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)
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)
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)
Solve your equation.
x = ……………………………
(Total 3 marks)
Expanding brackets and factorising
Expanding brackets
They will either say expand the brackets, or multiply out the brackets. It means the same thing.
There are three types of brackets questions:
1) Single brackets
2) Two brackets
3) Double brackets
We are going to deal with each type in turn.
1) Expanding single brackets
Method
The term (letter or number) outside the bracket multiplies each separate term inside the
bracket.
Worked examples
1) 4(x + 2y)
= 4x + 8y
2) a(2 – 3b)
= 2a – 3ab
2) Expanding two brackets
These questions will say expand and simplify, because after you have multiplied them out you can
collect like terms (the simplifying bit). You must use your number line when collecting the like
terms.
Worked examples
1) 2(2x + y) + 5(x – y)
= 4x + 2y + 5x – 5y
↑
We got this term because +5 X –y = –5y (+ x – = –)
Now we have some like terms that we can collect.
= 9x – 3y
2) 3(2a + 4b) – 4(5a – 2b) =
6a +12b – 20a + 8b
↑
We got this term because – 4 X –2b = +8b (–X – = +)
Again, we now collect like terms. = –14a + 20b
3) Expanding double brackets
This is where there are two sets of brackets again, but NOTHING IN BETWEEN THEM. The
method for solving these is completely different from the method used for solving a two bracket
question. We use a BOX METHOD. So when you are faced with a double brackets question in
the exam, I want you to chant BOX to yourself seven times then DRAW ONE!
Worked example and method
Before we do an example, it is crucial that you can remember and use the following rules for
multiplying + and – numbers.
+x+=+
─x─=+
+x─=─
─x+=─
So here’s the example:
Expand (2x + 3) (3x – 5)
Step One:
Draw a box and put the terms around it:
2x
3
3x
- 5 ← muppets put 5. Don’t forget the ─ sign!
Step 2: We multiply each term above and at the side of each box.
2x
Don’t put 5x2 or 6x. →
Only muppets do that.
-
3
6x2
9x
-10x
-15
3x
5
Now we write the terms as if we were reading from a book ( left to right, top to bottom) so its 6x2 +
9x – 10x – 15 (we don’t put a plus sign if there is already a – sign there).
If you have wrote it down in the order I have told you to do it in, then the middle 2 terms will simplify.
(use your number line).
So 6x2 – x – 15 is the answer.
Trick question that could be on the exam
Expand (x + 3)2
Now you rewrite this as (x + 3) (x + 3)
because if you think about it anything squared means multiplied by itself.
Then you draw your box and go on to solve it.
To summarise

If it’s single brackets, just multiply them out.

If it’s two brackets, then it requires two steps to answer it.
Step 1 – Multiply the brackets out
Step 2 – Collect any like terms

If it’s double brackets, then think (and chant) THE BOX. Do your box, then collect the
middle two terms.
The 2 golden rules for brackets:
1) RECOGNISE which type of bracket problem it is.
2) Take great care with MINUS SIGNS.
Factorising
Factorise means the reverse process of expanding brackets. In other words, putting expression
INTO brackets.
There are two types of factorising questions:
1) Standard factorising
eg) Factorise 4y – 6y2
Find the numbers, then letters that are common to both terms. For our example its 2 and y.
So 2y(
)
Now what goes into the brackets? Whatever 2y is multiplied by to get 4y. Obviously 2! Use the
same method to get the last term.
So 4y – 6y2 = 2y(2 – 3y)
2) Quadratic factorising
These have to be RECOGNISED. Look for a 3 term expression where you have a squared letter
term, a letter term, then a number term.
eg) y2 – 3y – 10
So how do we solve them?
Here goes:
They will factorise into DOUBLE BRACKETS. The letter at the start of each bracket is the letter in
the quadratic expression.
(y
)(y
)
Now what goes here and here?
