MATHEMATICS
FACULTY
ACULTY
GENERAL LEVEL
HOMEWORK AND
REVISION
FORMULAE LIST
C =πd
A = π r2
A = 2π rh
Circumference of a circle:
Area of a circle:
Curved Surface Area of a Cylinder:
Volume of a Cylinder:
V = π r 2h
V = Ah
Volume of a Triangular Prism:
Theorem of Pythagoras:
a
a 2 = b2 + c 2
b
c
Trigonometric ratios
In a right angled
triangle:
tan x o =
opposite
adjacent
sin x o =
opposite
hypotenuse
cos x o =
adjacent
hypotenuse
hypotenuse
opposite
xo
adjacent
Gradient:
Vertical
Height
PAGE
2
Horizontal
Distance
Gradient =
Vertical
Horizontal
Contents: Homework section
3
Page
5
6
10
12
14
15
17
18
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23
26
27
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31
32
33
34
35
38
40
41
42
44
45
46
PAGE
Topic
Calculations & Calculators
Shape and Space
Numbers Review 1
Money Matters
Similar Shapes
Letters and Numbers Review
Speed, Distance, Time
Brackets and Equations
Statistics and Probability 1
Pythagoras’ Theorem
Area and Volume
Formulae
Gradients and Straight Lines
Trigonometry
Numbers Review 2
Fractions, Decimals and Percentages
Fractions
Equations and Inequalities
Statistics and Probability 2
Shape and Space Review
House and Car Costs
Pairs of Straight Lines
Letters and Numbers Review
Proportion
Angles in a Circle
Areas and Volume Review
Contents: Worksheet/Revision section
PAGE
4
Topic
Rounding
Calculations and Rounding
Squares and Square Roots
Standard Form
Angles and Triangles
Shapes and Coordinates
Working with Percentages
Wages and Salaries
Overtime and Commission
Savings and Interest
Value Added Tax
Electricity Bills
Hire Purchase
Scale Drawings
Speed, Distance and Time
Algebra
Statistics
Pythagoras’ Theorem
Area, Perimeter and Volume
Tax and Travel
Formulae and Sequences
Probability - Multiple Outcomes
Gradients and Straight Lines
Trigonometry
Fractions, Decimals and Percentages
Solving Equations
Statistics
Best Buy
Pairs of Lines
Proportion
Variation
Ratio
Similarity
Circles
Probability
Surface Area and Volume
Formulae
Pythagoras’ Theorem (Practical Questions
Scientific Notation
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48
49
50
51
52
53
54
55
56
57
58
59
60
61
64
68
70
72
78
80
81
82
85
89
92
96
99
100
101
104
106
108
110
114
115
117
119
120
Calculations & Calculators
(b) 12 − 6 ÷ 3
Round each number to the nearest 10
(a)
55
(b) 123
(c) 24 ÷ 6 + 2
(d) (15 + 5) ÷ 4
(c) 1728
(d) 101 ⋅ 01
Q3.
Round each to the number of decimal places shown in brackets
(a)
3 ⋅ 76 (1)
(b) 18 ⋅ 324 (2) (c) 12 ⋅ 8735 (3) (d) 103 ⋅ 7839 (1)
Q4.
Use your calculator to find the following. Answer to 1 dp where necessary.
(a)
Q5.
Q6.
252 − 162
(b) 8 ⋅ 4 ÷ (9 ⋅ 6- 5 ⋅ 7) (c) 20 × (2 ⋅ 1 + 5 ⋅ 9) (d)
Write in standard form
(a)
20000
(b)
53400000
(c)
Write in full
(a)
3⋅ 2 × 10−4
8 × 103
(c)
(b)
0 ⋅ 0085
4 ⋅ 98 × 106
58
(1 ⋅ 2 × 14)
(d) 0 ⋅ 00000193
(d)
5 ⋅ 36 × 10−5
5
Q2.
Calculate:
(a)
2+3×5
PAGE
Q1.
Shape & Space
Q1.
Copy and fill in the missing angles in the diagrams below.
44o
68o
75o
40o
Q2.
Copy and complete the diagram so that the dotted lines are lines of symmetry.
Q3.
(a)
Plot the points A(2, 1), B (8, 1) and C (8, 5).
y
9
8
7
6
5
4
3
2
1
0
PAGE
6
(b)
(c)
1 2 3 4 5 6
7 8 9
Plot a fourth point D so that ABCD is a rectangle.
Write down the coordinates of D.
x
Q4.
Write down the name of each of these solids.
(a)
(b)
Q5.
(c)
(d)
By dividing the pentagon into triangles. or
otherwise, find the size of angle ABC.
A
B
Q6.
C
Find the surface area of this cuboid.
1.5 m
2m
4m
Q7.
Name, and calculate the sizes of, the angles marked
D
∗
G
P
C
65
B
Q
∗
F
o
∗
R
E
7
A
S
∗
H
PAGE
75o
159o
Q8.
(a)
From the diagram below, name
i. the supplement of ∠ABG
G
A
B
D
ii. the angle vertically opposite to ∠DEH
C
F
E
H
If ∠ABG = 120o, calculate the size of
i.
∠ABE
ii.
∠DEH
(b)
Q9.
iii.
the angle corresponding to ∠EBC
i v.
the angle alternate to ∠FEB
iii.
∠BEF
iv.
∠GBC
Find the area of this triangle:
7 cm
11 cm
5 cm
12 cm
Q10. From the rectangle shown, write down (a) the length of
A
i
AB
(b) the size of angle
i. ∠AOD ii. ∠DOC
O
58o
D
C
18 cm
Q11.
Write down the sizes of the angles marked with the letters a – h.
h
f
e
g
d a
c b
8
BC
B
8 cm
PAGE
ii.
28o
iii.
∠ODC
Q12. Copy and complete the diagrams so that each dotted line is an axis of symmetry.
Calculate the surface area of the carton shown.
11 cm
8 cm
9
10 cm
PAGE
Q13.
Numbers Review 1
Q1. Martin is reading a book with 200 pages. There are 360 words on each page.
How many words are there in total?
Q2.
Two bunches of grapes are weighed in grams. The weights are shown below.
249g
318g
Find
(a) the total weight of the 2 bunches
Q3.
Q4.
Q5.
Calculate
(a) 3 ⋅ 5 + 2 ⋅ 9
(b)
(b) 8 ⋅ 57 − 1⋅ 68
Calculate
(a)
40% of £210
(b)
the difference in their weights
(c) 6 ⋅ 7 × 8
(c)
¾ of 64
Copy and complete this table
1
5
(d) 7 ⋅ 56 ÷ 7
25% of 180g
9
10
0⋅7
0 ⋅ 25
12%
48%
Q6.
Bookworms Bookstore reduce the prices of their books by 331/3%.
How much would you pay for a book that was originally priced at £18?
Q7.
How many days are there in
(a)
a fortnight
(b)
Q8.
Q9.
(c)
9 weeks
How many hours and minutes between
(a)
2.30 pm and 5.45 pm
(b)
40 weeks?
10.15 am and 3.10 pm
The temperature in Aberdeen one morning is −2oC. During the day the
temperature rises by 9 degrees. What is the temperature now?
Q10. A coach leaves Milton Keynes at 1050 and arrives in Hamilton at 1745.
How long does the journey take?
Q11. Amanda collects music magazines which are printed monthly. Each magazine is
PAGE
10
5 millimetres thick. A shelf, 0⋅4 metres long, is filled with these magazines.
How many magazines are on the shelf?
Q12.
Calculate the area of the shape shown in the
diagram.
80 cm
25 cm
70 cm
Q13. The drum shown has a diameter of 24 cm.
Calculate (a) its circumference
(b) the area of the top
Surface
24 cm
Q14. Chris buys a 3-litre bottle of cola for when his friends come to visit. He has glasses which
hold 200 ml. How many glasses of cola will he get out of one bottle?
Q15. Brian gets a fish tank for his birthday.
It measures 30 cm by 60cm by 30 cm.
(a)
30 cm
Calculate the volume of the fish tank
in litres.
60 cm
30 cm
(b)
If Brian pours 36 litres of water into the empty tank, what will be the depth of the
water in the tank?
Q16. It costs £588 to hire scaffolding for 42 days. How much would it cost to hire the same
scaffolding for 36 days at the same rate per day ?
PAGE
11
Q17. Michael & David win £1600 on the lottery and are going to split it in the ratio 3:5.
How much does each get?
Money Matters
Q1.
A shop assistant receives a gross weekly wage of £146.15 for a 37 hour week.
What is the hourly rate ?
Q2.
Peter works as a plumber. He gets paid £4.20 per hour plus overtime.
Overtime is paid at time-and-a-half.
(a)
How much would he earn for
(i)
a 7-hour day?
(ii)
a 35-hour week?
(b)
(i)
What would he earn for
1 hour overtime?
(ii)
8 hours overtime?
Q3.
VAT is charged at 17⋅5%. How much VAT would be
paid on a music system costing £99.90 before VAT?
Round your answer to the nearest 1p.
Q4.
A mail order company sells a sofa for £469.95. It offers Hire Purchase
terms of deposit of £69.95 and 24 monthly payments of £21.50
Calculate
(a)
the total HP cost ?
(b)
how much you save by paying cash ?
Q5.
The White Horse Building Society pays interest of 4% per annum on
savings. How much interest will Sheila receive on savings of £270 after
(a)
1 year?
(b)
8 months?
Q6.
Copy and Complete this electricity bill
Northern Electricity Company
Charges
Present
Previous
29334
28202
Amount Due (£)
units @ 8⋅65p per unit
Standing charge
Sub-Total
VAT
PAGE
12
Total Due
12⋅50
Q7.
Tony is paid a basic monthly salary of £450 plus commission of 12% of his
total monthly sales.
Employee Number
0129
Employee Name
Tony Paterson
Tax Code
342H
Month
2
Basic Pay
450.00
Overtime
−
Commission
Gross Pay
Income Tax
243.70
Pension
National Insurance
Total Deductions
91.80
Net Pay
(a)
(b)
(c)
Q8.
Mark has an income of £23500 and allowances of £5300.
(a)
Calculate Mark’s taxable income.
(b)
Q9.
Calculate his commission in a month where his sales total £9000. Write this in his
pay-slip.
Calculate his gross pay and write it in the pay-slip.
Tony pays 8% of his gross pay into a pension fund. He also pays £91.80
National Insurance and £243.70 Income Tax this month.
Calculate his net pay for this month and complete the pay slip.
Use the table on the
right to calculate how
much tax Mark will pay
in a year.
Taxable Income
Lower rate (£1-£4300)
Basic rate (£4301 - £22800)
Higher rate (over £22800)
Rate of tax
20%
23%
40%
Soraya is travelling to Europe and changes £245 into Euros at the rate of £1 = €1.64
(a)
How many Euros does she receive ?
(b)
She spends 300 Euros. How much does she have left ?
(c)
When she returns she exchanges her Euros for British money. How much will she
get, to the nearest penny ?
Q10. Kevin earns £215 a week and pays 6% in superannuation.
How much superannuation will he pay?
PAGE
13
Q11. Calum wants to insure his life for £18000. The annual premium is £9.84 per £1000.
How much will he pay
(a)
per year
(b) per month
Similar Shapes
Q1.
Enlarge or reduce these shapes by the scale factor given.
×½
×2
Q2.
On a plan 1 centimetre represents 12 metres. How many metres does 7 cm represent?
Q3.
A model tower was made to a scale of 1 cm to 50 m. If the model’s height is 3 cm, what is
the height of the real tower?
Q4.
On a map the scale is 1:200000. What distance, in kilometres, is represented by 5 cm on
the map?
Q5.
These toy drums are similar. The
diameter of the small drum is 10 cm.
and the diameter of the large drum
is 15 cm.
If the height of the small drum is
24 cm, what is the height, x, of the
large drum ?
x
24 cm
10 cm
Q6.
15 cm
These two triangles are similar.
Calculate the length of the side
marked x.
x
10⋅2 cm
2⋅3 cm
PAGE
14
6⋅9 cm
Letters and numbers review
Q1.
Simplify:
(a)
x+x+x+x+x
(b)
k + 3k + 4k
(c)
12m + 9m − 2m
(d)
(e)
5x − 5x
(f)
3a + 5b − b − 2a
4p + 2q + 3 q
Q2.
How many points on (a) 1 star
(b)7 stars
(c) k stars?
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
If x = 5 and y = 3, find the value of
(a)
x+y
(b)
2x − 4
(c)
x2 + 6 y
Write without brackets
(a)
2(x + 5)
(c)
8(6 + w)
Calculate:
(a)
3 + (−2)
(b)
(b) −5 + 8
Solve these equations:
(a)
x+5=3
(b)
7(b − 9)
(c)
−4 − 6
y−4=1
(d)
(c)
z + 3 = −2
Solve these equations:
(a)
2x + 1 = 19
(b)
5y − 1 = 49
Solve :
(a)
7x + 5 = 3x + 21
(b)
15y − 9 = 7y − 1
Write down a formula for the
perimeter of this shape.
−12 + 5
y
x
x
Q10. Find a formula for the nth term of this sequence.
5, 13, 21, 29, 37, . . . .
y
y
x
x
y
Q11.
Solve these equations
(b)
9(x + 2) = 45
15
7(x − 1) = 21
PAGE
(a)
Q12.
(a)
Write down an expression for the area of this triangle.
x+2
6
(b)
The base and height of the triangle are in centimetres.
If the area of the triangle is 24 cm2, form an equation and solve it to find x.
Q13. Write down the solutions of these inequations from the set
{−4, −3, −2, −1, 0, 1, 2, 3, 4}
PAGE
16
(a)
x ≥ −2
(b)
y < −1
(c) z > 3
Speed, Distance, Time
Q1.
Stephen is doing a sponsored walk covering a distance of 20 km.
He can walk at an average speed of 5 km/h.
How long will it take him to finish the walk?
Q2.
An aeroplane flies at 420 miles per hour for 45 minutes.
Calculate the distance it has travelled.
Q3.
A bus leaves the bus station in Glasgow at 9.45 am.
It arrives in Edinburgh, a distance of 51 miles, at 11.15 am.
Calculate its average speed.
Q4.
The diagram shows Mr Jones' journey to
a business meeting.
He leaves his house at 10 am.
He stops for a break then continues on
his journey. He travels a total of 70 miles.
(b) What does one square on the vertical
axis stand for?
60
Distance (miles)
(a) What does one square on the
horizontal axis stand for?
70
(c) How far does he travel in the first part of the journey?
(d) Calculate his speed for the first part of the journey.
50
40
30
20
10
10am
1pm
11am
12noon
(e) How long does he take a break for?
(f) Was he travelling faster or slower in the second part of the journey?
(g) Calculate his average speed for the whole journey. (from when he leaves
his house until he arrives at his meeting)
17
A coach leaves London and travels to Aberdeen, a distance of 750 km, at an average speed
of 80 km/h. If the coach leaves London at 0815, what time will it arrive in Aberdeen?
PAGE
Q5.
Brackets & Equations
Q1.
Q2.
Q3.
Q4.
Q5.
Multiply out the brackets
(a)
3(x + 5)
(b)
7(y + 8)
(c)
4(a − 9)
(d)
9(b − 5)
Multiply out the brackets
(a)
x (x + 2)
(b)
a ( b + c)
(c)
m (8 − m)
(d)
y (b − 5)
Multiply out the brackets
(a)
5(3x + 4)
(b)
6(2b + c)
(c)
10(4 − 5d)
(d)
8(7y − 6)
Multiply out the brackets and simplify
(a)
3(x + 7) + 2x
(b) 5(2y + 3) − 6y
(c)
Factorise
(a)
12b + 8
(b)
x2 + 5 x
(c)
6b − 9c
(e)
2 y2 − 4 y
(d)
Q6.
3x + 5 = 29
(b)
4a − 9 = 7
5(x + 2) = 25
(b)
10(d − 8) = 20
Solve these equations by first multiplying out the brackets
(a)
Q9.
4ab2 − 6abc
Solve these equations by first multiplying out the brackets
(a)
Q8.
ab + ac
Solve these equations
(a)
Q7.
(f)
7(s − 4) + 13
3(2x − 4) = 6
(b)
4(5x + 2) = 108
(b)
3 − 2(4 − x) = 11
(b)
7(x + 1) = 2(x +11)
Solve these equations
(a)
4 + 3(x − 3) = 13
Q10. Solve these equations
PAGE
18
(a)
4(x − 3) = 2(x + 1)
Statistics & Probability 1
Q1.
(a)
List the countries in order, most popular first?
(b)
What fraction of the pupils preferred
(i)
Q2.
SPAIN
The pupils in a class were asked their favourite country for
going on holiday.
The results are shown in the pie chart
Spain
(ii)
France
(iii)
ITALY
FRANCE
USA
Italy
(iv)
USA ?
The table shows the numbers of different types of books borrowed from a
school library in one day.
Type of Book
Number
Sport
6
Crime
14
Horror
12
Romance
22
Sci-Fi
16
Show this information on a bar chart.
Q3.
Q4.
The ages of the players in a local football team are given below :
19
23
30
24
19
25
31
27
28
Calculate the mean, median, mode and range .
30
19
Simon’s temperature was measured every 2 hours over a 24-hour period. The
readings are shown in the line graph below.
Temperature(o
39
2400
2000
1600
1200
0800
0400
0000
37
What was the highest temperature?
(b)
When was this?
(c)
Between what two times did the temperature drop most?
(d)
When did his temperature finally return to normal (37oC)?
