20 x 60 = £12 100 So final bill is 60 - 12 = £48

eg 2) A car costs £2000 + VAT How much do you actually pay for the car? 17.5% of 2000 17.5 100

x 2000

= £350

So final cost = 2000 + 350 = £2350

1.

Work out 23% of £64

£ ............................. (Total 2 marks)

2.

The normal cost of a coat is £94 In a sale the cost of the coat is reduced by 36% Work out 36% of £94

£ ………………….. (Total 2 marks)

3.

Work out 70% of £340

£ .......................................... (Total 2 marks)

4.

Work out 45% of 800

............................ (Total 2 marks)

5.

Ann buys a dress in a sale. The normal price of the dress is reduced by 20%. The normal price is £36.80 Work out the sale price of the dress.

£ .......................... (Total 3 marks)

6.

William’s salary is £24 000 His salary increases by 4%. Work out William’s new salary.

£ ............................ (Total 3 marks)

7.

Martin had to buy some cleaning materials. The cost of the cleaning materials was £64.00 plus VAT at 17

1 %. 2

Work out the total cost of the cleaning materials.

£ ....................... (Total 2 marks)

8.

A jacket costs £50 plus VAT at 171/2%. Work out the total cost of the jacket.

£……………………. (Total 2 marks)

Basic algebra The basics a + a + a = 3a a x a x a = a3 Don’t get them confused! The above are fundamentals that you should know. Also, we get lazy and don’t use the x sign or the number 1 much. So 4y really means 4 x y and y really means 1y (1x y if you really want) Terms and expressions A term is a collection of numbers and letters multiplied together. egs) 4x or 5 or y or 4xy2 or 10abc All the above are terms (even if it’s just a letter or just a number) are separated by + and ─ signs. eg) 4x + 5 ─ y + 4y2 ─ 10abc This is now an algebra expression (with 5 terms in it). Don’t get an algebra expression confused with an algebra equation, equations have equals signs but expressions don’t. eg)

4x = 5x + 6 4x ─ 5x + 6

Equation. You solve this. Expression. You simplify this.

Simplifying expressions Method Collect up like terms, using you number line. Worked examples 1) 4x + 5x ─ 2x = 7x 2) 5x + 2y ─ x ─ 6y = 4x ─ 4y (check where we got the ─4y from in eg 2) We did 2y ─ 6y = ─ 4y. Muppets will put 8y, because they are ignoring the ─ sign and not using a number line. 3) 4p ─2q + 7p ─ 4q = 11q ─ 6q You can’t simplify 11p ─ 6q anymore. It’s not 5pq!

Note If you have terms with an x in between, you can collect terms. eg: 5p x 2q = 10pq Repeating again though: 5p + 2q has gone as far as it can go. It does not simplify to 7pq!

1.

Simplify (i)

c+c+c+c

.................................

(ii)

p×p×p×p

.................................

(iii)

2r × 5p

................................. (Total 3 marks)

2.

Simplify

5x + 3y  y + 2x

…………………………………. (Total 2 marks)

3.

(a)

Simplify

q+q+q+q ……………………. (1)

(b)

Simplify

7x + 3y + 2x  2y ……………………. (2) (Total 3 marks)

4.

(a)

Simplify x+x+x ............................ (1)

(b)

Simplify 2e × 3f ............................ (1) (Total 2 marks)

5.

Simplify 6x + 3y – x + 5y

............................. (Total 2 marks)

6.

(a)

Simplify 3p + 2q – p + 2q …………………….. (2)

(b)

Simplify 3y2 – y2 …………………….. (1)

(c)

Simplify 5c + 7d – 2c – 3d

…………………….. (2)

(d)

Simplify 4p × 2q …………………….. (1) (Total 6 marks)

7.

Simplify 3a + 5b + 6a – 2b

............................................... (Total 2 marks)

8.