Well it’s 2 numbers that multiply to make the last number (-10) and add to make the middle
number (-3)
Back to our example:
y2 – 3y – 10
(y )(y
)
The two numbers that multiply to make –10 could be:
+1
-10
-1
+10
+2
-5
-2
+5
But only one pair ‘add up’ to make -3. It’s +2, -5
So the answer is:
(y + 2) (y - 5)
1.
(a)
Expand
3(4x – 5)
......................................
(1)
(b)
Factorise 5y + 40
......................................
(1)
(Total 2 marks)
2.
Multiply out
(i)
4(2x – 3)
.............................
(ii)
y(y + 8)
.............................
(Total 2 marks)
3.
Factorise
3m + 15
……………………………
(Total 1 mark)
4.
(a)
Factorise
3t – 12
…………………………
(1)
(b)
Expand and simplify 4(3x + 1) – 3(2x – 1)
…………………………
(2)
(Total 3 marks)
5.
(a)
Expand and simplify
3(2x – 1) – 2(2x – 3)
…………………………
(2)
(b)
Factorise
y2 + y
…………………………
(1)
(Total 3 marks)
6.
Expand and simplify
(2x + 5)(3x – 2)
…………………….
(Total 2 marks)
7.
(a)
Expand and simplify
(x – 6)(x + 4)
……………………………
(2)
(b)
Factorise completely
12x2 – 18xy
……………………………
(2)
(Total 4 marks)
8.
(a)
Factorise p2 + 6p
…………………….
(2)
(b)
Expand and simplify (x + 7)(x – 4)
…………………….
(2)
(Total 4 marks)
9.
(a)
Expand and simplify (x + 5)(x – 3)
…………………….
(2)
(b)
Expand y(y3 + 2y)
…………………….
(2)
(c)
2
Factorise p + 6p
…………………….
(2)
(d)
Factorise completely 6x2 – 9xy
…………………….
(2)
(Total 8 marks)
10.
Factorise
x2  2x  15
………………………………..
(Total 2 marks)
11.
y2 + 3y – 10
Factorise
........................................
(Total 2 marks)
12.
(a)
Expand and simplify
(x + y)2
.............................................................
(2)
(Total 2 marks)
Graphs of linear equations
This is when you are given an equation and asked to plot it as a graph. You will do this via a table.
Worked example and method
Complete the table of values for y = 2x + 3
x
y
-2
-1
-1
0
1
5
2
They are usually kind in the exam and give you a couple of the answers like I have done here.
Now we substitute the x = –1 value into our equation.
So y = 2 X – 1 + 3
= – 2 +3
use your number line here!
Remember, 2 x –1 means +2 x –1 which is – 2
Because + x – = –
If you do the same with x = 0 and x = 2, you completed table would be:
x
y
-2
-1
-1
1
0
3
1
5
2
7
Check it yourself !
Each of these pairs x and y values are coordinates on the graph. With this type of equation you
will join the coordinate points using a ruler to make a straight line.
A nasty trick the examiner sometimes does:
Sometimes you are given the equation, and the graph for you to plot the line on. BUT NO TABLE
OF RESULTS. If this is the case, you will NEED TO DRAW YOUR OWN TABLE OF RESULTS.
You can choose any value of x you wish (-2 to 2 may be a good option).
1.
(a)
Complete the table of values for y = 2x – 3
x
3
y
9
2
1
0
1
2
3
5
3
(2)
(b)
On the grid, draw the graph of y = 2x – 3
y
3
2
1
–3
–2
–1 O
1
2
3
x
–1
–2
–3
–4
–5
–6
–7
–8
–9
(2)
(Total 4 marks)
2.
(a)
Complete the table of values for y = 3x + 2
x
2
1
0
1
1
y
2
5
(2)
(b)
On the grid, draw the graph of y = 3x + 2
y
8
7
6
5
4
3
2
1
–2
–1
O
1
2
x
–1
–2
–3
–4
(2)
(Total 4 marks)
3.