PAGE
(a)
19
Time
An express delivery firm charges by weight (W) and delivery time.
Q5.
No
Next day
delivery?
START
No
Is weight over
20 kg ?
charge = £50
Yes
Yes
charge = 2W + 10
No
Is weight over
20 kg ?
charge = £80
Yes
charge = 3W + 15
STOP
Calculate the cost of a parcel weighing
(a)
15 kg (next day)
32 kg
(c)
26 kg (next day)
Q6.
A small firm employs 10 people. The salaries of the employees are as follows :
£40 000
£18000
£15000
£9000
£15000,
£15000
£13000
£15000
£15000
£15000.
(a)
Calculate the mean, median and mode.
(b)
Which of the three measures best describes the average salary in the company?
The graph shows the time taken for a journey at different speeds.
Time (hours)
Q7.
(b)
Describe the correlation.
Speed (mph)
Q8.
The stem and leaf chart below shows the amounts of money spent by
customers in a shop :
PAGE
20
2
3
4
5
6
7
8
(a)
Write down
i.
2
0
0
0
0
1
0
3
1
1
0
0
3
6
3
1
3
1
1
3
8
the median
6
3
2
1
7
8
4
7
9
5
6
n = 33
ii. the range
9
21 represents 21
pence
(b)
Q9.
What is the probability that a customer chosen at random has spent less than 30
pence?
The table below shows the weights in kilograms of group of boys.
Show this information on a stem and leaf chart.
39
42
38
42
42
42
51
45
44
41
48
53
43
38
52
38
42
51
43
49
51
47
47
48
39
44
39
57
50
46
Q10. Darren and his friend are playing with a pack of cards which is missing the Ace of Spades
and the King of Hearts.
What is the probability that the first card dealt is
(a)
an Ace?
(b)
a black card?
(d)
the 4 of clubs?
21
a Queen?
PAGE
(c)
Pythagoras’ Theorem
Find the length of x in each of the triangles below.
Q1.
(a)
x
6 cm
24 mm
7 mm
Q2.
x
(b)
10 cm
A rectangular jigsaw measures 65 cm by 52 cm.
What length is its diagonal?
52 cm
65 cm
Q3.
y
(a)
(b)
12
11
10
9
8
(c)
7
6
5
4
3
2
1
PAGE
22
0
1 2 3 4
5 6 7 8 9 10 11 12
x
Plot the points A(3, 1) and B(10, 10)
Make a right-angled triangle and
mark in the lengths of the sides.
Calculate the length of AB, to 1 dp.
Area & Volume
Q1.
Find the area of each shape shown below. Show all your working.
(a)
(b)
10 cm
6 cm
6 cm
8 cm
(c)
5 cm
Q2.
Calculate the areas of the quadrilaterals shown below.
(a)
(b)
3 cm
6m
3 cm
8 cm
3 cm
9m
Q3.
Calculate the volumes of the following solid shapes.
(a)
6 cm
(b)
3 ⋅ 5 cm
12 cm
2 cm
6 cm
(c)
8 cm
5m
PAGE
23
20 m
Q4.
Calculate the volume of this solid shape.
7 cm
10 cm
7 cm
30 cm
10 cm
8 cm
Q7.
A rectangular tank is 1⋅ 5 m long, 30 cm broad and 20 cm high.
How many litres of water can it hold?
Q8.
A window is in the shape of a rectangle, 4m by 2m with a semicircle of diameter
4m on top.
Find the area of glass in the window.
2m
4m
Q9.
(a)
A box of chocolates is in the shape of a triangular prism.
Calculate its volume.
9 cm
10 cm
PAGE
24
20 cm
(b)
The box contains 63 chocolates each with a volume of 4 cm3.
What percentage of the volume of the box is unused?
Q10. A cylindrical tin holds a litre of liquid and has a diameter of 7 cm. Calculate its height.
Q11.
The end of the wooden mouldings used to make a
photograph frame is in the shape of a quarter–circle.
If a total length of 70 cm of mouldings is required for
a frame, find the volume of wood used.
1⋅5 cm
Q12. Mrs Gamp is going to cover the curved surface of a cylindrical umbrella stand with
waterproof fabric. The radius is 10 cm and the height is 60 cm.
Calculate the area of material required.
10 cm
PAGE
25
60 cm
Formulae
Q1.
A = 3b − 2c. Find the value of A when
(a)
b = 4, c = 5
Q2.
Find a formula for the nth term for each of the sequences below.
(a)
4, 7, 10, 13, . . .
(b)
(c)
b = 6, c = 9
(b)
b = 28, c = 11
(c)
3, 8, 13, 18, . . .
Q3.
26
b = 3, c = − 2
25, 29, 33, 37, . . .
The diagrams on the left
show tiling patterns for
doorsteps.
(a)
Draw the 4th pattern in the sequence
(b)
Complete the table below.
Number of
coloured tiles, C
Number of white
tiles, W
PAGE
(d)
1
2
3
4
5
6
(c)
Write down the formula for finding the number of white tiles if you know the
number of coloured tiles.
W=
(d)
How many white tiles would you need if you had 13 coloured tiles?
(e)
How many coloured tiles would you need if you had 24 white tiles?
Gradients & Straight Lines
Q1.
Find the gradients of the lines shown in the diagram below
c
a
b
d
e
f
Q2.
( a)
Copy and complete this table for y = 2x + 3
−3
x
−2
−1
0
1
2
3
−1
y
(b)
Plot the points on a grid, with x-axis from -5 to 5 and y-axis from -5 to 10, and
draw a straight line through them.
Car hire is charged by the day.
(a)
Write down the equation of the line.
(b)
Calculate the cost of hiring a car for
8 days.
(c)
Geri’s bill was £275. For how many days
did she hire a car?
15
12
10
7
5
2
1
2
3
4
Number of Days
5
6
27
0
PAGE
cost (£C)
Q3.
Q4.
Q
50
40
30
20
10
0
(a)
2
4
28
8
10
12
14
Find the equation of the line shown in the diagram above.
(b)
PAGE
6
Find the value of Q when N is 22.
N
Trigonometry
Q1.
Calculate x in each diagram below. Show all your working.
(a)
(b)
6⋅8 cm
10 cm
6 cm
53o
x
xo
(c)
7⋅9 cm
x
Q2.
27o
Jenny is standing 25 metres away
from the bottom of a church tower.
She looks up at the top at an
angle of elevation of 52o.
Calculate the height of the tower.
52o
25 m
An aircraft making a steady descent decreases height by 2 km in 18 km. What is the
angle of descent, xo ?
18 km
xo
29
2 km
PAGE
Q3.
Q4.
A ladder, which is 6⋅4 metres long, leans
against a vertical wall and makes an angle
of 67o with the ground.
67o
Calculate, to the nearest 0.1 m, how far the
bottom of the ladder is from the wall.
PAGE
30
Q5.
Eddie is flying his kite. The string
is at an angle of 32o to the
horizontal.
He lets out 30 metres of string.
How high is the kite above
the ground?
30 m
32o
h
Numbers Review 2
Q1.
Q2.
Q3.
Q4.
Calculate:
(a)
2+3×5
(b) 522 ÷ 6
(e)
6⋅8 ÷ 10
(i)
30% of £410 (j)
(f)
(c) √81
(d) £12.25 − £3.97
/3 of 51
(g) 2⋅19 × 1000
(h) 62
¾ of 56
(k) 75% of 180g
(l) 8 × £9.40
2
Change these units to the units shown in brackets
(a)
12 cm (mm)
(b) 5500 m (km)
(c)
9 mm (cm)
(d)
(f)
890 g (kg)
15 g (mg)
(e)
2⋅5 tonne (kg)
Write in standard form
(a)
1 450 000
(b) 0⋅000 79
(c)
245
Write in full
(a)
5⋅6 × 10−1
(b) 1⋅34 × 104
(c)
8⋅751 × 102
Q5.
Jamie leaves his house at 8.17 a.m. and arrives at school at 8.43 a.m. How long does it
take him to travel to school?
Q6.
An aeroplane takes ¾ of an hour to travel from one airport to another. If it’s average
speed was 440 km/h, how far apart were the airports ?
Q7.
Find the next 3 terms in each of these sequences.
(a)
1, 5, 9, 13, . . . .
(b) 20, 17, 14, . . . .
(c)
11, 34, 57, . . . .
Write these ratios in their simplest form.
(a)
18 : 12
(b) 2 m : 20 cm
(c)
24 minutes:6 hours
Share
(a) 200g in the ratio 9:1
(c) 36 m in the ratio 5:7
Q8.
Q9.
(b) £14 in the ratio 2:5
PAGE
31
Q10. The width of 6 identical textbooks is 13.8 cm. What is the width of 9 of the same books ?
Fractions, Decimals and Percentages
Q1.
Calculate
(a)
1% of 5000
(d)
1
(b) 10% of 120
(e)
/3 of 9⋅6
2
/5 of 85p
(c)
70% of 230
(f)
5
/8 of 40
Q2.
In a class of 30 pupils, 9 had brown eyes. What percentage is this?
Q3.
In a sale all prices are reduced by 35%.What would be the sale price of a mini disc player
which originally cost £200?
Q4.
Universal Video increased their hire charges by 12%.
What would it now cost to rent a video that used to cost £3 to hire?
Lord of the R ings
Q5. Copy and complete the table to show equivalent fractions decimals and percentages.
(Write the fractions in their simplest form)
fraction
decimal
PAGE
32
percentage
3
5
0⋅7
0⋅75
18%
36%
Fractions
Change the following to mixed numbers :
(a)
Q6.
6
1
8
(c)
(d)
83
7
2
1
2
2
1
10
(d)
9
7
9
2 1
+
5 10
(b)
1 1
−
2 6
(c)
(b)
5
1
3 +2
8
4
(c)
5 2
−
8 5
1
5
3 +1
2
6
6
7
1
−4
8
3
At a school, 1/8 of the time is spent in mathematics
classes and 3/20 is spent in English classes.
(a)
What fraction of the time is spent on English and mathematics together ?
(b)
If 1/20 of the time is spent in games, what fraction of the time is spent on all the
other subjects ?
Express as a single fraction in its simplest form
(a)
Q7.
(b)
53
6
Calculate
(a)
Q5.
(c)
Calculate each of the following, expressing your answer in its simplest form.
(a)
Q4.
22
5
Change to fractions :
(a)
Q3.
(b)
4 1
×
5 16
12 39
×
13 48
(b)
(c)
24 20
÷
35 21
Calculate
(a)
1
1
1 ×2
4
3
1
2
(b) 3 ÷ 1
3
4
(c)
3
1
3 ÷2
5
4
33
Q2.
3
2
PAGE
Q1.
Equations & Inequalities
Q1.
Solve these equations
(a)
5x = 20
(d)
(c)
y−9=1
(e)
5z + 9 = 4
(f)
6y − 9 = 2y + 5
Solve these inequalities, giving your answer from the set {−2, −1, 0, 1, 2, 3, 4, 5, 6, 7}
(a)
2x < 2
(b)
x+5≥8
(c)
3x + 8 ≤ 2
Q3.
Solve these inequalities
(a)
7x > 42
Q5.
5x − 3 ≤ 22
(c)
3x − 2 > −11
8k − 5 = 5k + 1
(b)
6(a − 1) = 4(a + 2)
(b)
5(y − 2) > 2(y + 4)
Solve these inequalities
(a)
Q6.
(b)
Solve these equations
(a)
9x + 2 ≤ 6x + 11
Solve these inequalities, giving your answer from the set {−3, −2, −1, 0, 1, 2, 3, 4, 5, 6}
(a)
34
z + 8 = 15
Q2.
Q4.
PAGE
2x − 12 = −3
(b)
7x − 3 > 2x − 23
(b)
9(y + 2) ≤ 7(y + 4)
Statistics and Probability 2
Q1.
80 people were asked their favourite country for
going on holiday.
The results are shown in the pie chart.
(a)
What fraction of the people preferred France?
SPAIN
ITALY
FRANCE
(b)
How many people preferred Spain?
(c)
The same number of people liked USA as liked Italy.
How many liked USA?
USA
Sam’s class held a competition to find out who could hold their breath the longest.
The results, in seconds, are shown below:
34
51
82
49
65
47
61 78 32 45 52
56 25 43 47 41
(a)
Draw a stem-and-leaf diagram to illustrate this data.
(b)
What is the median number of seconds for this data?
The stem and leaf chart below shows the amounts of money spent by
customers in a shop :
2
3
4
5
6
7
8
Write down
Q4.
44
39
(a)
2
0
0
0
0
1
0
3
1
1
0
0
3
6
3
1
3
1
1
3
8
6
3
2
1
7
8
4
7
9
5
6
n = 33
9
21 represents 21
pence
(b) the range
the median
The ages of the players in a local football team are given below :
19
23
25
24
19
(a) Calculate the mean and median .
25
31
27
28
30
33
(b) The two oldest players leave and are replaced by two players aged 18 and 26.
Calculate the mean and median age of the team now.
35
Q3.
56
63
PAGE
Q2.
The bar chart shows the different types of books borrowed from a school
library in one day.
(a)
There were also 16 science-fiction
books
borrowed that day.
22
Add a bar to the chart to show this.
20
Number of books
Q5.
18
16
14
12
10
8
6
4
2
sport
crime
(b)
What was the most popular type of
book borrowed?
(c)
How many more pupils chose
‘crime’ than ‘sport’ books?
horror romance
Type of book
(d)
Q6.
What is the probability that a pupil chosen at random had borrowed a horror book?
The table below shows the connection between the thickness of insulation in a
roof and the heat lost through the roof.
Heat loss in kilowatts (H)
1⋅5
1⋅8
3
4
4⋅4
4⋅6
Thickness in cm (T)
22⋅5
18
10
11
6⋅5
2⋅5
Draw a scatter diagram on the graph below
Thickness of insulation in centimetres (T)
(a)
25
20
15
10
5
0
PAGE
36
0
1
2
3
4
5
Heat loss from roof in kilowatts (H)
(b)
Draw the best fitting straight line through the points
(c)
Use your graph to estimate the heat loss from 15 centimetres of insulation
Q8.
The probability that it will rain is 83%. What is the probability that it won’t rain?
Q9.
The probability that Grampa Smith will beat Grampa Jones at dominoes is 2/5.
How many games would Grampa Smith expect to win if they play 30 games?
Q10. (a)
(b)
Complete this table to show all the possible combinations of flipping a coin and
rolling a dice.
1
2
H
(H, 1)
(H, 2)
T
(T, 1)
Calculate:
(i)
P(H, even number) (ii)
3
4
5
(iii)
P(H, 6)
6
P(T, factor of 6)
Q10. The table shows results after every 50 spins of a spinner.
number of wins
12
26
30
50
70
81
91
number of spins
50
100
150
200
250
300
350
relative frequency
PAGE
37
Copy and complete the table to show the relative frequencies.
Shape & Space Review
Q1.
Calculate the sizes of the missing angles in the diagrams below.
b
a
23o
d
34o
142
o
e
f
g h
120o
c
Q2.
Find the missing lengths and angles in the quadrilaterals shown below
5⋅7 cm
b
a
c
g h
41o
n
7 cm
37o
j
e
m k
35o
40o
d
f
Calculate x in the diagram below
Q3.
22 cm
PAGE
38
x
20 cm
Q4.
(a) Complete this diagram so that the
dotted line is a line of symmetry.
(b)
Complete this diagram so that X is a
centre of four-fold rotation symmetry.
X
Q5.
The diagram shows a scale drawing of a kitchen. The scale is 1cm to 0.5 m. Measure the
diagram and then calculate the real length of the wall marked k.
1 ⋅5 m
1m
3m
2m
k
Q6.
PAGE
39
These two picture frames are similar. The
larger is 15 cm wide and the smaller is
10 cm wide.
If the large frame is 12 cm high, what is the
height of the small frame?
House & Car Costs
Q1.
James buys a house costing £45 000.
The building Society will only give him a 90% mortgage.
How much does he need to pay for the deposit?
Q2.
The Barratt family is moving to a new home. It cost £68 000.
They had to pay a deposit of 5%.
(a)
How much will they need to take out a mortgage for?
(b)
Q3.
Q4.
If the mortgage payments are £6.40 per £1000, how much will they pay per month?
Arthur has a ring which is valued at £1200.
If it appreciates at the rate of 6% per year, how much will it be worth
in 3 years time?
Mr. & Mrs Scott are taking out a new insurance policy for their house and contents. The
rates they are quoted are:
BB
INSURANCE
Buildings £3 per £1000
Contents £5.80 per £1000
Buildings £2.80 per £1000
Contents £6.70 per £1000
If their house is worth £75 000 and they have contents valued at £12 000, which of the two
companies will give them the better deal?
PAGE
40
Q5.
Gemma buys a car for £6000. It depreciates by 30% in the first year and 20% the second
year, how much is it worth after 2 years?
Pairs of Straight Lines 1
(a)
y = 2x
Copy and complete the tables below for y = 2x and y = x + 4 .
x
−1
0
y=x+4
5
y
Q3.
(b)
Draw the graphs on a set of axes.
(c)
Write down the point where the lines cross.
(a)
On a set of axes, draw the graphs of x + y = 6 and y = x + 3.
(b)
Write down the coordinates of the point of intersection of the two lines.
Martin is looking for a job as a paper delivery boy. He sees two adverts. Pete’s Papers is
offering a wage of £5 plus 5p per paper delivered. Dave’s Daily Delivery offers 15 p per
paper delivered.