(a)

Simplify

y2 + y2 ……………….. (1)

(b)

Simplify

6c + 9d –3c – 5d ……………….. (2) (Total 3 marks)

Algebra substitution There are 3 reasons why students struggle with this type of question. 1) It’s algebra after all - and we know students think about that! 2) They struggle to identify what type of algebra it is. 3) MINUS SIGNS – This as the main reason why students who attempt this question get it horribly wrong. They just can’t seem to cope with them. I hope these notes will help you with that. Recognising the topic You’ll recognise it as algebra, but how will you know that it’s substitution? Well firstly you will be given an algebra expression. (not an equation with an = sign) Then you will be given the numerical values of the letters in the expression. You will then be asked to substitute the letters in order to work out the value of the expression. Example:

a 2 + 5bc

Calculate the value of:

if a = 4, b = 2, c = 1 Method Substituting gives: 42 + 5 x 2 x 1 = 16 + 10 = 26 Those dreaded MINUS SIGNS! Example:

If x = 3, y = ─4 and z = ─5 Find the value of x + y2 ─ 4z

Method Substituting gives: 3+ (─ 4)2 ─ 4 x ─ 5 Now (─ 4)2 = ─ 4 x ─ 4 = +16 and ─ 4 x ─ 5 = +20 so its 3 + 16 + 20 = 39 Key point If you substitute negative numbers into a calculator, always put them in brackets. Nobody knows why but if you don’t the calculator works it out wrong!

1.

C = 2p – 5q p = –3 q=4 Work out the value of C.

C = .......................... (Total 2 marks)

2.

P = 3a + 5b a = 5.8 b = –3.4 Work out the value of P. P = .......................... (Total 2 marks)

3.

P = x2  7x Work out the value of P when x = 5

P = …………… (Total 2 marks)

4.

2

P = x – 5x Find the value of P when x = – 4

P = .......................... (Total 2 marks)

5.

P = Q2  2Q Find the value of P when Q = 3

P = ....……………. (Total 2 marks)

6.

Here is a formula for the perimeter of a rectangle. Perimeter = (length × 2) + (width × 2) The length of a rectangle is 12 cm. Its width is 4 cm. Use the formula to work out the perimeter of this rectangle.

………………………. cm (Total 2 marks)

7.

Tayub said, “When x = 3, then the value of 4x2 is 144”. Bryani said, “When x = 3, then the value of 4x2 is 36”. (a)

Who was right? Explain why.

(2)

(b)

Work out the value of 4(x + 1)2 when x = 3.

................................. (1) (Total 3 marks)

Solving linear equations This is a huge topic. They are going to ask you a PAGE FULL OF these, trust me. As well as that they sneak into other areas of the maths curriculum. They are not easy but two things will help you: 1) Follow the METHOD 2) Practice hundreds of them! Recognise the question Algebra expressions don’t have an equal sign eg) 4x – 7y + 3x Algebra equations do have an equal sign. eg) 7x = 3x + 2 You simplify expressions and solve equations. So if you have decided that it’s an equation, you have another choice to make. Is it a linear equation? Well it will be linear when there are no powers on the letters. If there are, then you don’t solve them like we are going to do in this booklet. Examples X2 + 7x = 52 7x – 4 = 2x + 5

NOT linear Linear

So you now know how to IDENTIFY the question in the exam (a crucial skill indeed!) So how do you solve them? Here are some examples to demonstrate the method. Example 1 x +3=7 Now I bet that you know the value of x. Of course it’s 4. But we are going to start using the method even on the easy-peasy ones. Method 1) You want letters on the left hand side of the equal sign, and numbers on the right. (something like 4y you could argue is both, but you assume it’s a letter). 2) If it’s already on the correct side, just drop it down. 3) If it’s on the wrong side, then move it but change its sign. + goes to – x goes to ÷ Square goes to √ Back to our example: x+3 =7 x =7–3 x=4

Example 2 5y – 3 = 2y + 6 5y – 2y = 3 + 6 3y = 9 y=9 3 y=3 Linear equations with brackets With these you have to expand the brackets out first. Example 5(2a + 3) = 45 10a + 15 = 45 10a = 45 – 15 10a = 30 a = 30 10 a=3

1.

(a)

Solve the equation

5x = 30.

x = ………………… (1)

(b)

Solve the equation

y + 3 = 10.

y = ……………… (1) (Total 2 marks)

2.

Solve the equation

7x + 2 = 3x – 2

x = …………. (Total 3 marks)

3.

Solve

6y + 5 = 2y + 17

y = …………… (Total 3 marks)

4.

Solve 4y + 1 = 2y + 8

y =……………………… (Total 2 marks)

5.

Solve

2x + 7 = 6(x + 3)

x = .................... (Total 3 marks)

6.

(a)

Solve

7p + 2 = 5p + 8

p = ............................ (2)

(b)

Solve

7r + 2 = 5(r – 4)

r = ........................... (2) (Total 4 marks)

7.