(a)
Complete the table of values for y = 2x + 5
x
–3
–2
y
–1
0
1
1
2
5
(2)
(b)
On the grid, draw the graph of y = 2x + 5
y
10
9
8
7
6
5
4
3
2
1
–3
–2
–1
O
1
2 x
–1
–2
(2)
(Total 4 marks)
4.
On the grid, draw the graph of y = 3x + 1
y
11
10
9
8
7
6
5
4
3
2
1
–1
O
1
2
3
x
–1
–2
–3
–4
(Total 3 marks)
5.
Draw the graph of
y = 5x  2
on the grid below.
y
10
8
6
4
2
–2
–1
O
x
1
2
–2
–4
–6
–8
–10
(Total 3 marks)
Basic angle theory
If you learn the following basic angle rules, then at least you will have a fighting chance of
answering an exam question on angles. If you don’t – you’ve no chance.
1)
2)
3)
4)
Angles on a straight line = 180o
Angles around a point = 360o
Angles in any triangle = 180o
Angles in any quadrilateral (4 sided shape) = 360o
Worked example
Find x
Method
To find x we first need to find the angle next 110o. Angles on a straight line gives 70o because the
triangle is isosceles, we know that the other angle at the bottom of the triangle is 70 o. Now using
angles in a triangle, x = 40o
Now for the harder stuff
This is when we have a pair of parallel lines, and a line crossing both of them.
There is a relationship between all the angles (the ones I have labelled with letters), and also
NAMES of angle pairs that you have to remember.
From the previous diagram I have drawn, here are those relationships and names:
a=b
a=c
b=c
a+d=
corresponding angles
alternate angles
opposite angles
180o interior angles
Finding what the angles are and there relationship to each other is just common sense. (use your
eyes to see which angles are acute or obtuse). NAMING THEM has to be LEARNT.
1.
Diagram NOT accurately drawn
P
PQR is a straight line.
(i)
103º
Q
xº
R
Work out the value of x.
x = ................................
(ii)
Give a reason for your answer.
................................................................................................................................................
(Total 2 marks)
2.
Diagram NOT
accurately drawn
60°
100°
x°
(i)
Work out the size of the angle marked x°.
………………..°
(ii)
Give a reason for your answer.
..………………………………………………………………………………………
..………………………………………………………………………………………
(Total 2 marks)
3.
Diagram NOT accurately drawn
xº
(i)
205º
Work out the size of the angle marked x°.
................................°
(ii)
Give a reason for your answer.
................................................................................................................................................
(Total 2 marks)
4.
45º
122º
(i)
Diagram NOT accurately drawn.
yº
Work out the size of the angle marked y.
……………………………
(ii)
Give a reason for your answer.
………………………………………………………………………………………………
………………………………………………………………………………………………
(Total 2 marks)
Diagram NOT
accurately drawn
5.
x°
y°
48°
B
A
C
ABC is a straight line.
(a)
Work out the size of the angle marked x°.
………………°
(1)
(b)
(i)
Work out the size of the angle marked y°.
…………….°
(ii)
Give reasons for your answer.
…………………………………………………………………………………………
…………………………………………………………………………………………
(3)
(Total 4 marks)
R
6.
Diagram NOT
accurately drawn
y°
126° x°
P
PQ is a straight line.
(a)
Q
Work out the size of the angle marked x°.
..............................°
(1)
(b)
(i)
Work out the size of the angle marked y°.
..............................°
(ii)
Give reasons for your answer.
...................................................................................................................
...................................................................................................................
(3)
(Total 4 marks)
7.
A
Diagram NOT
accurately drawn
60°
B
(a)
(i)
60°
C
Find the size of angle C.
..........................°
(ii)
Triangle ABC is equilateral.
Explain why.
..................................................................................................................................
(2)
S
(b)
P
PQR is a straight line.
SQ = SR.
(i)
x°
Q
Diagram NOT
accurately drawn
50°
R
Work out the size of the angle marked x°.
..........................°
(ii)
Give reasons for your answer.
..................................................................................................................................
..................................................................................................................................