(a)
Copy and complete the table below
(b)
(c)
Q5.
5
y
Number of
papers (N)
Pete’s
Papers (£)
Dave’s
Daily (£)
Q4.
0
0
10
20
5
5⋅50
6
0
1⋅50
3
30
40
50
60
70
80
Draw graphs of both sets of wages.
Which company would you recommend Martin to take a job with? (Give reasons)
Solve these pairs of equations by substitution:
(a)
y = 2x + 3
y = 3x − 2
(b)
y = 4x
x + y = 10
Solve these pairs of equations by elimination:
(a)
x + y = 12
x−y=2
(b)
4x + y = 21
x+y=6
(c)
(d)
5x + 3y = 30
3x + y = 14
7x + 3y = 20
2x − y = 2
41
Q2.
−3
x
PAGE
Q1.
Letters & Numbers Review
Q1.
Q2.
Q3.
Q4.
Q5.
Q6.
Simplify:
(a)
2x − 3x + 5x
(b)
Multiply out the brackets
(a)
6(x − 5)
Factorise:
(a)
8y + 48
Calculate
(a)
−2 − 8
(b)
8y − 5y − 3y
(c)
2s + 3t − s + 5t
(b) 3(3a + 1)
(c)
5(7 − 4y)
(b) 15x − 20
(c)
6a + 9b
−7 + (−5)
(c)
18 − (−5)
−30 −(−11)
(d)
A = 3b − 2c. Find the value of A when
(a)
b = 5, c = 4
(b) b = 8, c = 12
(c)
b = 9, c = 15
Solve these equations
(a)
x + 5 = 13
(c)
9a + 2 = 5a − 6
(b) 6y = 3y + 15
a
Q7.
(a)
Write down the formula for the perimeter of this shape.
P=
(b)
a
a
b
b
Calculate P when a = 3 cm and b = 5 cm.
a
a
a
Q8. The volume of a pyramid is given by V = 1/3Ah, where A is the area of the base
and h is the height.
(a)
Calculate the volume if the area of the base is 49 cm2
and the height is 9 cm.
PAGE
42
(b)How many ice cubes with edges 3cm would you need to melt to fill the pyramid
with water?
Q9.
Coronet Wallcoverings make several designs for wallpaper borders one of which is shown
below. The pattern is made up of circle shapes and cross shapes.
(a)
Complete the table below.
Number of circle shapes (C)
2
Number of cross shapes (N)
2
(b)
3
4
5
6
7
6
Write down a formula connecting the number of cross shapes, N, and the number
of circle shapes, C.
N=
Q10. Solve these inequalities, taking your answers from the set {−3, −2, −1, 0, 1, 2, 3}
(a)
x≤2
(b) x + 5 > 6
(c)
x−1<0
Q11.
Solve these inequalities.
(a)
2x ≥ 14
(b)
2x − 5 < 7
(c)
7x − 2 < 5
(e)
5x + 1 ≤ 2x + 10
(f)
(d)
4x + 3 > −1
8x − 12 > 3x − 17
Q12. Match these graphs to the containers which release water from their bases at a steady rate.
2
B
Time
Time
43
Time
C
Depth of water
Depth of water
Depth of water
A
3
PAGE
1
Proportion
Q1.Jafar's heart beats at the rate of 84 beats per minute. How many beats will it make
in 5 minutes?
Q2.
A jet can cover a distance of 2436 miles in 3⋅5 hours. What is its rate of travel in miles per
hour?
Q3.
Copy and complete the table below for bicycle hire at £7.50 per day.
Number of days
1
2
6
9
15
Cost (£)
Q4.
Jack drives 400 km in 5 hours. At the same rate, how far could he drive in 8 hours?
Q5.
Paula paid £22 for 40 litres of petrol. How much would she pay for 47 litres?
Q6.
The mass of 20 cm3 of a metal is 30grams. What is the mass of 16 cm3?
Q7. (a)
Using this table of values, draw a graph of N against C.
PAGE
44
Number of tickets (N)
Cost (C)
(b)
Describe the graph.
(c)
Find a formula for C
5
20
10
40
15
60
20
80
25
100
Angles in a Circle
Q1.In each of the diagrams below AB is a diameter. Find the missing angles in each diagram.
ao
A
35
b
A
o
45
o
B
o
A
co
47
o
ho
go
72o
do
f
eo
B
o
B
Q2.Find the length of the diameter AB in each of the circles below, given the other 2 sides of
the triangle.
(a)
(b)
A
7 cm
8 cm
7 cm
A
3 cm
B
B
Q3.Use the symmetry properties of the circle to find the missing angles in the diagrams below.
In each diagram AB is a diameter.
A
A
m o no
ao bo
50
lo
28o
o
o
c
B
ro
go
B
Find x in each of the diagrams below.
5⋅7 cm
xo
60o
4 cm
45
8 cm
x
PAGE
Q4.
jo ko
h o io
Areas & Volumes Review
Q1.
Find the area of each shape below.
(a)
(b)
11 cm
9 cm
20 cm
Q2.
11 cm
Find each shaded area below.
9 ⋅5 m
9m
8m
4 ⋅5 m
9m
18 m
Q3.
Find the volumes of the solid shapes below.
7 cm
7m
17 cm
10 cm
11 m
50 cm
13 m
15 cm
Q4.
How many toy building bricks, 6 cm by 4 cm by 2 cm, would fit into a storage box 30 cm
by 16 cm by 10 cm?
Q5.
Calculate the volume of chocolate ice-cream in this
vanilla and chocolate desert in the shape of a cylinder,
with chocolate on the outside and vanilla inside.
6 cm
10 cm
PAGE
46
18 cm
Rounding
APPROXIMATION
round to 1 – decimal place
3.
4.
(a )
26
(b)
856
(c )
492
(d )
75
(e)
561
(f)
1242
(g)
565
(h)
798
Round each of the following numbers to the nearest whole number.
(a)
57 ⋅ 4
(b)
8 ⋅ 61
(c )
6 ⋅ 199
(d )
14 ⋅ 57
(e)
28 ⋅ 91
(f)
341 ⋅ 8
(g)
123 ⋅ 0
(h)
9 ⋅ 099
Round each of the following numbers to 1 – decimal place.
(one number after the point)
(a)
26 ⋅ 32
(b)
8 ⋅ 48
(c )
34 ⋅ 71
(d )
2 ⋅ 64
(e)
14 ⋅ 79
(f)
23 ⋅ 85
(g)
39 ⋅ 63
(h)
7 ⋅ 67
(i )
29 ⋅ 33
( j)
1 ⋅ 55
(k )
68 ⋅ 70
(l )
4 ⋅ 26
( m)
123 ⋅ 97
(n)
18 ⋅ 94
( o)
4 ⋅ 51
( p)
12 ⋅ 96
Round each of the following numbers to 2 – decimal places.
(two numbers after the point)
(a )
36 ⋅ 344
(b)
8 ⋅ 123
(c )
3 ⋅ 786
(d )
22 ⋅ 155
(e)
4 ⋅ 719
(f)
7 ⋅ 861
(g)
14 ⋅ 659
( h)
17 ⋅ 382
(i )
6 ⋅ 237
( j)
6 ⋅ 555
(k )
1 ⋅ 786
(l )
9 ⋅ 270
(m)
13 ⋅ 697
(n)
8 ⋅ 994
(o )
17 ⋅ 595
( p)
2 ⋅ 906
47
2.
Round each of the following numbers to the nearest 10.
PAGE
1.
Calculations & Rounding
** You need a calculator for this worksheet.
1.
2.
3.
Calculate each of the following rounding your answers to 1 – decimal place.
a)
2 ⋅ 31 × 6 ⋅ 4
b)
18 ÷ 7
c)
12 ⋅ 9 × 0 ⋅ 13
d)
4⋅7× 4⋅7
e)
5 ⋅ 232
f)
16 × 5 ⋅ 3
1⋅ 9
Calculate each of the following rounding your answers to 2 – decimal places.
a)
2 ⋅ 562 × 12 ⋅ 41
b)
79 ÷ 4.7
c)
17 ⋅ 91 × 10 ⋅ 13
d)
8 ⋅ 12 × 8 ⋅ 12
e)
2 ⋅ 77 2
f)
23 × 1 ⋅ 34
3⋅ 2
Change each of the following fractions to decimal fractions rounding your answers
to 2 – decimal places.
a)
4.
48
b)
11
19
c)
23
41
d)
1
13
d)
£167 ÷ 13
Round each of the following calculations to the nearest £1 .
a)
PAGE
3
7
£17 ÷ 4
b)
£233 ÷ 9
c)
£45 ÷ 11
5.
Round each of the calculations in question 4 to the nearest penny (1p).
6.
A man pays 7 boys £31 for digging his garden. The boys divide
the money equally between themselves.
a)
How much money, to the nearest penny, can each boy get ?
b)
How much money is left that can't be divided ?
Squares and Square Roots
** You need a calculator for this worksheet.
3.
a)
8
b)
23
c)
120
d)
13
e)
3
f)
45
g)
235
h)
1230
i)
8⋅9
j)
29 ⋅ 6
k)
0⋅7
l)
39 ⋅ 2
Find the value of each of the following , rounding your answers to 3 - decimal places .
a)
17
b)
288
c)
3478
d)
23 ⋅ 6
e)
85
f)
341
g)
31
h)
674
Calculate the value of x in each of the following , rounding off your answers to
2 - decimal places where necessary .
a)
x 2 = 32 + 4 2
b)
x 2 = 10 2 + 24 2
* show your working
c)
x 2 = 22 + 7 2
d)
x 2 = 12 2 + 4 2
x 2 = 7 2 + 32
e)
x 2 = 20 2 + 152
f)
x 2 = 14 2 + 9 2
= 49 + 9
= 58
g)
x 2 = 52 − 2 2
h)
x 2 = 17 2 − 14 2
i)
x 2 = 2 ⋅ 32 + 1 ⋅ 8 2
j)
x 2 = 6 ⋅ 7 2 − 3 ⋅ 82
k)
x 2 = 0 ⋅ 82 + 1 ⋅ 12
l)
x 2 = 1 ⋅ 32 − 0 ⋅ 7 2
m)
x 2 = 4 ⋅ 9 2 − 4 ⋅ 12
n)
x 2 = 7 ⋅ 32 + 9 ⋅ 2 2
x =
58
= 7 ⋅ 62
49
2.
Find the value of each of the following , rounding your answers to 2 - decimal places .
PAGE
1.
Standard Form
1.
2.
3.
4.
PAGE
50
5.
Express each of the following numbers in standard form :
(a)
234 000
(b)
650
(c)
8700
(d)
12 000 000 000
(e)
43
(f)
9 ⋅ 21
Write each of the following as an ordinary number :
(a)
2 ⋅ 4 × 102
(b)
3 ⋅ 61 × 101
(c)
7 ⋅ 003 × 104
(d)
5 ⋅ 8 × 107
(e)
6 ⋅ 04 × 103
(f)
2 × 100
Express each of the following numbers in standard form :
(a)
0 ⋅ 0045
(b)
0 ⋅ 0304
(c)
0 ⋅ 000 000 05
(d)
0 ⋅ 86
(e)
0 ⋅ 000 000 000 89
(f)
0 ⋅ 000345
Write each of the following as an ordinary number :
(a)
8 ⋅ 7 × 10−3
(b)
3 ⋅ 92 × 10−1
(c)
2 ⋅ 07 × 10−6
(d)
7 ⋅ 8 × 10−2
(e)
6 ⋅ 005 × 10−4
(f)
5 × 10−5
(a)
The distance between the planet Earth and the sun is a
mere 93 000 000 miles.
Write this number in standard form.
(b)
Jupiter's closest satellite is called Amalthea and is about 112 000 miles
from the centre of the planet.
Write this distance in standard form.
(c)
The average distance from the sun to the planet Pluto
is 3 ⋅ 7 × 109 miles.
Write this distance as an ordinary number.
(d)
The coefficient of linear expansion of brass is 1 ⋅ 9 × 10−7 per degree centigrade.
Write this coefficient as an ordinary number.
Angles and Triangles
1.
Calculate the size of each lettered angle below :
35o
65o
120
a
o
b
c
248o
155o
130o
e
115
o
145
90o
d
f
g
o
148o
38
o
h
m
k
n
p
l
i
2.
j
95o
Calculate the size of each missing angle in the triangles below :
105
49o
58o
a
o
c
b
35
28o
d
41o
Copy each triangle below and fill in all the missing angles :
20o
45o
42o
75o
(b)
51
(a)
(e)
(c)
(d)
PAGE
3.
44
o
42o
o
Shapes and Coordinates
You need a ruler for this worksheet.
1.
(a) Plot the points A(2,2) , B(7,2) and C(7,7) on a coordinate diagram.
(b) Given that ABCD is a square , complete the diagram and write down
the coordinates of the point D.
2.
(a) On a coordinate diagram plot the points P(1,3) , Q(8,3) and R(8,6).
(b) Given that PQRS is a rectangle , complete the diagram and write down the coordinates of
the point S.
3.
(a) On a coordinate diagram plot the points E(2,4) , F(4,1) and G(10,5).
(b) Given that EFGH is a rectangle , complete the diagram and write down the coordinates of
the point H.
4.
(a) On a coordinate diagram plot the points T(4,2) , U(7,3) and V(6,6).
(b) Given that TUVW is a square , complete the diagram and write down the coordinates of
the point W.
5.
(a) Plot on a coordinate diagram the points A(2,5) , B(-3,5) and C(-3,-2).
(b) Given that ABCD is a rectangle , complete the shape and write down the coordinates of D .
6.
(a) Plot on a coordinate diagram the points A(6,3) , B(3,5) and C(-3,3).
(b) Given that ABCD is a kite , complete the shape and write down the coordinates
of the point D.
7.
PAGE
52
8.
Repeat question 6. for the following sets of points :
(a)
A(-1,5) , B(2,3) and C(-1,-6).
(b)
A(2,5) , B(6,3) and C(2,-5).
(c)
A(-2,-4) , B(1,3) and C(5,3).
(a) Plot on a coordinate diagram the points P(3,5) , Q(5,2) and R(3,-1).
(b) Given that PQRS is a rhombus , complete the shape and write down the coordinates
of the point S.
Working with Percentages
Write each of the following percentages as a decimal fraction :
a) 32%
3.
4.
c) 20%
d) 8%
e) 3%
g) 7%
h) 12 12 %
i) 3 12 %
Calculate each of the following :
a) 36% of £24.00
b) 56% of £3000
c) 18% of £340
d) 8% of £15
e) 10% of £16.40
f) 86% of £12000
g) 20% of £34.50
h) 12% of £58
i) 2% of £85
j) 9% of £16
k) 6 12 % of £200
l) 12 12 % of £120
Calculate each of the following , rounding your answer to the nearest penny if necessary :
a) 23% of £45
b) 17% of £23.50
c) 45% of £12.75
d) 14% of £8.65
e) 3% of £24.34
f) 15% of £4.95
g) 8% of £8.50
h) 26% of £3.48
i) 16% of £340.10 j) 4% of £12.43
k) 78% of £13.93
l) 67% of £1.89
m) 18% of 94p
o) 5% of 23p
p) 76% of 94p
n) 23% of £0.47
The following items are to be reduced in price by 10% for a sale.
Calculate the sale price of each item :
Skate Board
£24
Clock Radio
£35
Video Camera
£462
5.
f) 90%
CD Player
£123.70
Increase the price of each item above by 8%.
Yo Yo
£12
Cool Shades
£16.50
53
2.
b) 87%
PAGE
1.
Wages and Salaries
Round all answers to the nearest penny where necessary
1.
2.
3.
4.
Calculate
i)
ii)
The weekly pay
The annual pay (to the nearest pound) ……... of each person below :
a)
Susan works a 40 hour week and gets paid £3.48 per hour.
b)
John gets paid £5.10 per hour and works a 35 hour week.
c)
Mark has an hourly rate of £7.85 and works a 44 hour week.
d)
Helen works a total of 28 hours each week and is paid £4.25 per hour.
For each person below calculate
i) their weekly pay ii) their hourly rate of pay.
a)
Tom earns £15400 per year and works a 38 hour week.
b)
Jane works 30 hours each week and has an annual salary of £18560.
c)
Brian has an annual salary of £26800 and works 42 hours each week.
d)
Andrea works 46 hours every week and has an annual salary of £28300.
Overtime is paid at "time-and-a-half" ( 1 ⋅ 5 times normal rate of pay).
a)
Graeme is paid £4.85 per hour and works a 40 hour week. How much will he earn
in a week where he works 46 hours ?
b)
Susan earns £3.20 per hour for a normal working week of 34 hours. How much will
she be paid for working 38 hours one week ?
c)
One week Tom works his normal 35 hours and an extra 12 hours overtime.
Calculate his pay for this week if his normal rate of pay is £8.40 per hour.
d)
In a particular week Michael works his normal 40 hours plus an extra 5 hours
overtime. Calculate his pay for this week if his normal rate of pay is £6.35 per hour.
Steven works a 36 hour week. His normal rate of pay is £4.95 per hour.
Calculate his total pay for a week where he works his normal hours + ……..
….. 4 hours overtime at "time-and-a-half" and 6 hours at "double-time"
PAGE
54
5.
Lucy works a 30 hour week. Her normal rate of pay is £3.50 per hour.
Calculate her total pay for a week where she works her normal hours + ……..
….. 8 hours overtime at "time-and-a-half" and 2 hours at "double-time"
Overtime and Commission
Overtime : Higher rates are paid to some workers who are
prepared to work evenings or weekends.