(a)

Solve 7x + 18 = 74

x = ……………………… (2)

(b)

Solve 4(2y – 5) = 32

y = ……………………… (2)

(c)

Solve 5p + 7 = 3(4 – p)

p = ……………………… (3) (Total 7 marks)

8.

Solve

5(2y + 3) = 20

y = ................................. (Total 3 marks)

9.

Solve

5x – 3 = 2x + 15

x = ………………………….. (Total 2 marks)

10.

Nassim thinks of a number. When he multiplies his number by 5 and subtracts 16 from the result, he gets the same answer as when he adds 10 to his number and multiplies that result by 3. Find the number Nassim is thinking of.

…………………………… (Total 4 marks)

11. A

2x

B

2x

10

C

Diagram NOT accurately drawn In the diagram, all measurements are in centimetres. ABC is an isosceles triangle. AB = 2x AC = 2x BC = 10 (a)

Find an expression, in terms of x, for the perimeter of the triangle. Simplify your expression.

……………………… (2)

The perimeter of the triangle is 34 cm. (b)

Find the value of x.

x =……………………… (2) (Total 4 marks)

12.

The perimeter of this triangle is 19 cm. All lengths on the diagram are in centimetres.

(t + 4) (t + 3)

(t – 1)

Diagram NOT accurately drawn Work out the value of t.

t = …………………………… (Total 3 marks)

13.

x + 90

x + 20

Diagram NOT accurately drawn

x + 10

2x

The sizes of the angles, in degrees, of the quadrilateral are

(a)

x + 10 2x x + 90 x + 20 Use this information to write down an equation in terms of x.

.......................................................... (2)

(b)

Use your answer to part (a) to work out the size of the smallest angle of the quadrilateral.

.....................................  (3) (Total 5 marks)

(x + 4) cm

14.

A

D (2x – 1) cm

B

C

Diagram NOT accurately drawn ABCD is a parallelogram. AD = (x + 4) cm, CD = (2x – 1) cm. The perimeter of the parallelogram is 24 cm. (i)

Use this information to write down an equation, in terms of x.

……………………………………………………. (ii)

x = …………………………… (Total 3 marks)

Averages and range There are 3 types of averages that you may be asked to work out:

MEAN

MODE

MEDIAN

The mean This is where you add up the numbers and divide by how many numbers there is: Find the mean of 5, 9, 6 and 8 5+ 9 + 6 + 8 = 28 There are 4 numbers so 28 ÷ 4 = 7 Sometimes though the exams are a little nastier than this and ask you other types of questions to do with the mean. eg) 3 numbers have a mean of 4 what are they? If three numbers have a mean of 4 then when you add them up divide by 3 you get 4. This means that they add up to (3 x 4) 12. So it’s any 3 numbers that add up to 12 5,5,2

4,4,4

6,1,5

7,8,-3 the list is endless

If you are given the mean and how many numbers remember the sum of the numbers is the mean times by the number of numbers. The median This where you put the numbers in order and find the middle one: Find the median of 5, 7, 2, 4 and 8

2

4

5 7 8

Find the median of 5, 7, 2, 4, 8 and 4

2

4

4

5 7 8

What number is this? It’s halfway between 4 and 5 It is obviously 4.5

The mode This is the most popular number (not how many friends it has but the one that occurs the most) Find the mode of 6, 7, 5, 4, 6, 7, 6, 4, 2, 6 6 turns up in this list most so 6 is the mode Find the mode of 7, 10, 11, 6, 7, 9, 10 7 and 10 turn up the most so 7 and 10 are the mode Find the mode of 9, 1, 5, 3, 7, 2, 6 All the numbers turn up once so there is NO mode The range This is just the spread of data and it is found by: Biggest Number – Smallest Number

1.

Five boxes are weighed. Their weights are given below. 3 kg,

11 kg,

5 kg,

20 kg,

11 kg,

Write down the range of their weights. …….………….. kg (Total 1 mark)

2.

Work out the median of these 15 numbers. 3, 8, 8, 6, 4, 2, 8, 9, 4, 5, 1, 5, 7, 8, 9

……………. (Total 2 marks)

3.

Here are the test marks of 6 girls and 4 boys.

(a)

Girls:

5

3

10

2

Boys:

2

5

9

3

7

3

Write down the mode of the 10 marks. …………………………… (1)

(b)

Work out the median mark of the boys.

…………………… (2)

(c)

Work out the range of the girls’ marks.