(3)
(Total 5 marks)
a°
8.
Diagram NOT
accurately drawn
138°
65°
Work out the value of a.
a = .............................
(Total 3 marks)
9.
Diagram NOT accurately drawn
yº
81º
37º
xº
Write down the size of
(i)
x,
x = ………
(ii)
y.
y = ………
(Total 2 marks)
10.
108º
xº
yº
43º
Diagram NOT accurately drawn
(a)
Work out the value of x.
x = ………….
(1)
(b)
Work out the value of y.
y = ………….
(2)
(Total 3 marks)
Diagram NOT
accurately drawn
A
11.
100°
D
2 x°
x°
47°
C
B
ABCD is a quadrilateral.
Work out the size of the largest angle in the quadrilateral.
……………..°
12.
(Total 4 marks)
S
(a)
Diagram NOT accurately drawn
xº
P
Q
50º
R
PQR is a straight line.
SQ = SR.
(i)
Work out the size of the angle marked x°
..........................°
(ii)
Give reasons for your answer.
...............................................................................................................................
...............................................................................................................................
(Total 3 marks)
(b)
D
62º
E
yº
Diagram NOT accurately drawn
64º
F
G
DE is parallel to FG.
(i)
Find the size of the angle marked y°.
..........................°
(ii)
Give a reason for your answer.
...............................................................................................................................
(Total 2 marks)
A
13.
C
B
38º
Diagram NOT accurately drawn
yº
xº
D
E
F
G
ABC is parallel to DEFG.
BE = EF.
Angle ABE = 38°.
(a)
(i)
Find the value of x.
x = .............................
(ii)
Give a reason for your answer.
......................................................................................................................................
(2)
(b)
Work out the value of y.
y = .............................
(2)
(Total 4 marks)
D
14.
C
B
Diagram NOT accurately drawn
x
103°
y
A
F
E
ABCD and AFE are straight lines.
BF is parallel to CE.
Angle CBF = 103°.
AB = AF.
(a)
(i)
Find the size of angle x.
(ii)
Give a reason for your answer.
......................................
...........................................................................................................................
...........................................................................................................................
(2)
(b)
Find the size of angle y.
......................................
(2)
(Total 4 marks)
B
A
15.
C
56º
Diagram NOT accurately drawn
x
D
E
F
y
G
BEG and CFG are straight lines.
ABC is parallel to DEF.
Angle ABE = 56°
EF = EG
(a)
(i)
Write down the size of the angle marked x
x = ………..…..°
(ii)
Give a reason for your answer.
.....................................................................................................................................
.....................................................................................................................................
(2)
(b)
Work out the size of the angle marked y
Give reasons for your answer.
y = ……………..°
(3)
(Total 5 marks)
Volume
The volume of a 3-D shape is the amount of space it takes up.
Cubes and cuboids
To find the volume of these solids is easy.
Its just
length X width X height
Example
Find the volume of
4cm
5cm
8cm
Volume is 8 x 5 x 4 = 160cm3
Note – Always include the units of measurement. For volume it’s a small 3 (for area a small 2)
Prisms
A prism is any 3-D shape that has the same cross sectional area across its length. To find
volume, you use the following formula:
Volume = area of cross section x length
Example
Step 1
Find the cross sectional area (a triangle in this case)
Cross sectional area = 8 x 5 = 20cm2
2
Now just multiply this by how far back the shape goes (10cm)
So volume = 20 x 10 = 200cm3
Back to front questions
If you are unlucky one of these types of questions may turn up on the exam. What I mean by a
back to front question is one where the answer is given, but they want to know what part of the
question was. For our volume work it means they give you the volume of the shape, but one of
the lengths of the shape, but one of the lengths of the shape has to be found.
Volume = 180m3
Applying your formula for cuboids:
Vol = length x width x height
180 = 10 X x X 3
180 = 30x
Using your equations skills:
180 = x so x = 60 m
30
1.
Diagram NOT
accurately drawn
30 cm
5 cm
The diagram shows a cuboid. 20 cm
Work out the volume of the cuboid.