The following rates apply for all the questions below :Evenings ....... time-and-a-half
Saturdays ....... time-and-a-half
Sundays ........ double-time
1.
a) Helen is paid a basic rate of £4.60 per hour and works a 36 hour week.
Calculate her total pay for the week she works ....
her basic 36 hours + 4 hours of evening work + 6 hours work on Sunday.
b) John works a 40 hour week and is paid £5.20 per hour. Calculate his total pay for the
week where he works an extra 8 hours on Saturday and 2 hours on Sunday.
c) Susan works for an advertising agency and is paid £8.10 per hour.
Calculate her total pay for each of the following weeks ......
Week 1 - 30 basic hours + 5 hours on Saturday + 4 hours on Sunday.
Week 2 - 26 basic hours + 10 hours of evening work + 2 hours on Saturday.
2.
Michael works for Bee and Queue and is paid £3.80 per hour.
He works a basic 5–day week from 9.30am until 5.30pm , Monday to Friday.
How much will Michael earn in the week where he works his normal week plus overtime
from 10.00am until 4.30pm on Sunday ?
Commission : People who "sell" for a living are usually paid commission calculated as a percentage
of their total sales.
3.
Calculate the commissions on each sale below :
a) 5% on a sale of £460
b) 3% on a sale of £1200
c) 10% commission on a sale of
Mr Greig is a door-to-door salesperson. He is paid a weekly wage of £140 plus a commission
of 12% on his total weekly sales. How much will he earn in a week where his sales total £1050 ?
5.
Miss Jones sells cars. She works a basic 38 hour week and is paid £3.45 per hour.
She is also earns a 2% commission on every car she sells.
Calculate her total pay for the week where she sells three cars totalling £14200 in value.
PAGE
4.
55
£625
Savings and Interest
1.
The Royal Bank of Scone has an interest rate of 4% p.a. (per annum).
How much interest would you receive on the following amounts
of money at the end of 1 year ?
(a)
2.
£600
(b)
£1400
(c)
£480
(d)
£80.50
Pegasys Building Society has three different savings schemes.
Each scheme has its own interest rate and a particular rule for withdrawing your money.
Super Saver Scheme : Interest 5% p.a. - monthly withdrawals allowed.
Bumper Savings Scheme : Interest 6% p.a. - 3 months notice of withdrawal.
Flexible Savings
(a)
: Interest 3% p.a. – no notice of withdrawal
How much interest would you receive in a year on :(i) Savings of £750 in the "Bumper Scheme" ?
(ii) Savings of £2340 in the "FlexibleScheme" ?
(iii) Savings of £320 in the "Super Saver Scheme" ?
(iv) Savings of £8160 in the "Flexible Scheme" ?
(b)
3.
Calculate the interest on :
56
(a)
(c)
(e)
PAGE
John wants to split his savings of £1200 .
He decides to put half in the "Super Saver Scheme" and half in "Flexible Savings.
Calculate the total interest he will receive in a year from his two accounts.
£960 at 7% p.a. for 6 months.
£440 at 3% p.a. for 2 months.
£3450 at 2% p.a. for 1 month.
(b)
(d)
(f)
£1356 at 5% p.a. for 4 months.
£972 at 9% p.a. for 9 months.
£576 at 2 12 % p.a. for 7 months.
4.
Mr White invests £3200 at 6% p.a.
Five months later he decides to draw out his interest to help pay for
a new camera. How much does he draw out ?
5.
Miss Gray invests £1500 at 8% p.a.
Eight months later she decides to lift out her interest to help pay for
a night out. How much will she draw out ?
Value Added Tax
(VAT)
Take 17 12 % as the rate of VAT for all the following questions.
1.
2.
Calculate the VAT due on each item below :
(a) A cycle costing £80
(b) A stereo costing £130
(c) A toy costing £8
(d) A radio costing £82
(e) A gas bill costing £68
(f) A holiday flight costing £640
For each article below, calculate :
(a)
the VAT
(b)
the total cost.
£26
£60
£12
£230
£48
£148
£180
Copy and complete the following bills :
£
6 dozen nails @ £0.80 per doz.
5 tins of paint @ £4.60 per tin
10 rolls of wallpaper @ £6.80 per roll
Subtotal
+ VAT @ 17 12 %
Subtotal
+ VAT @ 17 12 %
TOTAL
TOTAL
(b)
(a)
Which of these two boats is the cheapest to buy ?
£16400
plus VAT
£19280
including VAT
57
4.
£
8 starters @ £2.40 each
6 main courses @ £8.10 each
8 sweets @ £3.60 each
PAGE
3.
Electricity Bills
Calculate the entries A , B , C , D and E for each bill below :
1.
Plugs Electricity Co.
Meter Reading
Present
27458
Previous
27158
Charges
A units at 8.11p
Standing Charge
Subtotal
VAT at 17.5%
TOTAL
2.
Present
58458
Previous
57123
Charges
A units at 6.31p
Standing Charge
Subtotal
VAT at 17.5%
TOTAL
Present
27636
Previous
26992
Charges
A units at 8.48p
Standing Charge
Subtotal
VAT at 17.5%
TOTAL
Present
4762
Previous
4012
Charges
A units at 6.47p
Standing Charge
Subtotal
VAT at 17.5%
TOTAL
Present
27297
58
Amount (£)
B
8.72
C
D
E
Amount (£)
B
12.40
C
D
E
Shock Electricity Co.
Meter Reading
PAGE
B
10.40
C
D
E
Fuse Electricity Co.
Meter Reading
5.
Amount (£)
Pegasys Electricity Co.
Meter Reading
4.
B
8.40
C
D
E
Sparks Electricity Co.
Meter Reading
3.
Amount (£)
Previous
27200
Charges
A units at 9.32p
Standing Charge
Subtotal
VAT at 17.5%
TOTAL
Amount (£)
B
8.63
C
D
E
Hire Purchase
1.
For each of the following , calculate :
i)
the total hire purchase price ;
ii)
the difference between the HP price and the cash price.
Computer
For Sale - £9000
Cash : £780
or Deposit of £50
+ 24 payments of
£35
HP Terms ......
Deposit £2000 + 36
payments of £245
£350
or Dep. £30
+ 9 payments
of £40
Cash £760 or
£70 down + 18
payments of £42
Cash ...... £6450
or Deposit of 10% +
48 monthly instalments
Camera
£675 or
£28 down !
+ 12 instalments
of £59.50
HP Terms
15% down plus
30 × £6.58
of .... £136
Cash £190
An electric guitar has a cash price of £340.
The hire purchase terms are ........
(a)
(b)
Calculate the total HP price.
How much would you save by paying cash ?
59
12% deposit + 24 monthly payments of £14.20
PAGE
2.
Scale Drawings
1.
For each pair of pictures below
i) State the enlargement scale factor (k).
ii) Calculate the length marked x .
5cm
7mm
21mm
10cm
8cm
12mm
x
x
12mm
8mm
5mm
x
14cm
x
16cm
48cm
6cm
24cm
12cm
8cm
x
16cm
x
24cm
2.
For each pair of pictures below
i) State the reduction scale factor (k).
ii) Calculate the length marked x.
76mm
19mm
10cm
5cm
60mm
x
x
18cm
30cm
28mm
x
PAGE
60
24cm
27cm
x
24mm
18mm
Calculation of Distance
** You need a calculator for this worksheet.
REMEMBER ......
D = S ×T
1.
Calculate the distance travelled for each
journey below. Remember the working and the units !
How far have you gone if you travel for .....
(b)
6 hours at a speed of 65 mph ?
(c)
2 12 hours at a speed of 87 km/h ?
(d)
40 minutes at a speed of 300 metres per minute ? (answer in kilometres)
(e)
30 minutes at a speed of 48 mph ?
A plane flies at a maximum speed of 460 km/h.
(a)
How far will it travel in 7 hours at maximum speed ?
(b)
The pilot wants to fly to Rio a distance of 5900 km.
Can he complete the journey within 13 hours ? Explain your answer.
A luxury cruiser has a maximum speed of 28 km/h.
(a)
How far can the boat sail in 3 14 hours at top speed ?
(b)
On a journey from one island to another the cruiser has to navigate
between two reefs, breaking the crossing into three stages.
Stage 1 : 2 hours at half-speed.
Stage 2 : 4 12 hours at full speed.
Stage 3 : 3 hours at one-quarter speed.
Calculate the total distance travelled on the journey.
61
3.
4 hours at a speed of 50 km/h ?
PAGE
2.
(a)
Working with Time and Speed
1.
2.
PAGE
62
3.
Express each of the following in hours only, rounding your answers to 2 decimal places
where necessary:
a)
4 hours
c)
28 minutes
b)
5 hours
48 minutes
40 minutes
d)
2 hours
6 minutes
e)
1 hour
f)
37 minutes
g)
12 hours
h)
7 hours
51 minutes
i)
2 hours
21 minutes
j)
22 hours
8 minutes
k)
9 hours
55 minutes
l)
17 hours
42 minutes
a)
A car travels a distance of 340 km in 4 hours 36 minutes.
Calculate its average speed.
b)
A plane travels a distance of 490 km in 1 hours 15 minutes.
Calculate its average speed.
c)
A car travels a distance of 58 miles in 48 minutes.
Calculate its average speed.
d)
A boat travels a distance of 86 km in 9 hours 6 minutes.
Calculate its average speed.
e)
A man runs a distance of 60 km in 5 hours 17 minutes.
Calculate his average speed.
f)
A boat travels a distance of 274 km in 1 days 3 hours 20 minutes.
Calculate its average speed.
a)
A man travels a distance of 340 km in his car. If the time taken for the journey
is 5 hours 8 minutes, calculate his average speed for the journey.
b)
A woman travels 54 miles to her work.
She leaves home at 0910 and arrives at her work at 1015.
Calculate the average speed for her journey.
c)
A lorry driver leaves Manchester at 1420 and arrives in London at 1735.
18 minutes
43 minutes
i)
How long did his journey take (in hours and minutes).
ii)
Calculate his average speed, to the nearest kilometre per hour, if the
distance he travelled was 130 miles.
Calculation of Time
3.
a)
2.4 hours
b)
3.6 hours
c)
1.35 hours
d)
8.33 hours
e)
9.21 hours
f)
4.75 hours
g)
12.5 hours
h)
3.18 hours
i)
6.123 hours
j)
18.457 hours
k)
2.32 hours
l)
5.64835 hours
a)
A car travels a distance of 340 km at an average speed of 64 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
b)
A plane travels a distance of 3123 km at an average speed of 278 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
c)
A car travels a distance of 58 miles at an average speed of 48 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
d)
A boat travels a distance of 186 km at an average speed of 18 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
e)
A man runs a distance of 32 km at an average speed of 20 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
f)
A boat travels a distance of 179 km at an average speed of 13 km/h.
Calculate the time taken for the journey, giving your answer to the nearest minute.
a)
A man travels a distance of 340 km in his car. If his average speed for the
journey is 54 km/h, calculate the time taken for his trip to the nearest minute.
b)
A woman travels 34 miles to her work.
Her average speed for the journey is 42 mph.
Calculate the time taken to reach her work.
c)
A plane can fly at an average speed of 462 km/h.
It flies from Glasgow to London , a distance of 598 km.
Calculate the time taken for the flight, giving your answer to the nearest minute.
d)
A train leaves Edinburgh at 0915.
It travels the 180 km to Dundee at an average speed of 78 km/h.
At what time will it arrive in Dundee ?
63
2.
Express each of the following times in hours and minutes, rounding to the nearest minute
where necessary :
PAGE
1.
Removing Brackets
1.
2.
3.
PAGE
64
4.
Remove the brackets.
For example
3( x + 4) = 3x + 12
a)
4( c + 2)
b)
2( e + 4)
c)
5( f + 6)
d)
3(t + 8)
e)
7( g + 3)
f)
9( w + 1)
g)
6( h + 6)
h)
8( p + 2)
i)
3(2 + y )
j)
7(1 + k )
k)
5(5 + z )
l)
4( 7 + u )
m)
9(1 + e)
n)
3(2 + w)
o)
8(12 + r )
p)
10(7 + m)
Remove the brackets :
a)
5(3c + 2)
b)
2( 2e + 4)
c)
6(4 f + 6)
d)
3(2t + 8)
e)
2(8 g + 3)
f)
6(4 w + 1)
g)
7(5h + 6)
h)
8(3 p + 2)
i)
3(2 + 2 y )
j)
7(1 + 9k )
k)
5(5 + 10 z )
l)
4( 7 + 7u )
m)
9(1 + 3e)
n)
3(2 + 6 w)
o)
3(12 + 2r )
p)
4( 7 + 5m )
Expand these brackets :
a)
c(c + 5)
b)
e ( e + 2)
c)
f ( f + 4)
d)
t (t + 7)
e)
e( g + 3)
f)
p ( w + 1)
g)
a ( h + 6)
h)
r ( p + 2)
i)
y (2 + y )
j)
c(1 + k )
k)
z (5 + z )
l)
h( 7 + u )
m)
e(1 + e)
n)
p (2 + w)
o)
x(12 + x)
p)
m( 7 + m)
Expand each of the following brackets :
a)
3(c − 2)
b)
5(e − 4)
c)
5( f + 4)
d)
2(t − 7)
e)
g ( g − 3)
f)
v( w + 1)
g)
h(h − 6)
h)
p(4 − p )
i)
3(2 − y )
j)
p(1 − k )
k)
a(b + c)
l)
x( x − y )
Simplifying Expressions
3.
a)
3x + 2 x
b)
4p + 2p + 6p
c)
8a − 3
d)
5m + 3m − 2m
e)
3v + v
f)
4y + 6y − y
g)
5a + 4b + 3a
h)
9f −4f +6
i)
8x + 3 + 2 x
j)
4c + 6 + 3
k)
5m + 3 + 4 m
l)
4y + 5 + 2y
m)
8 + 3x − 4
n)
7 d + 6 − 3d
o)
5 y + 6z + y
p)
6a + 5b − 2a
q)
12 + 7 x − 7
r)
5 g + 6h + 4 g
s)
5r + 8 − 2
t)
6 x + 3 + 3x
u)
8y − 4 + y
Remove the brackets and simplify where possible :
a)
3(c + 2) + 7
b)
2( e + 4) − 5
c)
6( f − 6) + 2 f
d)
4(t + 8) − 7
e)
7( g − 3) + 5 g
f)
8( w − 1) − 3w
g)
6( h + 2) + 9
h)
9( p + 3) + 5 p
i)
3(2 + f ) − 4
j)
5(1 + k ) + 4k
k)
5(5 + p ) − 2 z
l)
4(7 − u ) − 15
m)
6(1 + e) + e
n)
3(6 + w) + w
o)
8(11 + q ) − 4r
p)
19(7 + k ) − 60
Expand and simplify :
a)
6(3 g + 2) + 7
b)
2( 2e + 4) − 3
c)
7(4c + 5) − 20c
d)
3(2t + 8) − t
e)
3(8 f + 3) − 4
f)
3(4a + 1) − 4
g)
2(5k + 6) + 24k
h)
8(3h + 2) − 24h
i)
5(2 + 2t ) + 3t
j)
4(1 + 9u ) + 2u
k)
6(4 + 10 z ) − 16
l)
5(7 + 7u ) − 28
n)
3(2 + 6 w) + 4
o)
3(10 + 2d ) + 5d
p)
5(6 + 5 x) + x
m)
9(1 + 3e) + 7e
65
2.
Write each of the following in a shorter form :
PAGE
1.
Solving Equations
1.
2.
3.
Solve each of the following equations :
a)
5 x + 3 = 23
b)
7m − 11 = 24
c)
3 y + 14 = 17
d)
10e − 6 = 14
e)
8k − 11 = 21
f)
12 p + 1 = 37
a)
6 x = 3x + 27
b)
4a = 2a + 16
c)
7v = v + 30
d)
14w = 9w + 40
e)
12 x = 3x + 54
f)
10m = 3m + 56
a)
6a + 3 = 2a + 19
b)
7 x + 5 = 5 x + 17
c)
4 y − 2 = y + 19
d)
8 p − 1 = 2 p + 29
e)
11 f + 4 = 3 f + 52
f)
12 x − 4 = x + 51
Solve :
Solve :
PAGE
66
Solve each of the following equations :
PUPIL
(1)
7 x + 3 = 31
( 2)
4a − 1 = 11
(3)
7 y = 5 y + 12
( 4)
8d = 5d + 33
(5)
4 x + 3 = 2 x + 17
( 6)
6 x − 5 = x + 40
( 7)
10 p − 1 = 3 p + 20
(8)
2 x + 16 = 5 x − 2
Common Factors
3.
a)
2x + 6 = 2( x + )
b)
5a + 20 = 5 ( a + )
c)
4m − 24 = 4 ( − )
d)
3 f − 6 = 3( − )
e)
5x + 5 y = 5 ( + )
f)
6 p − 12q = 6 ( − )
g)
3d − 12e = 3 ( − )
h)
14 + 7 k = 7 ( + )
Factorise :
a)
2x + 8
b)
3m + 12
c)
4y − 4
d)
5p + 5
e)
8w − 16
f)
7u + 21
g)
10 z − 20
h)
6h + 24
i)
2d − 12
j)
5r + 5s
k)
3k − 3l
l)
7w + 7 x
m)
4u + 8v
n)
6r − 18s
o)
2e + 20 f
Factorise :
a)
4 x + 10
b)
6 g − 15
c)
4f +2
d)
8y − 4
e)
12e + 8
f)
6m + 21
g)
10a − 6
h)
9h + 12
i)
6r − 14
j)
4q + 18
k)
8 + 18 g
l)
12m − 9
67
2.