…………………… (1)

(d)

Work out the mean mark of all 10 students.

…………………… (2) (Total 6 marks)

4.

Here are 10 numbers. 3 2 5 4 2 4 6 2 1 2 Find the mode of these numbers. ………………. (Total 1 mark)

5.

Jalin wrote down the ages, in years, of seven of his relatives. 45, (a)

38,

43,

43,

39,

40,

39

Find the median age.

....................................... (1)

(b)

Work out the range of the ages. ....................................... (1)

(c)

Work out the mean age.

....................................... (2) (Total 4 marks)

6.

Here are the shoe sizes of 6 students. 2

10

7

6

10

9

Work out the median shoe size.

.......................... (Total 2 marks)

7.

Tom recorded the shoe size of five of his friends. Here are his results. 8 (a)

9

3

4

7

Work out the median shoe size. ............................................ (2)

Another friend has a shoe size of 8 (b)

Work out the median shoe size of all six friends of Tom.

............................................ (2) (Total 4 marks)

Perimeter and area of shapes The main problem here is that you will get these things mixed up all the time. Read this guide 97 times and it might cure your problem. Perimeter This is the distance all the way around the outside of a 2-D shape. You ADD UP all the lengths of the sides. This includes sides that do not have a length written on them. eg)

Area This is the amount of space inside the shape. If you have squares within a shape, you can count the squares in order to find the area of the shape. This is rarely the case though. You will more than likely need to use a formula. Area of squares/rectangles

Area of triangles

Note 1 – Everyone forgets the divide 2 bit. Note 2 – The base doesn’t have to be on the bottom, but the base and perpendicular height have to be at right angles.

Area of parallelograms/rhombus

Area of trapeziums Well first of all how do you recognise a trapezium? It belongs to the family of 4 sided shapes we call quadrilaterals, but other than that it is a sad little shape with hardly anything going for it. The only thing it has is a pair of parallel sides.

An easy way to remember which shapes you divide two with is (TTT) Triangles, Trapeziums divide by Two

Area of compound shapes These are complicated shapes that can be split into simpler shapes that you know the area of. Let’s do a worked example

The key behind this type of question is to show all your working out steps.

1.

The diagram shows a rectangular carpet.

5m

Diagram NOT accurately drawn 2m

Work out the area of the carpet. .......................... (Total 2 marks)

2.

Here is a rectangle. Diagram NOT accurately drawn

3 cm 4 cm (a)

Work out the area of the rectangle.

................................. cm2 (2)

(b)

Work out the perimeter of the rectangle.

................................ cm (1) (Total 3 marks)

3.

A shaded shape is shown on the grid of centimetre squares. (a)

Find the area of the shaded shape. …………………………… (2)

(b)

Find the perimeter of the shaded shape. ……………………………cm (1) (Total 3 marks)

4.

(a)

Work out the area of this rectangle. Diagram NOT accurately drawn

4.5 cm 2.5 cm

………………………cm2 (2)

A square has an area of 324 cm2. (b)

Work out the length of one side of the square. Diagram NOT accurately drawn Area 324cm2 ………………………cm (2) (Total 4 marks)

5.

Mary’s floor is a rectangle 8 m long and 5 m wide. She wants to cover the floor completely with carpet tiles. Each carpet tile is square with sides of length 50 cm. Each carpet tile costs £4.19 Work out the cost of covering Mary’s floor completely with carpet tiles.

£ .............................. (Total 3 marks)

6.

The diagram shows a rectangular field. Diagram NOT accurately drawn 54.5 m

35.5 m

The length of the field is 54.5 m. The width of the field is 35.5 m. The field is for sale. Mrs Fox wants to buy the field. She also wants to plant a hedge along the perimeter. The field costs £11.44 per square metre. Each metre length of hedge costs £4.81 £ ................................. Mrs Fox has £23 000 Has Mrs Fox enough money to buy the field and plant the hedge? You must show the working you use to make your decision.

(Total 6 marks)

7.

5 cm

8 cm

Work out the area of the shaded parallelogram in cm2

.…………… cm2 8.

A

7 cm

(Total 1 mark)

B

Diagram NOT accurately drawn 8 cm

D

13 cm

C

ABCD is a trapezium. Angle A = 90°. Angle D = 90°. AB = 7 cm. AD = 8 cm. DC = 13 cm. Work out the area of the trapezium.

..................................... cm2 (Total 2 marks)

9.