.................................. cm3
(Total 2 marks)
2.
Diagram NOT accurately drawn
3 cm
2.8 cm
8.7 cm
6 cm
12.3 cm
Box A
9 cm
Box B
Maxine has two boxes in the shape of cuboids.
Box A measures 12.3 cm by 6 cm by 3 cm.
Box B measures 9 cm by 8.7 cm by 2.8 cm.
Maxine wants to use the box with the greater volume.
Give the letter of the box Maxine should use.
You must show all your calculations.
.....................................
(Total 3 marks)
2 cm
3.
Diagrams NOT accurately drawn
3 cm
3 cm
This cuboid is made up of a number of small cubes.
Each small cube has side 1 cm.
1 cm
1 cm
1 cm
Work out the volume of the cuboid
…………………………
(Total 2 marks)
4.
10 cm
Diagram NOT accurately drawn
4 cm
12 cm
The diagram shows a box in the shape of a cuboid.
(a)
Work out the volume of the box.
…………………………. cm3
(2)
The box is full of sugar lumps.
Each sugar lump is in the shape of a cuboid.
Each lump is 1 cm by 1 cm by 2 cm.
(b)
Work out the number of sugar lumps in the box.
……………………………..
(1)
(Total 3 marks)
5.
Ben fills a container with boxes.
Each box is a cube of side 0.5 m.
The container is a cuboid of
length 9 m,
width 4 m and
height 3 m.
Work out how many boxes will fit exactly into the container.
………………
(Total 3 marks)
6.
Bob has 12 toy bricks.
Each toy brick is in the shape of a cube.
Each cube has sides of length 2 cm.
toy
brick
2 cm
2 cm
2 cm
Diagram NOT accurately drawn
Bob builds a solid cuboid.
He uses all 12 toy bricks.
Write down the length, width and height of one cuboid that Bob can build.
length ............................ cm
width ............................. cm
height .............................cm
(Total 2 marks)
7.
Diagrams NOT accurately drawn
80 cm
10 cm
Light
Bulb
Carton
6 cm
6 cm
30 cm
30 cm
A light bulb box measures 6 cm by 6 cm by 10 cm.
Light bulb boxes are packed into cartons.
A carton measures 30 cm by 30 cm by 80 cm.
Work out the number of light bulb boxes which can completely fill one carton.
..........................
(Total 3 marks)
8.
A cuboid has
(a)
a volume of 40 cm3
a length of 5 cm
a width of 2 cm
Work out the height of the cuboid.
...................... cm
(2)
Diagrams NOT
accurately drawn
(b)
20
0c
m
100 cm
50
cm
100 cm
20 cm
20 cm
A carton measures 200 cm by 100 cm by 100 cm.
The carton is to be completely filled with boxes.
Each box measures 50 cm by 20cm by 20 cm.
Work out the number of boxes which can completely fill the carton.
.....................................
(3)
(Total 5 marks)
9.
represents 1 cm3
Diagrams NOT accurately drawn
In this solid prism, the volume of each small cube is 1 cm3.
(i)
Find the area of the top face of the prism.
................................ cm2
(ii)
Work out the volume of the prism.
................................ cm3
(Total 3 marks)
10.
The diagram shows a wedge in the shape of a triangular prism.
Diagram NOT
accurately drawn
15 cm2
10
cm
The cross section of the prism is shown as a shaded triangle.
The area of the triangle is 15 cm2.
The length of the prism is 10 cm.
Work out the volume of the prism.
..........................
(Total 3 marks)
11.
The diagram shows a prism.
Diagram NOT
accurately drawn
9.3 cm 2
cm
10
The cross section of the prism is a triangle of area 9.3 cm2.
The length of the prism is 10 cm.
Work out the volume of the prism.
State the units of your answer.
…………… ..……….
(Total 3 marks)
12.
Diagram NOT
accurately drawn
5 cm
4 cm
7 cm
3 cm
Calculate the volume of the triangular prism.