Copy and complete each of the following :
PAGE
1.
Mean, Median & Mode
** You need a calculator for this worksheet.
1.
2.
68
i) the range ;
ii) the average (mean).
(a)
22
13
12
14
12
31
11
22
16
(b)
8
5
6
6
3
9
7
4
9
(c)
18
43
76
45
87
55
(d)
8⋅3
2⋅6
5⋅2
12 ⋅ 8
1⋅ 7
17 ⋅ 5
1
5
9
9⋅3
For each set of numbers below :
i) arrange the numbers in order from lowest to highest;
ii) write down the mode and the median number.
(a)
6
8
6
7
8
5
9
(b)
20
32
70
76
21
70
18
(c)
12
9
12
13
8
4
4
(d)
45
36
22
13
12
12
1
6
12
1
6
12
7
44
22
17
33
22
4
3.
For each set of numbers in question 2 above, calculate the mean value rounding your answers to
1 decimal place where necessary.
4.
Find the median of each set of numbers below :
5.
PAGE
For each set of numbers below, calculate :
(a)
23
12
13
24
36
42
24
48
(b)
8
22
3
25
38
24
(c)
34
56
22
68
34
46
76
78
(d)
2
3
2
1
3
8
6
9
14
15
At a golf tournament the 23 golfers taking part posted the following first round scores ......
72
74
76
72
69
76
72
77
70
71
72
86
70
68
(a)
(b)
(c)
State the range of the scores.
Calculate the mean first round score.
Find the median and mode scores.
68
76
84
73
68
73
76
72
64
Frequency Tables
** You need a calculator for this worksheet.
Data has been collected from various sources and the following frequency tables constructed ......
C
A
Number of Close Friends
B
Tomato Plant Heights
Family Size
height (cm) frequency
6
7
8
9
10
11
No. of
Friends
Family Size frequency
frequency
4
6
6
7
11
5
1
2
3
4
5
6
7
frequency
1
2
3
4
5
6
2
2
5
13
10
3
1
3
7
5
9
4
2
E
D
Running 400 metres
time (sec)
frequency
65
66
67
68
69
70
71
5
4
6
9
12
12
6
Shoe Sizes
Size
frequency
4
5
6
7
8
9
10
11
12
1
3
7
15
11
8
5
2
3
Draw a Bar Graph (or Histogram).
(b)
Calculate the mean value.
(c)
State the mode and determine the median .
PAGE
(a)
69
For each of the frequency tables above .......
Pythagoras' Theorem
* You need a calculator for this worksheet.
1.
Calculate the length of the side marked x in each triangle below, rounding your answers to 1d.p.
where necessary :
x
8
5
(b)
(a)
(c)
x
17
12
4
x
14
14 ⋅ 5
(d)
6⋅4
6
(f)
(e)
x
x
9
32
x
16
4⋅3
(g)
4
x
42
(i)
(h)
1⋅ 7
x
x
0⋅8
2.
40
Calculate the height of each tree, rounding your answers to 1 – decimal place.
15 m
h
h
27 m
56 m
h
PAGE
70
13 m
23 m
48 m
Distance Between Two Points
** You need a calculator for this worksheet.
1.
Calculate the distance between each pair of points below :
Round your answers to 1-decimal place where necessary.
Plot each pair of points on a coordinate diagram and join them with a line.
Construct a right angled triangle and use Pythagoras' Theorem to calculate
the distance between the points.
2.
(a)
P(2,1) and
Q(5,3)
(b)
A(1,3) and
B(6,5)
(c)
E(3,3) and
F(5,8)
(d)
R(1,6) and
S(8,1)
(e)
M(0,2) and
N(5,5)
(f)
G(7,2) and
H(2,10)
(g)
K(9,0) and
L(2,7)
(h)
U(1,1) and
V(6,13)
Calculate the distance between each pair of points below :
Round your answers to 1-decimal place where necessary.
A(-3,4)
(c)
and
B(4,1)
(b)
C(4,-6) and
D(-2,-2)
E(3,4) and
F(-2,-6)
(d)
G(-2,1)
H(7,-5)
(e)
I(0,-4)
J(-2,8)
(f)
K(-4,-3)
(a)
Triangle PQR has corner points P(1,5) , Q(3,8) and R(6,1).
and
and
and
L(6,1)
Calculate the lengths of the three sides PQ , QR and RP.
Repeat part (a) for the triangle with corners P(-3,7) , Q(-1,-4) and R(5,2).
71
(b)
PAGE
3.
(a)
Area and Perimeter (1)
Calculate the area and the perimeter of each rectangle below :
1.
8cm
5m
(c)
9cm
(b)
(a)
2m
5cm
7cm
3.8cm
6mm
2.
1.8m
(e)
14mm
(d)
(f)
3.2cm
5m
Calculate the shaded area in each diagram below :
0.8m
9cm
4cm
6mm
2cm
15mm
5cm
2.4m
2m
13mm
(a)
(b)
(c)
22mm
3.
0.4m
Calculate the area of each composite shape below :
(a)
(b)
6cm
3m
(c)
2.4m
24cm
5cm
4cm
3cm
14cm
9cm
2.4m
14cm
0.8m
PAGE
72
4.
A carpet fitter is called out to fit a carpet in a rectangular room measuring
2 ⋅ 4 m by 6 m .
Calculate (a) The area of carpet needed for the room.
(b) The length of fixing strip to go round the edge of the carpet.
Area and Perimeter (2)
** You need a calculator for this worksheet
1.
Calculate the cost of re-glazing each of the broken windows below, given that glass costs
£13.50 per square metre to replace.
(a)
(b)
2.4m
1.8m
(c)
0.9m
1.4m
0.8m
2.
1.2m
Carpet costs £14.80 per square metre. Calculate how much it would cost to carpet each of the
rooms shown below.
(b)
(a)
6m
8.6m
3m
5m
3.
Fixing strip for carpets costs £1.60 per metre. Calculate the cost of the fixing strip to go round the
edge of each carpet above.
4.
Office carpet costs £23.50 per square metre.
5m
3.8m
3.8m
The plan of a small office complex is shown
opposite.
Calculate the total cost of carpeting the
three rooms.
Room 1
Room 2
4m
Room 3
6m
13m
A man decides to paint the four inside walls of his
garage and to re-concrete the floor.
The paint costs £1.20 per square metre to apply and the
concrete £6.80 per square metre.
Calculate the total cost of his DIY.
6m
3m
73
3m
PAGE
5.
Area and Perimeter (3)
** You need a calculator for this worksheet
Remember :
C = π d or
Area and Circumference of a Circle
1.
2.
(a) diameter 10cm
(b) diameter 8mm
(c) diameter 1.2m
(d) d = 7cm
(e) d = 25cm
(f) radius 5cm
(g) radius 11mm
(h) radius 0.9m
(i) r = 12cm
(j) r = 1.8m
Calculate the circumference of each circle below.
(b) r =7mm
(c) r = 12cm
4.
Calculate the area of each circle in question 2.
5.
Calculate the area of each semi-circle below.
20cm
7.
74
3cm
(e) r = 17cm
29cm
15cm
Calculate the circumference of this circle
(b)
Calculate its area.
A circle has a radius of 28mm.
(a)
8.
(d) r = 0.9m
A circle has a diameter of 36cm.
(a)
PAGE
28cm
6m
Calculate the area of the circles with the following radii.
(a) r = 4cm
6.
( π = 3 ⋅ 14 )
Calculate the circumference of the circle with .......
16cm
3.
A = π r2
2π r
Calculate its area
(b)
Calculate its circumference.
Calculate the perimeter of each semi-circle in question 5.
(f) r = 32mm
Area and Perimeter (4)
1.
(More Practice)
Calculate the circumference and the area of each
circle below.
(a)
(b)
(c)
32mm
.
4.8cm
(d)
2.
3.
.
1.5cm
19cm
(e)
2.4m
(a)
Calculate the area of the circle with radius 42 cm.
(b)
Calculate the circumference of the circle with diameter 6.2 m.
(c)
Calculate the area of the circle with diameter 16.2 cm.
(d)
Calculate the circumference of a wheel of radius 40 cm.
The diagram shows a rectangular steel plate with five holes,
each with a radius of 4cm, drilled through it.
72cm
Calculate the shaded area.
36cm
4.
The diagram shows a rectangular steel plate with
four holes, of radius 6cm, drilled through it.
70cm
Calculate the shaded area.
50cm
(a)
Calculate the circumference of each wheel.
(b)
How many turns will the small wheel make for one turn of the large wheel ?
75
The "Penny-Farthing" bicycle shown opposite was all the rage
when it first appeared. The large front wheel has a radius of 98cm
and the small back wheel a radius of 14cm.
PAGE
5.
Volume of a Cuboid
Volume of a cube ..... V = l
(l × l × l )
Volume of a cuboid ... V = lbh ( l × b × h )
3
** You need a calculator for this worksheet.
1.
Calculate the volume of each cuboid below :
(a)
(b)
(c)
2cm
3cm
8cm
6m
3cm
10cm
4m
8cm
5m
(g)
12mm
(d)
(e)
2cm
2cm
(f)
6cm
25mm
22cm
7m
40mm
1cm
7m
14cm
7m
2.
Calculate the volumes of the cuboids measuring :
(a)
(c)
(e)
3.
12cm by 8cm by 9cm
50cm by 20cm by 5cm
11mm by 9mm by 2mm
(b)
(d)
(f)
18mm by 12mm by 3mm
15m by 7m by 8m
4 ⋅ 3 cm by 2 ⋅ 2 cm by 10cm
(c)
14cm
Calculate the volumes of the cubes of side :
(a)
4.
6cm
(b)
4mm
5.
3000 cm3
1460 cm3
2400 cm3
480 cm3
(b)
(f)
(c)
(g)
12600 cm3
320000 cm3
1 litre = 1000 cm3
600 cm3
2565 cm3
(d)
(h)
Calculate the volume of water in each fish tank below, giving your answer in litres :
(b)
(a)
(c)
9cm
8cm
76
23mm
Convert each of the following volumes in cubic centimetres into litres :
(a)
(e)
PAGE
(d)
10cm
30cm
14cm
6cm
12cm
40cm
25cm
Volume of a Cylinder
Volume of a Cylinder ..... V = π r h
2
** You need a calculator for this worksheet
1.
Calculate the volume of each cylinder below :
2m
4cm
(a)
(b)
5cm
(c)
(d)
7cm
18mm
5m
10cm
11mm
2.
Calculate the volume of each cylinder below :
6cm
12mm
30mm
5cm
14cm
1.8m
7m
5cm
3.
The drinks can opposite is cylindrical in shape.
Calculate its volume (in ml) if it has a diameter of 6cm and a
length of 11.68cm . Give your answer to the nearest millilitre.
4.
Six cola-cans each with a diameter of 6.8cm and a height of
9.183cm are sold together in an economy pack.
Calculate the total volume of cola in the six-pack.
Answer to the nearest millilitre.
5.
A container for holding coffee is cylindrical in shape.
Given that it has a diameter of 8cm and a height of 15cm,
calculate its volume in cubic centimetres.
Calculate the capacity of the drum to the nearest litre.
77
An oil drum has a diameter of 66cm and a height of 105.3cm.
PAGE
6.
Income Tax
Allowances
Personal allowance £3500
Married allowance £1500
any other allowances will be
given within the question
The information opposite can be
used when needed in this worksheet.
1.
2.
Tax Rate
Taxable Income
Lower rate 20%
Basic rate 25%
the first £3000
£3001 - £24000
Copy and complete the table below :
Q.
Income
Allowances
(a)
£18000
£3500
(b)
£26000
£5500
(c)
£21500
£5000
Taxable Income
Copy and complete each of the following :
(a)
Taxable Income
£16000
(b)
Tax ⇒ £3000 at 20% =
£13000 at 25% =
Total Tax
(c)
Taxable Income
Total Tax
(d)
Tax ⇒ £3000 at 20% =
£20000 at 25% =
Total Tax
£8000
Tax ⇒ £3000 at 20% =
£5000 at 25% =
=
£23000
Taxable Income
=
Taxable Income
£32000
Tax ⇒ £3000 at 20% =
£21000 at 25% =
£8000 at 40% =
Total Tax
3.
and
=
=
John Henderson is a married man with a gross income of £25600 p.a. . He claims the personal
married allowances and has a further £800 in mortgage allowance. Calculate :
(a) his total allowances.
4.
78
PAGE
(c) the income tax he pays.
Ian McStay is a farmer with an income of £26000. He is married and claims
the married allowance + additional allowances totalling £4500. Calculate :
(a) his total allowances.
5.
(b) his taxable income.
(b) his taxable income.
(c) the income tax he has to pay.
Susan Moffat is a computer analyst with an income of £43000. She is single and
claims allowances of £3000 over and above her personal allowance. Calculate :
(a) her total allowances.
(b) her taxable income. (c) the income tax she pays.
Holiday Travel
Use the currency table opposite to answer the questions on this worksheet
unless instructed otherwise.
Exchange Rate
equivalent to £1 .....
1.
Change each of the sums of money into the currency indicated :
(a)
(c)
(e)
(g)
(i)
2.
4.
£550
£80
£100
£285
£2400
(escudos)
(pesetas)
(marks)
(drachmas)
(marks)
francs
dollars
lire
escudos
drachmas
pesetas
Swiss fr.
marks
Change each of these sums of money into pounds sterling :
(a)
(c)
(e)
(g)
(i)
3.
£240 (francs)
(b)
£60 (lire)
(d)
£1200 (dollars)
(f)
£28
(francs)
(h)
£990 (swiss francs) (j)
9.65
1.55
2700
280
340
205
2.20
2.45
579 francs
13500 lire
651 dollars
1200 francs
84 swiss francs
(b)
(d)
(f)
(h)
(j)
12040 escudos
79950 pesetas
490 marks
16800 drachmas
56 marks
Mr and Mrs Wilson are taking their caravan to France.
(a)
They changed their £1400 spending money into francs. How many francs did they get?
(b)
When they returned home they had 1737 francs left. How much did they receive
in pounds sterling for their francs?
John and Graeme are off on a camping holiday to Italy. John has £750 spending
money and Graeme has £680.
(a) They changed their spending money into lire. How many lire did each receive?
A salesman travelling from Spain to Switzerland notices the same motorbike for sale, first in
Spain, and then in Switzerland.
In Spain it is priced at 194 750 pesetas and in Switzerland 2156 francs.
In which country is the motorbike the cheapest and by how much?
6.
In anticipation of going ashore, a tourist on board a ferry changed £65 into francs at the rate
of 9.80 francs to the pound. She did not spend any money. When she returned to the
ferry she changed her money back at the rate of 9.98 francs to the pound.
How much did she lose on the deal?
7.
In France petrol costs 7.2 francs per litre.
(a)
How much is this in British money if the exchange rate is 9.6 francs to the pound.
(b)
How much is this a gallon, in British money, if 1litre =
7
32
of a gallon ?
PAGE
5.
79
(b) John returned home with 56700 lire and Graeme with 22950 lire.
How much did each boy spend in pounds?
Formulae & Sequences
All working must be shown. Formulae should be written out and substitutions made !
1.
2.
3.
F = 5a + 4 .
A formula is given as
(a)
Find the value of F when .... i) a = 3
(b)
What value of a would make F equal to 54 ?
A formula is defined as
ii )
a=7
iii ) a =
1
2
.
E = 3f + g.
Find the value of E when ... (a)
f=4
and
g=6.
(b)
f=6
(c)
f=2
and
g=5.
(d)
f=
(e)
f = 12
(f)
f=
and
g=7.
and
g=1.
1
3
and
g=4.
1
2
and
g = 2 12 .
A formula is defined as P = r s − 2k .
(a) Find P when .... r = 3 , s = 6 and k = 4.
(b) Find P when .... r = 2 , s = 4 and k =
3.
(c) Find P when .... r =8 , s = 2 and k = 5.
4.
(d) Find P when .... r = 3 , s = 8 and k = 12.
The first formula of motion can be defined as follows ..... a =
v − u
t
,
where a is the acceleration, v is the final velocity, u is the initial velocity and t is the time.
(a) Find a when v = 20 , u = 12 and t = 2.
5.
The second formula of motion takes the form ..... s = u t +
moved.
(a) Find s when u = 3, t = 4 and a = 2.
6.
PAGE
80
7.
(b) Find a when v = 400 , u = 8 and t = 16.
1
2
a t 2 , where s is the distance
(b) Find s when u = 2, t = 7 and a = 6.
Find a formula connecting the variables in each table below :
(a)
a
F
1
4
2
7
3
10
4
13
5
16
(b)
g
P
1
3
2
8
3
13
4
18
5
23
(c)
x
E
5
13
6
15
7
17
8
19
9
21
(d)
h
Q
3
19
4
27
5
35
6
43
7
51
Find a formula for the nth term of each of the following sequences and hence find the 20th term
of each sequence.
(a)
3 , 7 , 11 , 15 , ........
(b)
14 , 26 , 38 , 50 , .......
(c)
2 , 8 , 14 , 20 , ........
Probability – Multiple Outcomes (Tree Diagrams)
1.
2.
3.
4.
A coin is tossed twice.
(a)
Draw a tree diagram to show all the possible outcomes.
(b)
How many possible outcomes are there?