The diagram shows a trapezium of height 3 m. 2m Diagram NOT accurately drawn 3m

6m

Find the area of this trapezium State the units with our answer.

................................ (Total 3 marks)

10 cm Diagram NOT accurately drawn

10. 4 cm

4 cm 2 cm

2 cm 8 cm 4 cm 2 cm

The diagram shows 3 small rectangles inside a large rectangle. The large rectangle is 10 cm by 8 cm. Each of the 3 small rectangles is 4 cm by 2 cm. Work out the area of the region shown shaded in the diagram. ………………………cm2 (Total 3 marks)

11.

20 cm

Diagram NOT accurately drawn 9 cm

4 cm The diagram shows a shape. Work out the area of the shape.

8 cm

…………………………… cm2 (Total 4 marks)

12.

The diagram shows a 6-sided shape made from a rectangle and a right-angled triangle. 2 cm

Diagram NOT accurately drawn

7 cm 12 cm 6 cm

Work out the total area of the 6-sided shape.

...........................cm2 (Total 3 marks)

13.

The diagram shows a Tangram.

E

E Diagram accurately drawn

D B

C

A B

The Tangram is a large square that is made up from one square A, two triangles B, one parallelogram C, another square D and two small triangles E. The total area of the Tangram is 64 cm2. Find the area of (i)

square A,

........................... cm2 (ii)

triangle B,

........................... cm2 (iii)

parallelogram C.

........................... cm2 (Total 4 marks)

Probability

Probability questions usually require numbers not words for answers. Unless they clearly want you to use words: eg: What is the probability that Mr Harrisons hair will be an afro by June? Choose from: Likely, unlikely, impossible. This type of question is rare though, and its much more likely that they want a numerical answer. This can be a fraction, decimal or percentage (usually a fraction). So if you have a question like: What is the probability of getting a 4 on a dice? and you put unlikely for the answer, then you would get no marks. Example A bag contains 8 counters, 5 are blue and the rest are yellow a) prob (a blue counter)

= 5 8

b) prob (a yellow counter) = 3 8 c) prob (a pink counter)

= 0 (there are no pink ones!)

Note - All probabilities are between 0 and 1 The probability of the opposite happening This is easy: It’s just the rest of the probability that’s left over. eg) If the probability that Mr Caudwell will tell a well funny joke is 0∙3, what is the probability that his joke is not funny? Answer: 0 ∙ 7

(1 − 0 ∙3)

1.

On the probability scale below, mark (i)

with the letter S, the probability that it will snow in London in June,

(ii)

with the letter H, the probability that when a fair coin is thrown once it comes down heads,

(iii)

with the letter M, the probability that it will rain in Manchester next year.

0

1 (Total 3 marks)

2.

Here is a fair 4-sided spinner.

4 2

6

8

The spinner has four sections numbered 2, 4, 6 and 8. The spinner is to be spun. It will land on one of the sections. On the probability scale below mark, with a letter, the probability that the spinner will land (i)

on 2 (use the letter A),

(ii)

on an odd number (use the letter B),

(iii)

on a number greater than 3 (use the letter C).

0

1 2

1

(Total 3 marks)

3.

Michael picks one number from Box A. He then picks one number from Box B. Box A

Box B 2

7

8 1

4 6

5

List all the pairs of numbers he could pick. One pair (1, 2) is shown. (1, 2) ...……………………………………………………………………………………... .……………………………………………………………………………………………... .……………………………………………………………………………………………... (Total 2 marks)

4.

A bag contains some beads which are red or green or blue or yellow. The table shows the number of beads of each colour. Colour Number of beads

Red

Green

Blue

Yellow

3

2

5

2

Samire takes a bead at random from the bag. Write down the probability that she takes a blue bead.

............................ (Total 2 marks)

5. A B

B

A

C B

B A

The diagram shows a fair spinner in the shape of a rectangular octagon. The spinner can land on A or B or C. Marc spins the spinner. Write down the probability that the spinner will land on A.

............................. (Total 2 marks)

6.

Kerry has a bag of beads. 2 of the beads are red. 4 of the beads are blue. The other 9 beads are green. Kerry is going to take a bead at random from the bag. Work out the probability that she will take a blue bead.

………… (Total 2 marks)

7.

Natasha says the probability that she will be late for school next Monday is 1.5 Natasha is wrong. Explain why. .......................................................................................................................................................... .......................................................................................................................................................... (Total 1 mark)

8.