...................................
(Total 4 marks)
13.
4 cm
24 cm
Diagram NOT accurately drawn
A cylinder has a height of 24 cm and a radius of 4 cm.
Work out the volume of the cylinder.
Give your answer correct to 3 significant figures.
…………………………… cm3
(Total 2 marks)
14.
A can of drink is in the shape of a cylinder.
The can has a radius of 4 cm and a height of 15 cm.
Diagram NOT
accurately drawn
15 cm
4 cm
Calculate the volume of the cylinder.
Give your answer correct to 3 significant figures.
…………………………
(Total 3 marks)
15.
350 cm
1.2 cm
Diagram NOT accurately drawn
The diagram shows a piece of wood.
The piece of wood is a prism of length 350 cm.
The cross-section of the prism is a semi-circle with diameter 1.2 cm.
Calculate the volume of the piece of wood.
Give your answer correct to 3 significant figures.
………………………… cm3
(Total 4 marks)
Surface area
Area applies to flat 2-D shapes.
Volume applies to solid 3-d shapes.
Now surface area applies to solid 3-d shapes. Confused? That’s understandable.
Surface area is the area of all the faces on a solid shape added together.
DON’T CONFUSE IT WITH VOLUME!
There are 3 main types of surface area question:
1) Cubes and cuboids
2) Triangular prisms
3) Cylinders
We will do an example of each type
1)
C
3cm
B
A
4cm
5cm
Area of A = 5 X 3 = 15cm2
Opposite face of A has the same area (15cm2)
Area of B = 4 X 3 = 12cm2
Opposite face of B has same area (12cm2)
Area of C = 5 X 4 = 20cm2
Opposite face of C has same area (20cm2)
15 x 2 +12 x 2 + 20 x 2 = 94cm2
If you were asked for the volume it would just be:
5 X 4 X 3 = 60cm3
length X width X height
DON’T CONFUSE THEM!
Note
If you were asked to find the surface area of a CUBE, that would be easy. Just find the area of
one face and multiply your answer by 6 (because all 6 faces on a cube are equal in area).
2)
Area of triangle at end = 4 X 5 = 10m2
2
Area of opposite triangle at end = 10m2
Notice that the triangle at the end of our triangular prism is equilateral. This makes this question
easier than if the triangle was isosceles.
So the area of each rectangular face = 4 X 10 = 40m2
So total surface area = 10 x 2 + 40 X 3 = 140m2
3)
You are unlucky if this one turns up in the exam. It’s a cylinder by the way.
Area of circle at end = π x r2
= π x 62
= 113.1cm2
Opposite circle has same area (113.1cm2)
The hard bit is calculating the curved surface area. Imagine folding a piece of paper into a roll.
You have formed a cylinder. Now think in reverse. You started with a rectangle (your piece of
paper) whose area is length x width. Applied to our cylinder this is:
Length of cylinder X circumference of circle
=8xπxD
= 8 x π x 12
= 301.6cm2
So total surface area = 113.1 x 2 + 301.6
= 527.8 cm2
1.
Diagram NOT
accurately drawn
4.5 m
2.8 m
3.2 m
The diagram represents a large tank in the shape of a cuboid.
The tank has a base.
It does not have a top.
The width of the tank is 2.8 metres.
The length of the tank is 3.2 metres.
The height of the tank is 4.5 metres.
The outside of the tank is going to be painted.
1 litre of paint will cover 2.5 m2 of the tank.
The cost of paint is £2.99 per litre.
Calculate the cost of the paint needed to paint the outside of the tank.
£ ..................................
(Total 5 marks)
2.
Diagram NOT accurately drawn
The diagram shows a solid cylinder.
The radius of the cylinder is 54 cm.
The height of the cylinder is 10 cm.
Calculate the curved surface area of the cylinder.
Give your answer correct to three significant figures.
……………….
(Total 3 marks)
3.
Diagram NOT accurately drawn
18 cm
12 cm
The diagram shows a solid cylinder.