(c)
Calculate
i) P(two tails)
ii) P(1 head and 1 tail).
A coin is tossed three times.
(a)
Draw a tree diagram to show all the possible outcomes.
(b)
How many possible outcomes are there?
(c)
Calculate
i) P(three heads)
ii) P(two heads and one tail).
A three-sided spinner is coloured Green, Red and Yellow.
(a)
If it is spun twice, draw a tree diagram to show all the possible outcomes.
(b)
How many possible outcomes are there?
(c)
Calculate i) P(two greens) ii) P(two colours the same) iii) P(two different colours).
A three-sided spinner is coloured Green, Red and Yellow.
If it is spun three times ............
How many possible outcomes are there?
(b)
Calculate i) P(three greens) ii) P(three colours the same) .
Four young boys take part in a lucky dip. They will win one of the two toy cars shown below.
They have an equal chance of winning either one.
Jeep
Racing car
(a) Draw a tree diagram to show all the possible outcomes. How many possible outcomes are
there?
(b) What is the probability that all the boys win the racing car?
(d) Calculate the probability that one wins the jeep and the other three win the racing car.
81
(c) What is the probability that two win the racing car and two win the jeep?
PAGE
5.
(a)
Gradients and Straight Lines (1)
1.
Calculate the gradient of each ladder below :
(b)
(a)
(c)
2m
3m
8m
4m
10 m
2m
(d)
(f)
(e)
12 m
1m
6m
8m
8m
2m
2.
Calculate the gradient of each line below, leaving your answer as a fraction in its simplest
form where necessary.
(b)
(c)
(a)
(d)
(e)
(g)
(f)
(h)
(j)
PAGE
82
(i)
Gradients and Straight Lines (2)
A straight line has as its equation y = 2 x .
(a)
Copy and complete the table for this line.
x
y
(b)
(b)
4
5
10
Copy and complete the table for this line.
0
1
3
2
3
4
5
Plot the points from the table on a coordinate diagram and draw the line through them.
A straight line has as its equation y = 12 x .
(a)
Copy and complete the table for this line.
x
y
(b)
0
2
1
4
6
8
4
10
Plot the points from the table on a coordinate diagram and draw the line through them.
A straight line has as its equation y = 13 x .
(a)
Copy and complete the table for this line.
x
y
(b)
5.
3
Plot the points from the table on a coordinate diagram and draw the line through them.
x
y
4.
2
A straight line has as its equation y = 3 x .
(a)
3.
1
2
0
0
3
6
9
3
12
Plot the points from the table on a coordinate diagram and draw the line through them.
A straight line has as its equation y = x .
(a)
Copy and complete the table for this line.
x
y
(b)
1
1
2
3
4
5
5
6
Plot the points from the table on a coordinate diagram and draw the line through them.
83
2.
0
PAGE
1.
Gradients and Straight Lines (3)
1.
A straight line has as its equation y = x + 2 .
(a)
Copy and complete the table for this line.
x
y
(b)
2.
(b)
(b)
5
0
1
3
2
3
4
9
5
Plot the points from the table on a coordinate diagram and draw the line through them.
Copy and complete the table for this line.
0
4
2
4
6
8
8
10
Plot the points from the table on a coordinate diagram and draw the line through them.
A straight line has as its equation y = 14 x + 5 .
(a)
Copy and complete the table for this line.
x
y
(b)
0
4
6
8
12
Plot the points from the table on a coordinate diagram and draw the line through them.
A straight line has as its equation y = 3x − 2 .
(a)
Copy and complete the table for this line.
x
y
84
4
Copy and complete the table for this line.
x
y
PAGE
3
5
A straight line has as its equation y = 12 x + 4 .
(a)
5.
2
Plot the points from the table on a coordinate diagram and draw the line through them.
x
y
4.
1
A straight line has as its equation y = 2 x + 1 .
(a)
3.
0
2
(b)
1
1
2
3
4
10
5
Plot the points from the table on a coordinate diagram and draw the line through them.
Trigonometry (1)
Remember
SOH CAH TOA
1.
Calculate the angle marked x in each triangle below :
11
(a)
(b)
3
(c)
8
10
x
x
17
x
5
(d)
(e)
(f)
14
9
4
21
19
x
x
12
x
Calculate the angle marked x in each triangle below :
4
(a)
(b)
(c)
x
16
9
14
x
3
x
15
(d)
(e)
12
(f)
34
5
x
7
x
x
27
5
4
5.6
(h)
(i)
8.9
x
12.8
x
7.3
x
3.2
85
(g)
PAGE
2.
Trigonometry (2)
Remember
1.
SOH CAH TOA
Calculate the length of the side marked x in each triangle below :
(a)
(b)
16
(c)
x
x
7
32o
24o
12
45
o
x
(d)
(e)
(f)
14
x
20
x
78o
50o
48
o
x
11
2.
Calculate the side marked x in each triangle below :
(a)
15
(b)
x
(c)
9
x
25o
32o
x
42o
3
(d)
(e)
(f)
20
26
PAGE
86
x
x
35o
20o
x
80
7
o
Trigonometry (3) - Angles & Sides
Calculate the value of x in each triangle below :
8
3
5
xo
xo
11
(b)
(a)
(e)
x
8
(d)
28o
(c)
9
26o
x
19
x
75o
(h)
2.7
16
xo
4.5
xo
(f)
(g)
11
27
(i)
x
7
(j)
58
xo
o
11
37
xo
(l)
x
4.7
20o
12
17
xo
(k)
3.5
72o
xo
(m)
(n)
x
87
27
8
PAGE
12
Trigonometry (4)
1.
To test the stability of a bus a tilting platform is used.
It is known that a bus will topple if the angle between the platform and the ground is greater that
20o.
Which of the buses below would topple?
Each answer must be accompanied with the appropriate working.
(a)
(b)
(c)
4m
2m
8m
4.8 m
14 m
10 m
(e)
(f)
(h)
3.6 m
7.45 m
2m
6.2 m
(g)
2.7 m
2.
(d)
2.6 m
3m
9.8 m
6.8 m
8.43 m
To comply with building regulations a roof must have
an angle of between 22o and 28o to the
horizontal (see diagram opposite).
x
x must lie between 22o and 28o
Which of the roofs below comply, and which do not comply,
with building regulations?
(a)
(b)
1.8 m
2.1 m
4m
6m
(c)
(d)
1.4 m
4.5 m
3.6 m
8m
(e)
(f)
2.2 m
1.5 m
7.4 m
PAGE
88
6.2 m
(g)
1.9 m
6.8 m
(h)
2.4 m
12.3 m
Fractions, Decimals and Percentages (1)
(a)
1
3
(d )
3
4
(g)
2
3
( j)
5
6
( m)
11
20
5.
( e)
5
8
of £15.96
( h)
9
10
of £5.10
(k )
3
8
of £2540
( n)
9
16
26% of £90
42% of 60 kg
19% of £64
8% of 4500 g
78% of £1500
(b)
(e )
( h)
(k )
( n)
54% of
17% of
5% of
80% of
4% of
of 48 cm
(c )
1
7
of £36.40
(f)
7
9
of 58 ⋅ 5 grams
of 45 kg
(i )
3
7
of £10.92
of 984 mm
(l )
9
12
of £1.08
of 480 tonnes
( o)
5
17
of 25 ⋅ 5 kg
(c )
(f)
(i )
(l )
(o )
13% of £45
21% of 85 cm
65% of £880
94% of £360
7% of 1200 tonnes
of 65 kg
of £136
300 g
£10
£340
£250
£12
Calculate each of the following rounding your answers to the nearest penny.
(a )
(d )
(g)
( j)
(m)
4.
1
5
Calculate :
(a )
(d )
(g)
( j)
(m)
3.
(b)
of £96
36%
47%
12%
81%
71%
of
of
of
of
of
£13.20
89 p
£18.30
£3.45
£1.53
(b)
(e)
( h)
(k )
( n)
24% of
57% of
4% of
9% of
3% of
£12.71
£10.43
£341.20
£2.57
£12.08
(c )
(f)
(i )
(l )
(o )
18% of
41% of
5% of
34% of
57% of
£6.35
51 p
£834.65
88 p
97 p
Change each of the following fractions to percentages.
Round your answer to the nearest percent when necessary.
(a )
4
5
(b)
3
4
(c )
7
25
(d )
7
10
(e)
17
100
(f)
19
20
(g)
5
9
( h)
3
11
(i )
18
23
( j)
5
12
(k )
4
32
(l )
8
13
(m)
2
15
( n)
3
7
(o )
6
31
( p)
38
365
(q)
48
95
(r )
6
29
John's schedule marks are shown in the table below :
Subject
Mark
Maths
45 out of 60
English
64 out of 72
Tech
40 out of 65
Science
38 out of 55
Art
75 out of 90
History
27 out of 40
French
63 out of 95
%
(a)
Copy and complete the table by calculating John's "percentage mark" for each subject.
Round each answer to the nearest percent where necessary.
(b)
Which was John's best subject ?
(c)
Which was his worst ?
89
2.
Calculate :
PAGE
1.
Fractions, Decimals and Percentages (2)
** You need a calculator for this worksheet.
1.
Increase each of the following by 15%.
(a)
£250
(b)
160kg
(c)
25cm
(d)
£36
(e)
2100g
(f)
210oC
(g)
£8
(h)
£3500
2.
Decrease each of the amounts in Q1 by 20%.
3.
The nine workers in a small factory are given different percentage wage rises dependant upon
their length of service. The table below represents their weekly wages.
Copy and complete the table .......
Name
4.
(e)
90
PAGE
£230
4%
Steven Higgins
£168
6%
Susan Marshal
£210
4%
Stewart Aitken
£145
2%
Pamela Grant
£360
Neil McShane
£225
3.5%
6%
James Mackie
£235
8%
Lorna Graham
£210
Pat Lavery
£468
4.5%
5%
(b)
Increase
£9.20
New Wage
£239.20
i) the fraction shaded; ii) the percentage shaded .
(c)
(d)
(f)
(g)
Calculate the percentage of vowels in each word below.
(a)
6.
% Increase
John Hughes
For each diagram below, write down
(a)
5.
Old Wage
(b)
(c)
(a)
In a class of thirty pupils, 6 were absent. Calculate the percentage absent.
(b)
A machine produces 300 heating elements in a morning. Six are found to be defective.
What percentage of the elements are defective ?
(c)
A small farm has 160 sheep. During a severe storm the farmer loses 8 sheep.
What percentage of the sheep got lost ?
Fractions, Decimals and Percentages (3)
1.
VAT is charged at 17 12 % . Calculate the VAT on each item below.
(a)
A stereo costing £230
(b)
A fridge costing £148
(c)
A cooker costing £456
(d)
A watch costing £68
(e)
A computer costing £650
(f)
A gold ring costing £134
2.
Find the total cost of each item in Q1 after the VAT has been added.
3.
A man places £2300 in a savings account which has an annual interest rate of 4%.
4.
5.
(a)
How much interest will he earn in the first year ?
(b)
Assuming he does not touch his money, how much does he now have in the bank
at the beginning of year two ?
(c)
Hence calculate the interest he will get at the end of year two.
A woman places £22100 in a Post Office savings account which has an annual interest rate of 5%.
(a)
How much interest will she earn in the first year ?
(b)
Assuming she does not touch her money, how much does she now have in the bank
at the beginning of year two ?
(c)
Hence calculate the interest she will get at the end of year two.
Steven places £800 in a Building Society at an annual interest rate of 3%.
How much will he have in his account after two years ?
6.
Susan invests £800 in a Building Society at an annual interest rate of 6%.
How much will she have in her account after two years ?
7.
Mr Banks places £1300 in a savings account which has an annual interest rate of 2%.
How much will he have in his account after three years ?
How much will she have in her account after three years ?
91
Miss Anderson places £5600 in a Building Society at an annual interest rate of 7%.
PAGE
8.
Solving Equations (1)
1.
2.
PAGE
92
3.
Solve each of the following equations :
(a)
5 x + 8 = 18
(b)
2t + 3 = 11
(c )
4m + 1 = 13
(d )
3 y + 4 = 22
(e)
6a + 5 = 17
(f)
7 d + 3 = 31
(g)
6h + 1 = 25
(h)
2 p + 6 = 20
(i )
3 x + 2 = 26
( j)
8a + 5 = 21
(k )
9 x + 2 = 38
(l )
10 y + 7 = 37
Solve each of the following equations :
(a)
5 x = 3 x + 12
(b)
6t = 3t + 15
(c)
4m = 2m + 16
(d )
7 y = 3 y + 20
(e)
6a = a + 30
(f)
7 d = 2d + 35
(g)
6h = 2h + 24
( h)
9 p = 3 p + 18
(i )
8 x = 5 x + 27
( j)
12a = 4a + 16
(k )
9 x = 6 x + 33
(l )
10 y = y + 36
Solve each of the following equations :
(a )
5 x + 2 = 3x + 8
(b)
6t + 3 = 3t + 15
(c)
8m + 1 = 3m + 21
(d )
7 y + 4 = 5 y + 12
( e)
6a + 5 = 2a + 29
(f)
7 d + 1 = 2d + 16
(g)
6h + 4 = 4h + 30
( h)
10 p + 6 = 4 p + 24
(i )
8 x + 2 = x + 16
( j)
8a + 5 = a + 26
(k )
9 x + 2 = 6 x + 32
(l )
10 y + 7 = 6 y + 35
(m)
12 p + 3 = 2 p + 43
( n)
11h + 1 = 2h + 19
( o)
15 x + 3 = 3 x + 51
( p)
16 y + 7 = 8 y + 23
(q)
7 d + 12 = 4d + 36
(r )
11a + 4 = 5a + 40
(s)
8m + 9 = m + 51
(t )
9 x + 4 = 7 x + 40
(u )
6 x + 8 = 4 x + 11
Solving Equations (2)
3.
(a )
4 x − 4 = 16
(b)
2t − 3 = 9
(c )
4m − 5 = 15
(d )
6y − 7 = 5
(e)
8a − 1 = 31
(f)
7 d − 2 = 19
(g)
6h − 4 = 32
(h)
2 p − 1 = 29
(i )
7u − 4 = 24
( j)
8 y + 7 = 23
(k )
6 x + 1 = 43
(l )
12 y + 3 = 27
Solve each of the following equations :
(a)
5 x + 12 = 7 x
(b)
6t + 18 = 9t
(c )
4m + 20 = 9m
(d )
28 + 5 y = 12 y
(e)
6 + 3y = 5y
(f)
70 + 4 x = 14 x
(g)
6h + 42 = 12h
( h)
8r + 15 = 11r
(i )
5 x + 45 = 14 x
( j)
10a = 2a + 16
(k )
7 x = 5 x + 30
(l )
12 y = y + 55
Solve each of the following equations :
(a )
5 x − 2 = 3x + 8
(b)
7t − 3 = 3t + 17
(c )
8m − 1 = 5m + 20
(d )
7 m − 1 = 5m + 15
(e )
8k − 5 = 3k + 35
(f)
6d − 1 = 2d + 19
(g)
6h − 7 = 4h + 21
( h)
9y − 8 = 4y + 2
(i )
12 x − 1 = 3x + 26
( j)
7 x − 2 = x + 34
(k )
11x − 2 = 7 x + 34
(l )
3y − 7 = y + 1
( m)
2p + 3 = p + 4
( n)
7 h + 6 = 2h + 21
(o )
5 x + 9 = 3 x + 33
( p)
14 y − 7 = 8 y + 23
(q)
7 k + 2 = 4k + 38
(r )
10a − 4 = 6a + 40
(s)
8a + 12 = a + 33
(t )
5 x − 4 = 3 x + 38
(u )
6x + 8 = 4x + 4 ?
93
2.
Solve each of the following equations :
PAGE
1.
Solving Equations (3)
1.
2.
PAGE
94
3.
Solve each of the following equations :
a)
4(c + 2) = 20
b)
2(e + 4) = 12
c)
5( f + 6) = 40
d)
6( g + 3) = 24
e)
7( w + 1) = 35
f)
2(h + 6) = 30
g)
5(a − 1) = 10
h)
4( p − 2) = 16
i)
6( x − 3) = 12
j)
3(2 + y ) = 21
k)
5(1 + k ) = 25
l)
8(5 + z ) = 48
m)
9(1 + e) = 45
n)
6(2 + w) = 24
o)
10(2 + r ) = 80
Solve each of the following equations :
a)
4(3c + 1) = 28
b)
2(2e + 4) = 12
c)
6(4 f + 2) = 60
d)
2(2t + 4) = 20
e)
4(5 g + 3) = 52
f)
7(4w + 1) = 35
g)
3(5h − 1) = 27
h)
2(3 p − 1) = 28
i)
4(2 y − 3) = 12
j)
3(1 + 2k ) = 21
k)
2(5 + 3z ) = 22
l)
2(7 + 7u ) = 28
m)
2(1 + 3e) = 20
n)
3(2 + 6w) = 24
o)
4(10 + 2r ) = 80
Solve each of the following equations :
a)
3(a + 2) = a + 12
b)
4( x + 3) = 2 x + 30
c)
5(m + 3) = 2m + 24
d)
7(d + 1) = 3d + 15
e)
8(h + 3) = 3h + 29
f)
6( y + 1) = 2 y + 34
g)
4(a + 1) = 2(a + 8)
h)
7( x + 2) = 4( x + 5)
i)
8(a + 2) = 4(a + 28)
j)
5(d − 1) = 3d + 7
k)
6( x − 2) = 3x + 3
l)
7(u − 1) = 3u + 21
m)
8( w − 1) = 6( w + 4)
n)
7( x − 2) = 4( x + 1)
o)
4(2 x − 3) = 2( x + 15)
Equations (Extension Examples)
3.