A company makes hearing aids. A hearing aid is chosen at random. The probability that is has a fault is 0.09 Work out the probability that a hearing aid, chosen at random, will not have a fault.

………………… (Total 1 mark)

9.

A train can be on time or early or late. The probability that the train will be on time is 0.69 The probability that the train will be early is 0.07 Work out the probability that the train will be late.

…………………. (Total 2 marks)

10.

Mr Brown chooses one book from the library each week. He chooses a crime novel or a horror story or a non-fiction book. The probability that he chooses a horror story is 0.4 The probability that he chooses a non-fiction book is 0.15 Work out the probability that Mr Brown chooses a crime novel.

……………………. (Total 2 marks)

11.

Each day, Anthony travels to work. He can be on time or early or late. The probability that he will be on time is 0.02 The probability that he will be early is 0.79 Work out the probability that Anthony will be late. ...................................... (Total 2 marks)

12.

A box contains bricks which are orange or blue or brown or yellow. Duncan is going to choose one brick at random from the box. The table shows each of the probabilities that Duncan will choose an orange brick or a brown brick or a yellow brick. Colour

Orange

Probability

0.35

Blue

Brown

Yellow

0.24

0.19

Work out the probability that Duncan will choose a blue brick.

…………………………… (Total 2 marks)

13.

A school snack bar offers a choice of four snacks. The four snacks are burgers, pizza, pasta and salad. Students can choose one of these four snacks. The table shows the probability that a student will choose burger or pizza or salad. Snack Probability

burger

pizza

0.35

0.15

pasta

300 students used the snack bar on Tuesday. Work out an estimate for the number of students who chose pizza. ................................. (Total 2 marks)

14.

Four teams, City, Rovers, Town and United play a competition to win a cup. Only one team can win the cup. The table below shows the probabilities of City or Rovers or Town winning the cup. City

Rovers

Town

United

0.38

0.27

0.15

x

Work out the value of x.

......................... (Total 2 marks)

15.

A box contains sweets which are red or green or yellow or orange. The probability of taking a sweet of a particular colour at random is shown in the table. Colour

Red

Green

Yellow

Probability

0.25

0.1

0.3

Orange

Sarah is going to take one sweet at random from the box. Work out the probability that Sarah will take an orange sweet.

.......................... (Total 2 marks)

16.

A bag contains some sweets. The flavours of the sweets are either strawberry or chocolate or mint or orange. Sarah is going to take one sweet at random from the bag. The table shows the probability that Sarah will take a strawberry sweet or a mint sweet or an orange sweet. Flavour Probability

Strawberry

Chocolate

0.32

Mint

Orange

0.17

0.2

Work out the probability that Sarah will take a chocolate sweet.

............................. (Total 2 marks)

17.

The probability that a biased dice will land on a six is 0.4. Marie is going to throw the dice 400 times. Work out an estimate for the number of times the dice will land on a six.

…………………………… (Total 2 marks)

18.

The probability that a biased dice will land on a four is 0.2 Pam is going to roll the dice 200 times. (a)

Work out an estimate for the number of times the dice will land on a four.

......................... (Total 2 marks)

19.

20 000 adults live in Mathstown. The probability that one of these adults, chosen at random, will vote in an election is 0.7 Work out an estimate for the number of these adults who will vote in an election. ..................................... (Total 2 marks)

20.

A school snack bar offers a choice of four snacks. The four snacks are burgers, pizza, pasta and salad. Students can choose one of these four snacks. The table shows the probability that a student will choose burger or pizza or salad. Snack Probability

burger

pizza

0.35

0.15

pasta

One student is chosen at random from the students who use the snack bar. (a)

Work out the probability that the student (i)

................................. (ii)

chose pasta.

................................. (3)

300 students used the snack bar on Tuesday. (b)

Work out an estimate for the number of students who chose pizza.