The cylinder has a diameter of 12 cm and a height of 18 cm.
Calculate the total surface area of the cylinder.
Give your answer correct to 3 significant figures.
………………………… cm2
(Total 4 marks)
Pythagoras Theorem
Recognising the question
Pythagoras theorem can only be applied to right angled triangles. It can be confusing because
trigonometry is also applied to right angles triangles. So we really need to be clear about when to
use which.
It’s Pythagoras when:
Two sides are given and a third side is asked for. IT DOES NOT INVOLVE ANGLES IN ANY
WAY (except a right angle). This means neither asked for nor given
It’s Trigonometry when:
An angle is either given or asked for.
So now you know.
The Method
SQUARE IT
SQUARE IT
ADD or SUBTRACT IT
(depending on whether you are finding the longest side or one of
the shorter sides)
SQUARE ROOT IT
In every right angled triangle the shortest sides make the right angle and the longest side is
always opposite the right angle (the hypotenuse!)
Worked example
42 = 16
32 = 9
x is the long side so add these 16 + 9 = 25
√25 = 5cm
8m
x
5m
82 = 64
52 = 25
X is one of the short sides so subtract these 64 – 25 = 39
√39 = 6.24m (2dp)
1.
3.2 cm
X
Y
Diagram NOT
accurately drawn
1.7 cm
XYZ is a right-angled
Z triangle.
XY = 3.2 cm.
XZ = 1.7 cm.
Calculate the length of YZ.
Give your answer correct to 3 significant figures.
…………………………. cm
(Total 3 marks)
2.
P
Q
15 cm
19 cm
R
Diagram NOT accurately drawn
PQR is a right-angled triangle.
Angle PQR = 90°.
QR = 15 cm.
PR = 19 cm.
Work out the length of PQ.
Give your answer correct to 3 significant figures.
.......................... cm
(Total 3 marks)
P
3.
Diagram NOT accurately drawn
6 cm
Q
4 cm
R
PQR is a right-angled triangle.
PR = 6 cm.
QR = 4 cm.
Work out the length of PQ.
Give your answer correct to 3 significant figures.
.....................................cm
(Total 3 marks)
4.
A
B
12.6 cm
4.7 cm
C
Diagram NOT accurately drawn
AC = 12.6 cm.
BC = 4.7 cm.
Angle ABC = 90°.
Calculate the length of AB.
Give your answer correct to 3 significant figures.
………………….. cm
(Total 3 marks)
P
5.
Diagram NOT
accurately drawn
5.7 cm
In triangle PQR,
QR = 9.3 cm.
PQ = 5.7 cm.
Angle PQR = 90°.
Q
9.3 cm
R
Calculate the length of PR.
Give your answer correct to 3 significant figures.
........................................ cm
(Total 3 marks)
6.
L
3.7 m
6.3 m
M
N
Diagram NOT accurately drawn
Angle MLN = 90°.
LM = 3.7 m.
MN = 6.3 m.
Work out the length of LN.
Give your answer correct to 3 significant figures.
LN = ……….…………….. m
(Total 3 marks)
7.
Diagram NOT
accurately drawn
B
A
17 cm
10 cm
D
C
ABCD is a rectangle.
AC = 17 cm.
AD = 10 cm.
Calculate the length of the side CD.
Give your answer correct to one decimal place.
................................... cm
(Total 3 marks)
A
8.
7 cm
B
Diagram NOT accurately drawn
7 cm
M
8 cm
C
Work out the length, in centimetres, of AM.
Give your answer correct to 2 decimal places.
…………………… cm
(Total 3 marks)
9.
A
2.1 m
D
3.2 m
1.9 m
B
C
Diagram NOT accurately drawn
ABCD is a trapezium.
AD is parallel to BC.
Angle A = angle B = 90.
AD = 2.1 m,
AB = 1.9 m,
CD = 3.2 m.
Work out the length of BC.
Give your answer correct to 3 significant figures.
………………………… m
(Total 4 marks)