(a)
5 x − 4 = 26
(b)
7t + 3 = 3t + 15
(c )
7m − 2 = m + 16
(d )
8v − 24 = 5v
(e)
9d − 15 = 6d
(f)
9y +1 = 2y +1
(g)
7x −1 = x + 5
( h)
12m + 3 = 11m + 6
(i )
5v + 3 = 2v + 3
Solve each of the following equations :
(a )
3 x = 12 − x
(b)
5m = 24 − 3m
(c )
y = 21 − 2 y
(d )
5t = 42 − t
(e)
2a = 20 − 2a
(f)
6 x = 40 − 4 x
(g)
2 y + 1 = 21 − 3 y
( h)
p − 3 = 21 − 5 p
(i )
8r − 5 = 45 − 2r
( j)
6 + x = 12 − 2 x
(k )
14 + 4a = 26 − 2a
(l )
2 + 6d = 24 − 5d
( m)
1 + 3c = 13 − c
( n)
9 + x = 27 − 5 x
(o )
6 + 4 x = −2 x + 12
( p)
3 x + 5 = −4 x + 19
(q)
5v − 1 = −3v + 15
(r )
8 + 7 x = −2 x + 35
Solve each of the following equations :
(a)
5 x − 7 = −2
(b)
3x − 12 = −3
(c)
7 y − 15 = −1
(d )
8v − 8 = 6v − 2
(e)
4h − 10 = 2h − 4
(f)
6a − 16 = a − 6
(g)
3x − 11 = x − 5
(h)
5m − 18 = m − 6
(i )
8e − 30 = 2e − 6
( j)
2 x − 12 = −3x − 2
(k )
5 y − 20 = −2 y − 6
(l )
3a − 9 = −2a − 4
(m)
7 x − 13 = − x − 5
(n)
4k − 24 = −2k − 12
(o)
3c − 18 = −c − 2
95
2.
Solve each of the following equations (a warm up) :
PAGE
1.
Statistics
1.
-
Stem-and-Leaf Diagrams
A sample of tomato plants are measured for height. Their heights are recorded
to the nearest centimetre.
The stem-and-leaf diagram shows the results.
Height of Plant (cm)
(a)
(b)
(c)
(d)
How many plants were in the sample?
What height is the tallest plant?
Write out level 5 in full.
What fraction of the plants were more
than 50cm tall?
2
3
4
5
6
1
0
4
6
3
n = 16
2.
2
1 represents 21cm
Susan decided to visit various shops in her surrounding area in order to
compare the price of an identical CD player.
Her results, shown below, are given to the nearest pound.
£68
(a)
(b)
3.
3
2 2 7
5 6 8 9 9
7 9
£75
£73
£80
£75
£79
£81
£66
£71
£92
£83
£75
£78
Construct a stem-and-leaf diagram to represent this data.
What was the median price?
A factory has a small workforce of eleven people. The owner decides to compare absence
rates (in days) over the last two years.
The results are shown in the back-to-back stem-and-leaf diagram below.
Absences (days)
last year
this year
7 6 0 3 9 9
5 1 1 1 7
8 5 1 2 4 6
7 2 0 3 3
4 2 4 1 5
n = 11
(a)
(b)
(c)
4.
box.
0
3 represents 3 days
n = 11
What is the largest number of absences recorded?
State the median of the absences for "last year" and "this year".
Compare the absences and comment.
Two makes of matches are being compared,"Brighto" and "Sparky", they both cost the same per
14 boxes of each type are sampled to find the number of matches in a box. Here are the results.
PAGE
96
Brighto
(a)
(b)
48 45 47 39 52 36 58
41 38 39 46 50 61 37
Sparky
38 42 49 39 62 56 52
40 58 49 29 51 64 57
Construct a back-to-back stem-and-leaf diagram to represent this information.
Which make of match, if any, is a better buy? Give a reason for your answer.
Frequency Tables (Mean,Median & Mode)
1.
Fifty arrows were fired at a target. The outside ring scored 1 point and the centre ring
was worth 10 points. The results are shown below.
4
1
9
1
4
4
1
5
5
8
5
8
6
4
4
1
7
4
1
9
8
2
4
1
4
9
(a)
Copy and complete the frequency table below
Score (x)
1
2
3
4
5
6
7
8
9
10
7
2
2
4
7
5
3
4
9
8
5
4
Tally
5
1
6
10
Frequency (F)
2
8
6
4
2
10
4
8
F × x
Totals
Calculate the mean score.
(c)
Determine the median score and state the modal score.
For each of the frequency tables below ......
Copy the table and add an extra column for F × x .
Calculate the mean value from the table.
Determine the median and state the mode.
(a)
(b)
x
5
6
7
8
9
10
11
F
3
5
4
7
8
2
1
x
12
13
14
15
16
17
18
19
20
(c)
F
1
2
2
12
4
6
9
3
1
x
20
21
22
23
24
25
F
12
13
13
8
15
9
97
(i)
(ii)
(iii)
PAGE
2.
(b)
Scatter Diagrams
1.
Plot each of the following sets of points on a separate coordinate diagram and comment
on the correlation (if any).
SET 1
PAGE
98
SET 4
2.
x
y
3
4
1
7
9
3
2
7
6
5
9
8
3
7
3
3
1
9
7
5
2
2
6
5
8
7
2
6
x
y
1
1
3
3
3
3
5
7
8
9
10
11
11
11
12
13
13
13
9
4
2
5
6
4
5
6
5
6
7
1
6
8
6
8
7
5
SET 2
SET 5
x
y
3
4
10
11
7
12
13
2
9
5
8
11
10
6
11
1
3
14
7
6
4
5
5
3
3
6
5
5
4
4
3
7
3
8
1
2
x
y
1
2
3
4
5
6
6
8
9
11
11
12
13
13
6
1
4
5
6
4
3
4
3
6
1
1
1
2
SET 3
SET 6
y
Where possible, insert a line of best fit on the appropriate diagram.
o
x
y
2
7
5
10
9
9
1
3
4
6
6
7
2
3
8
2
5
7
9
6
4
11
5
8
x
y
1
2
3
4
5
6
6
7
7
8
8
9
8
2
3
3
5
5
6
2
7
7
8
9
.
. . .. .
.
.
.
.
.
.
.
.. . .
.
x
Best Buy
Which item is the best buy in each group below ?
(a)
(b)
3 litres
£1.68
2 litres
£1.20
3 litres
£1.74
1 litre
£0.62
(c)
1 ⋅ 5 litres
84p
0 ⋅ 8 litres
48p
(d)
350 g
£5.60
550 g
£8.20
200 g
£3.60
1 kg
£1.80
(e)
750 g
£1.20
200 g
44p
500 g
£1.00
(f)
0 ⋅ 7 litres
40p
2 litres
£1.39
150ml
£1.20
4 litres
£2.56
(g)
250ml
£1.75
500ml
£2.50
(h)
0 ⋅ 8 kg
£4.20
0 ⋅ 5 kg
£2.80
0 ⋅ 3 kg
£1.59
320 g
£1.40
200 g
£1.05
(j)
1 ⋅ 5 kg
£3.06
Twin pack
( 2 × 3 ⋅ 5 kg )
£11.90
250ml
80p
550ml
£1.60
950ml
£2.95
99
6 kg
£10.50
2 ⋅ 5 kg
£4.80
650ml
£1.99
PAGE
(i)
600 g
£2.50
Pairs of Lines
1.
(a)
A straight line has as its equation y = x .
Copy and complete the table below for this line.
x
y
0
1
1
4
6
7
7
10
Plot the points from the table on a coordinate diagram and draw the line through them.
(b)
A second straight line has as its equation y = 12 x + 2
Copy and complete the table for this line.
x
y
0
2
3
4
6
8
6
10
Plot the points from this table on the same coordinate diagram and draw the line.
2.
(c)
Write down the coordinates of the point where the two lines cross.
(a)
A straight line has as its equation y = 13 x .
Copy and complete the table below for this line.
x
y
0
3
6
9
3
12
Plot the points from the table on a coordinate diagram and draw the line through them.
(b)
A second straight line has as its equation y = x − 4 .
Copy and complete the table for this line.
x
y
4
5
1
6
7
8
4
9
Plot the points from this table on the same coordinate diagram and draw the line.
(c)
3.
Write down the coordinates of the point where the two lines cross.
Answer the same questions, as in Q1., for each of the following pair of lines.
x + y = 12
x
y
0
3
6
8
y = 12 x + 3
x
y
0
4
8
12
2 x + y = 10
x
y
x
y
0
2
4
5
0
4
7
8
PAGE
100
(a)
(b)
y = x +1
12
10
Direct Proportion
1.
Fred walked at a steady rate of 5km/h for 7 hours. The table below represents his distance,
from the start, at the end of each hour.
Time (t) hours
Distance (D) km
0
1
5
(a) Copy and complete the table.
2
3
4
20
5
6
7
(b) Draw a graph of distance (D) against time (t).
(c) Explain why the relationship between D and t is directly proportional.
2.
300g of flour is used to make 6 cakes. How much flour is needed to make:
(a)
3.
12 such cakes?
(b)
3 cakes?
(c)
9 cakes?
Eight bars of chocolate cost £3.36. Calculate the cost of:
(a) 1 bar of chocolate
(b) 3 bars
(c) 11 bars.
4.
A stack of six identical books weighs 1⋅ 38 kg. How much would a stack of 10 books weigh?
5.
(a)
(b)
(c)
(d)
6.
4 CD's cost £35.92 and 3 cassettes cost £15.78. Find the total cost of .......
4 cakes cost £3.12. Find the cost of 9 cakes.
The height of 12 stacked CD cases is 136 ⋅ 8 mm. Calculate the height of 7 such cases.
A row of 24 staples measures 14 ⋅ 4 mm. How long would a row of 38 staples be?
The weight of 3 baskets of fruit is 5 ⋅ 4 kg. Calculate the weight of 5 baskets.
(a) 7 CD's and 2 cassettes.
7.
(b) 3 CD's and 5 cassettes.
Carpet is priced relative to its area.
A rectangular carpet measuring 5m by 4m costs £264.
(a) Calculate the cost for 1 square metre of this carpet. (the cost per sq.m)
A bedroom carpet measuring 4m by 7m costs £180.60.
How much would the same type of carpet measuring 5m by 8m cost?
9.
A car uses 15 litres of petrol to travel 210 miles. How much petrol would the car use for a
journey of 378 miles at the same rate of consumption?
10.
Fifteen books cost £123. How many books could you buy for £73.80?
11.
For £250 you receive 2750 francs. How much would 1364 francs cost you in pounds sterling?
PAGE
8.
101
(b) How much would a carpet measuring 8m by 6m cost?
PAGE
102
Inverse Proportion
1.
If 20 men can load a ship in 4 days, how long would it take 10 men ?
2.
If 16 men can load a ship in 6 days, how long would it take 12 men ?
3.
If 8 men can load a ship in 12 days, how long would it take 16 men ?
4.
If 6 men can load a ship in 5 days, how long would it take 15 men ?
5.
If 4 men can build a house in 40 days, how long would it take 10 men to build the same house ?
6.
If 2 men can build a house in 60 days, how long would it take 12 men to build the same house ?
7.
If 9 men can build a house in 8 days, how long would it take 4 men to build the same house ?
8.
If 12 men can build a house in 24 days, how long would it take 18 men to build the same house ?
9.
A fort has enough food to feed 60 men for 15 days. How long would the food last if there were
100 men in the fort ?
10.
A fort has enough food to feed 80 men for 24 days. How long would the food last if there were
60 men in the fort ?
11.
A town under seige has enough food to feed 500 people for 30 days. How long would the food last
if there were 300 people in the town?
12.
A fort has enough food to feed 80 men for 12 days. How long would the food last if an extra 16
men arrived at the fort ?
13.
A fort has enough food to feed 160 men for 6 days. How long would the food last if an extra 80
men arrived at the fort ?
14.
A fort has enough food to feed 50 men for 10 days. How long would the food last if 30 men
left the fort ?
Proportion (Direct & Inverse)-Mixed Exercise
In 5 hours an electric fire uses 20 units of electricity. How many umits will it use in :
(a)
(b)
8 hours
(c)
20 hours ?
A car can travel 26km on 2 litres of petrol. How far can it travel on :
(a)
1 litre
(b)
6 litres
(c)
12 12 litres of petrol ?
3.
12 dinner plates cost £29.40. How much would you pay for 16 plates ?
4.
20 metres of copper piping costs £28.40. How much would 17 metres cost ?
5.
If 8 men take 12 days to dig a ditch how long would it take 6 men to dig the same ditch ?
6.
A builder employed 14 men to landscape a garden. It took them 6 days. How long would it have
taken 12 men to landscape the same garden ?
7.
At a constant speed a train can travel 497 km in 7 hours. How far could it travel in 10 hours ?
8.
Two dozen oranges cost £3.12. How much would you pay for 8 oranges ?
9.
A woman worked for 9 hours and was paid £61.20. How much would she be paid if she worked
for 16 hours ?
10.
A farmer has enough feed to last his 20 cows 16 days. How long would the feed last if he had
64 cows ?
11.
A car can complete a journey in 6 hours at an average speed of 65 km/h. How long would it
take to complete the same journey at an average speed of 78 km/h ?
12.
Six bottles of wine is the exact amount you need to give 21 people one glass each.
(a)
How many bottles would you need to give 56 people one glass each ?
(b)
How many people could you give a glass of wine if you had 32 bottles ?
13.
A town, with a population of 144, is under seige. It has enough food to last the people 24 days.
If they take in an extra 48 people how long will the food supply now last ?
14.
A gang of 36 dockers can unload a ship in 8 hours. If 4 of the dockers are ill, and don't show
for work, how long will it now take to unload the ship ?
103
2.
1 hour
PAGE
1.
Variation 1
1.
In each table below " y varies directly as x " .
For each table ........
(a)
2.
1
Write down a formula connecting y and x.
ii)
Copy and complete the table.
2
12
t
P
1
6
5
30
104
PAGE
(b)
6
x
y
18
7
42
(b)
m
E
120
36
200
60
10
45
450
135
2
12
4
6
8
10
(b)
P
.
t2
Copy and complete the table where P = k t 2 .
(a)
M varies directly as d . When d = 8 , M = 24. Find M when d =14 .
(b)
E varies directly as t .
(c)
P varies directly as u . When u = 80 , P = 64.
(a)
F varies directly as s 2 .
(b)
T varies directly as
(a)
5.
4
In the table below "P varies directly as the square of t ".
t
P
4.
3
Explain why each table below shows "direct variation" between the two variables.
(a)
3.
x
y
i)
Find k given that k =
When t = 16 , E = 56.
When s = 2 , F = 20.
Find E when t = 20 .
Find P when u = 260 .
Find F when s = 5 .
m . When m = 25 , T = 15 . Find T when m = 81 .
6.
The cost, C (£) , of painting a wall "varies directly" as the length of the wall , l (metres).
A wall 12 metres long costs £57.60 to paint. How much would it cost to paint a wall 17 metres
long ?
7.
The volume V cm3 of a gas "varies directly" as its temperature t (degrees Kelvin).
When the temperature is 90 degrees K , the volume of the gas is 585 cm3.
Find the volume of the gas when the temperature is 140 degrees K.
83.7
Variation 2
(a)
x
y
1
(c)
x
y
10
3
4
22
55
30
80
(b)
x
y
4
(d)
x
y
5.6
7
6
7
21
30
8.4
8
15
Explain why each table below shows "direct variation" between the two variables.
(a)
3.
2
8
t
P
2
12
8
48
10
60
(b)
m
E
260
13
400
20
1060
53
In the table below "P varies directly as the square of t ".
t
P
(a)
(b)
4
32
6
7
10
16
P
.
t2
Copy and complete the table where P = k t 2 .
Find k given that k =
4.
E varies directly as h . When h = 4 then E = 20 .
Find E when h = 14 .
5.
(a)
(b)
(c)
(d)
M varies directly as d .
E varies directly as t .
P varies directly as u .
L varies directly as e .
6.
(a)
F varies directly as s 2 .
2
When
When
When
When
d = 10 , M = 35. Find M when d = 24 .
t = 6 , E = 30.
Find E when t = 24 .
u = 60 , P = 15. Find P when u = 280 .
e = 3.6 , L = 720. Find L when e = 6.8 .
When s = 3 , F = 36.
Find F when s = 7 .
When p = 4 , G = 8.
Find G when p = 6 .
(b)
G varies directly as p .
(c)
T varies directly as
m . When m = 36 , T = 18 . Find T when m = 64 .
(d)
B varies directly as
a . When a = 9 , B = 15 .
Find B when a = 16 .
7.
The cost, C (£) , of painting a wall "varies directly" as the length of the wall , l (metres).
A wall 18 metres long costs £43.20 to paint. How much would it cost to paint a wall 30 metres
long ?
8.
The volume V cm3 of a gas "varies directly" as its temperature t (degrees Kelvin).
When the temperature is 200 degrees K , the volume of the gas is 1300 cm3.
Find the volume of the gas when the temperature is 500 degrees K.
9.
The swing time, T seconds, of a pendulum varies directly as the square root of its length, l cm.
When the length is 60 cm the time of swing is 3 seconds.
Calculate the time of swing for a pendulum 84cm long.