................................. (2) (Total 5 marks)

Number sequences We are mainly looking at how to form general formulas for number sequences. These formulas are called nth term formulas. There is a clear set method for how you derive (find) an nth term formula for a particular sequence. An example will show you the method clearly, so here goes: Example Find the nth term formula for the sequence 4, 7,10,13,16, ………… Method 1) Find how you get from one term to the next. (I call it the gap number) Here it is +3 N.B Numbers in sequences are often called terms. 2) Multiply your gap number by n, so we are now at 3n. 3) Write down how you get from your gap number to the first number (term) in your sequence. For this example to get from 3 to 4 it’s +1 So our answer is 3n +1 How do we use this formula? Well we can use it to find any term in your sequence. This therefore makes it a very powerful tool. So for our example, if we wanted to find to 50 th term in the sequence, using our nth term formula makes it easy. (You substitute the n in the formula for 50). So 50th term = 3 x 50 + 1 = 151 More examples eg 2) -10,-15,-20,-25 …… nth term = – 5n – 5 …… nth term = 4n – 24

eg 3) -20,-16,12,-8

The gap is actually positive because although the question is riddled with minus signs, the sequence is increasing Fractional sequences eg 4)

1/

2,

nth term =

1/

5,

1/

8,

1/

11,

1/ ………………. 14

eg 5)

3/

4,

4/

6,

5/

8,

6/

10,

For this type of example, form an nth term formula for the numerator and an nth term formula for the denominator. nth term =

Two special sequences These are: Square number sequences Cube number sequences Watch for them, because you use none of the above methods!! Square numbers 1, 4, 9, 16, 25, 36,…………. nth term = n2

obvious? Cube numbers 1, 8, 27, 64, 125,…………. nth term = n3

obvious? Note You can have sequences with gaps that are not constant. eg) 5, 7, 10, 14, 19……. There is a formula to find the nth term for these sequences, but it is very complicated so I have decided not to include it in this booklet. Don’t worry, it is rarely asked and would only be answered by geeky Albert Einstein types.

1.

Here are the first 4 terms of an arithmetic sequence 3

7

11

15

Find an expression, in terms of n, for the nth term of the sequence.

………………………… (Total 2 marks)

2.

The first four terms of an arithmetic sequence are 21

17

13

9

Find, in terms of n, an expression for the nth term of this sequence.

............................. (Total 2 marks)

3.

The first five terms of an arithmetic sequence are 2

7

12

17

22

Write down, in terms of n, an expression for the nth term of this sequence.

.................................... (Total 2 marks)

4.

The first five terms of an arithmetic sequence are 2

9

16

23

30

Find, in terms of n, an expression for the nth term of this sequence.

............................................ (Total 2 marks)

5.

Here are the first 5 terms of an arithmetic sequence. 6,

11,

16,

21,

26

Find an expression, in terms of n, for the nth term of the sequence.

................................................. (Total 2 marks)

6.

Here are the first four terms of a number sequence. 2 (a)

7

12

17

Write down the 6th term of this number sequence.

............................................ (1)

The nth term of a different number sequence is 4n + 5 (b)

Work out the first three terms of this number sequence.

............

............

............ (2) (Total 3 marks)

7.

Here are the first five terms of an arithmetic sequence. 1 (a)

3

7

11

15

Find, in terms of n, an expression for the nth term of this sequence. ………….……………………..

In another arithmetic sequence the nth term is 8n  16

(2)

John says that there is a number that is in both sequences. (b)

Explain why John is wrong. …………………………………………………….…………………………..….….. ……………………………………………………………………………………...... …………………………………………………………………………………….…. (2) (Total 4 marks)

8.

Barry and Kath are studying a number pattern. The first three numbers in the number pattern are

1, 2, 4

Barry says that the next number is 8. Kath says the next number is 7. Explain why both Barry and Kath could be right. ........................................................................................................................................ ........................................................................................................................................ ........................................................................................................................................ (Total 2 marks)

9.

Here are some patterns made from dots.

Pattern number 1

Pattern number 2

Pattern number 3

Pattern number 4

Write down a formula for the number of dots, d, in terms of the Pattern number, n.

(Total 2 marks)

10.

Here are some patterns made from sticks.

Pattern number 1

Pattern number 2

Pattern number 3

Complete the table. Pattern number

Number of sticks

1

6

2

10

3

14

4

18

5 n (Total 3 marks)

11.

The table shows some rows of a number pattern. Row 1

1

=

1 2 2

Row 2

1+2

=

23 2

Row 3

1+2+3

=

3 4 2

Row 4

1+2+3+4

Row 8 (a)

In the table, complete row 4 of the number pattern. (1)

(b)

In the table, complete row 8 of the number pattern. (1)

(c)

Work out the sum of the first 100 whole numbers. ..................................... (1)

(d)

Write down an expression, in terms of n, for the sum of the first n whole numbers.

................................. (2) (Total 5 marks)