105
2.
In each table below " y varies directly as x " .
For each table ........ i)
Write down a formula connecting y and x.
ii)
Copy and complete the table.
PAGE
1.
Ratio 1
1.
2.
(a)
Divide £50 in the ratio 3 : 7 .
(b)
Divide 80kg in the ratio 3 : 7 .
(c)
Divide £35 in the ratio 5 : 2 .
(d)
Divide 240 g in the ratio 4 : 1 .
(a)
Three boys, Harry, James and Bill divide £120 in the ratio 1 : 3 : 8 .
How much does each boy get ?
(b)
Three girls, Susan, Beth and Jill divide £56 in the ratio 2 : 5 : 7 .
How much does each girl get ?
3.
Graeme and Fred invest £3400 in a new company.
(a)
(b)
If the money each of them put in was in the ratio 3 : 7 , how much
did Fred invest in the new company ?
They decide to split the profits in the same ratio as their investment.
If they made £6200 profit, how much of the profit will Graeme get ?
£
4.
The ratio of boys : girls in a class is 4 : 5. If there are 27 pupils in the class, how many
girls are there.
5.
(a)
In a piece of jewellery the ratio of gold to silver is 5 : 2.
If the jewellery contains 40 grammes of gold, what weight of silver does it contain ?
(b)
An lottery win was shared between three brothers, Dave, Frank and Pat, in the ratio 1 : 3 :
4.
If Pat received £824, how much did each of the other two brothers receive ?
PAGE
106
6.
Two farmers, Bill and Dan, decided to split a herd of cows in the ratio 5 : 7.
(a)
If Dan's share was 42 cows, how many cows did Bill get ?
(b)
How many cows were there altogether ?
(c)
A third farmer, George, came along and the three farmers decided to split the herd in
the ratio (B : D : G) 3 : 4 : 1.
How many cows will each farmer get
Ratio 2
Divide £48 in the ratio 3 : 5 .
(b)
Divide £100 in the ratio 7 : 3 .
(c)
Divide £56 in the ratio 1 : 6 .
(d)
Divide £50 in the ratio 4 : 1 .
(e)
Divide £120 in the ratio 5 : 3 .
(f)
Divide £75 in the ratio 8 : 7 .
(g)
Divide £36 in the ratio 4 : 5 .
(h)
Divide £240 in the ratio 5 : 7 .
(a)
Three boys divide £88 in the ratio 1 : 3 : 7 . How much does each boy get ?
(b)
Three girls divide £48 in the ratio 2 : 3 : 11 . How much does each girl get ?
(c)
Three men divide £60 in the ratio 3 : 4 : 5 . How much does each man get ?
(d)
Three girls divide £96 in the ratio 1 : 2 : 5 . How much does each girl get ?
3.
John and David inherit £3400. If they divide the money in the ratio 2 : 3 , how much
does each person receive.
4.
The ratio of boys : girls in a class is 3 : 5. If there are 32 pupils in the class, how many
girls are there.
5.
The ratio of sand : cement in a certain concrete is 7 : 4. If a cement mixer has been filled with
33 bags, how many of the bags were sand ?
6.
(a)
The ratio of cats : dogs in an animal hospital is 1 : 5.
If there are 8 cats, how many dogs are there ?
(b)
In a school show the ratio of girls : boys is 2 : 1.
If there are 24 girls, how many boys are there ?
(c)
In a necklace the ratio of diamonds : emeralds is 3 : 4.
If there are 16 emeralds, how many diamonds are there ?
(d)
An estate was shared between three brothers, Tom, John and Dave, in the ratio 2 : 3 : 5.
If Tom received £2400, how much did each of the other two brothers receive ?
7.
Three friends, Xena,Gabrielle and Joxar, have found a treasure chest full of gold coins.
They decide to split the coins in the ratio 5 : 3 : 1.
(a)
If Gabrielle was to receive 24 coins, how many coins would the others get ?
(b)
How many coins are there altogether ?
(c)
Before they can share out the coins, Calisto arrives, and persuades
the friends to split the coins in the ratio (X : G : J : C) 9 : 5 : 4 : 6 .
How many coins will each person now receive ?
107
2.
(a)
PAGE
1.
Similarity and Area
1.
For each pair of pictures below
(a)
i) State the enlargement scale factor for the lengths (kL).
ii) State the scale factor for the areas (kA).
iii) Calculate the area of the larger shape.
(b)
10cm
5cm
96mm2
2
16cm
12mm
(c)
36mm
(d)
12mm
8mm
40mm2
14cm
33.6cm
50cm2
(e)
(f)
6cm
24cm
120cm2
22cm2
16cm
28.8cm
2.
For each pair of pictures below
i) State the reduction scale factor for the lengths (kL).
ii) State the scale factor for the areas (kA).
iii) Calculate the area of the smaller shape.
(a)
(b)
70cm2
10cm
76mm
19mm
5cm
3648mm2
(c)
(d)
30cm
400mm2
PAGE
108
24cm
150cm2
24mm
18mm
Similarity and Volume
1.
(kL).
For each pair of similar pictures below
i) State the enlargement scale factor for the lengths
ii) State the scale factor for the volumes (kV).
iii) Calculate the volume of the larger solid.
(a)
(b)
48cm3
12mm
36mm
4cm
V = 216mm3
8cm
(c)
(d)
12mm
8mm
72mm3
8cm
19.2cm
V = 20cm3
(e)
(f)
3cm
3
9cm
12cm
400cm3
16cm
22.4cm
2.
For each pair of similar pictures below
(a)
i) State the reduction scale factor for the lengths (kl).
ii) State the scale factor for the volumes (kv).
iii) Calculate the volume of the smaller solid.
(b)
368cm3
10cm
5cm
144mm3
2mm
8mm
(c)
V = 912cm3
(d)
230mm3
9mm
15mm
PAGE
18cm
109
24cm
The Circle 1
** You need a calculator for this worksheet.
Remember :
C = πd
( π = 3 ⋅ 14 )
The Circumference of a Circle
The circumference of a circle is the distance around the outside of the circle
Example 1 .... Calculate the circumference of
the circle with diameter 18cm.
working ........
Example 2 .... Calculate the circumference of
the circle with radius 3mm.
C = πd
working ........ If r = 3 then d = 6
= 3 ⋅14 × 18
= 56 ⋅ 2 cm (to 1 d . p.)
C = πd
= 3 ⋅ 14 × 6
= 18 ⋅ 8 mm (to 1 d . p.)
1.
2.
3.
Calculate the circumference of the circle with .......
(a) diameter 12 cm
(b) diameter 14 mm
(c) diameter 2 ⋅ 4 m
(d) d = 4 cm
(e) d = 30 cm
(f) d = 6 ⋅ 8 mm
(g) d = 80 m
(h) d = 17 cm
Calculate the circumference of the circle with .......
(a) radius 11 cm
(b) radius 23 mm
(c) radius 1⋅ 5 m
(d) radius 2 cm
(e) r = 14 mm
(f) r = 2 ⋅ 1 cm
(g) r = 6 ⋅ 2 m
(h) r = 9 cm
Calculate the circumference of each circle below.
(a)
10cm
4.
(c)
(b)
110
2 ⋅ 4 cm
16 ⋅ 2 cm
36mm
Calculate the circumference of each circle below.
(c)
(b)
PAGE
(d)
(a)
.
13mm
.
7cm
.
2.7m
The Circle 2
A = π r2
The Area of a Circle
( π = 3 ⋅ 14 )
The Area of a circle is the amount of flat space inside the circle
Example 1 .... Calculate the area of the
circle with radius 7cm.
working ........
Example 2 .... Calculate the area of
the circle with diameter 6mm.
A = π r2
working ........ If d = 6 then r = 3
= 3 ⋅ 14 × 7 2
= 153 ⋅ 9 cm2
A = π r2
= 3 ⋅ 14 × 32
(to 1 d . p.)
= 28 ⋅ 3 mm2
1.
(to 1 d . p.)
Calculate the area of the circle with .......
2.
(a) radius 12 cm
(b) radius 21 mm
(c) radius 1⋅ 5 m
(d) radius 6 cm
(e) r = 15 mm
(f) r = 2 ⋅ 1 cm
(g) r = 6 ⋅ 2 m
(h) r = 11 cm
Calculate the area of the circle with .......
3.
(a) diameter 14 cm
(b) diameter 28 mm
(c) diameter 1⋅ 4 m
(d) d = 6 cm
(e) d = 40 cm
(f) d = 7 ⋅ 4 mm
(g) d = 36 m
(h) d = 15 cm
Calculate the area of each circle below.
(a)
.
18cm
.
(b)
.
(c)
26cm
1.5m
Calculate the area of each circle below.
(c)
(a)
12cm
24mm
(d)
22cm
9cm
111
(b)
PAGE
4.
The Circle 3
A circle has a diameter of 14 ⋅ 8 cm .
1.
Calculate
(a)
The circumference of the circle.
(b)
The area of the circle.
(b)
The circumference of the
A circle has a radius of 5 ⋅ 1 m .
2.
Calculate
(a)
The area of the circle.
circle.
24cm
3.
(a)
(b)
4.
Calculate the area of the rectangular
steel plate shown opposite.
10cm
A hole of radius 6cm is drilled through the plate.
Calculate the shaded area
Calculate the shaded area in each shape below.
(a)
16cm
20mm
(b)
8mm
6cm
11cm
14mm
8mm
5.
Calculate the area of each shape .
(a)
(b)
4m
PAGE
112
15cm
8m
18cm
Angles and Circles
.
1.
marks the centre of each circle
Theorem 1
Calculate the size of each of the
angles marked with letters in the
diagrams below.
33o
Theorem 2 A tangent meets a radius at right angles.
b
a
.
The angle in a semicircle is a right angle.
40o c
72o
.
d
f
.
50o
64
o
g
.
h
e
2.
Find the angles marked with letters.
35o
c
.
25
o
.
a
.
40o
.
e
d
f
g
60o
b
LOOK for ..................
24o
* Isosceles triangles
* Angles in a triangle add up to 180o
* Angle in a semicircle is a right angle
* Angle between a tangent and a radius is 90o
Find the size of each lettered angle in the diagrams below :
a
b
d
g
.
.
f
50o
e
k
20o
h
i
.
j
.
70o
l
113
c
.
68o
PAGE
3.
30o
Statistics
1.
(a)
(b)
(c)
(d)
(e)
(f)
5.
People will holiday in space within the next 10 years.
You can hold your breath for 10 minutes.
You will watch TV tonight.
Your teacher will be boring next week.
You are reading this question.
The next baby born in Scotland is a boy.
green
(b)
red
(c)
red or green
(d)
blue?
a one (b)
an even number
(c) more than a two
a factor of nine
(e) less than one?
If the spinner is spun, calculate :
(a)
P(7)
(b)
P(1)
(c)
P(even)
(d)
P(odd)
(e)
P(prime)
(f)
P(less than 4)
(g)
P(6)
(h)
P(factor of 24) (i)
P(less than 10)
A letter is chosen at random from the word STATISTICS . Write, as a decimal fraction :
(a) P(A)
6.
unlikely
even chance
In throwing an ordinary die, what is the probability of obtaining :
(a)
(d)
4.
certain
likely
Five coloured beads are placed in a bag, two are red and three are green.
If a bead is drawn from the bag at random, calculate the probability that it is :
(a)
3.
Simple Probability
Use the words below to describe the probability of each statement happening. (It's your choice?)
impossible
2.
-
(b) P(I)
(c) P(not a T)
(d)
P(vowel)
The probability of a bus arriving on time at a certain bus stop is
(e) P(not a vowel)
1
4
.
(a) What is the probability of it not arriving on time?
(b) Out of 64 buses arriving at that bus stop, how many are likely to be on time?
7.
The probability of a cat having a litter of more than eight kittens is 0 ⋅ 24 .
(a) What is the probability of a cat having a litter of eight or less kittens?
PAGE
114
(b) Out of 75 female cats, how many would you expect to have a litter of more than eight kittens?
8.
There are seven blue beads and 6 yellow beads in a bag. A blue bead is drawn from the bag and
not replaced. What is the probability that the next bead drawn is also blue?
Surface Area & Volume of a Cuboid
For each of the following solids below, calculate
(i) its surface area (A);
(ii) its volume (V).
(a)
(b)
4cm
5cm
4cm
8cm
1cm
3cm
(c)
1cm
2cm
7cm
(d)
CUBE
(e)
4mm
4cm
7mm
12mm
(g)
(f)
6cm
CUBE
16cm
PAGE
8mm
115
10cm
Volume of a Prism
Calculate the volume of each prism below.
6m
(a)
(b)
3m
4.6m
3.4m
2.8m
5m
3.2m
4m
18cm
(c)
(d)
30cm
28cm
10cm
16cm
(e)
12cm
11cm
4cm
(f)
3cm
0.6m
0.3m
5.4cm
8cm
0.9m
PAGE
116
0.5m
1.5m
rectangular hole measuring 0.2m by 0.4m
1.2m
Formulae 1
All working must be shown.
A formula is given as E = p 2 + 2 .
Find the value of E when .... i) p = 2
A formula is given as
iv) r = 1 .
ii) h = 6
iii) h = 2
iv) h = 7 ?
ii) s = 5
iii) s = 10
iv) s = 1 .
ii) x = 6
iii) x = 8
ii) p = 3
iii) p = 5
iv) p = 10 .
ii) t = 4
iii) t = 3
iv) t = 10 .
ii) k = 6
iii) k = 2
iv) k = 1 ?
iv) x = 7 .
A formula is given as
.... i) p = 2
H = t 2 + 2t + 1 .
Find the value of H when
9.
iii) r = 6
A formula is given as L = 2 p 2 − 6 .
Find the value of L when
8.
ii) r = 4
A formula is given as W = 25 + 3 ( x 2 ) .
Find the value of W when .... i) x = 2
7.
iv) e = 2.
A formula is given as T = 2 ( s 2 ) + 4 .
Find the value of T when .... i) s = 3
6.
iii) e = 8
A formula is given as G = 45 − h 2 .
Find the value of G when .... i) h = 4
5.
ii) e = 4
A formula is given as Q = 36 − r 2 .
Find the value of Q when .... i) r = 3
4.
iv) p = 1 .
A formula is given as T = e2 + 6 .
Find the value of T when .... i) e = 3
3.
iii) p = 6
.... i) t = 2
T = k 2 + 3k − 6 .
Find the value of T when
.... i) k = 3
117
2.
ii) p = 3
PAGE
1.
Formulae 2
All working must be shown.
1.
A formula is given as E = 3 p + q .
Find the value of E when ....
2.
118
ii)
iv)
d = 6 and e = 3
d = 12 and e = 8
i)
iii)
r = 2 and s = 5
r = 4 and s = 4
ii)
iv)
r = 3 and s = 10
r = 6 and s = 20
i)
ii )
iii )
u = 3 , a = 2 and t = 4
u = 6 , a = 3 and t = 7
u = 2 , a = 8 and t = 10
i)
ii )
iii )
p = 4 and t = 3
p = 5 and t = 2
p = 8 and t = 0 ⋅ 5
i)
ii )
iii )
a = 4 , b = 6 and c = 4
a = 5 , b = 2 and c = 3
a = 6 , b = 4 and c = 8
A formula is given as A = 2 l h + 2 l b + 2 b h .
Find the value of A when
PAGE
d = 5 and e = 2
d = 8 and e = 5
A formula is given as W = a b − 3 c .
Find the value of W when ....
7.
i)
iii)
A formula is given as C = 20 + 4 p t .
Find the value of C when ....
6.
p = 6 and q = 3
p = 3 and q = − 6
A formula is given as V = u + a t .
Find the value of V when ....
5.
ii)
iv)
A formula is given as F = 7 r − 2 s .
Find the value of F when ....
4.
p = 4 and q = 2
p = 5 and q = 1
A formula is given as T = 2 d − 3 e .
Find the value of T when ....
3.
i)
iii)
....
i)
ii )
iii )
l = 6 , b = 3 and h = 2
l = 5 , b = 4 and h = 6
l = 8 , b = 7 and h = 4
Pythagoras' Theorem (Practical Questions)
Answers should be rounded to 1-decimal place where necessary.
1.
A ladder leans against a wall as shown in the diagram
opposite.
6m
From the information given calculate the length of the ladder.
3m
2.
Calculate the perimeter of each shape below.
5m
12cm
12m
8cm
7m
17cm
3.
13m
Blackpool light decorations are suspended above the street by wire cables as shown below.
4.6m
4m
3.8m
3m
4.6m
6m
Calculate the total length of cable in each diagram.
Q
Three trees are situated as shown with angle PQR = 90o.
4.
Calculate the distance between the trees Q and R.
(careful !)
P
16m
R
Calculate the total length of each mountain bike ramp shown below.
5m
4m
2.7m
9m
2.6m
1.7m
8m
119
3.5m
PAGE
5.
12m
Scientific Notation
1.
2.
3.
4.
PAGE
120
5.
(Basic Practice)
Write each of the following numbers in scientific notation.
(a)
2 300
(b)
425 000
(c)
120
(d)
67 000 000
(e)
500
(f)
41 000
(g)
84
(h)
5 000 000 000
(i)
3 450 000
(j)
1 903
(k)
346 000 000
(l)
28 000 000 000 000 000
(m)
63 ⋅ 4
(n)
121⋅ 7
(o)
4 007
(p)
6
For each of the following numbers
i) write it out in figures ;
ii) write it in scientific notation.
(a)
4 thousand
(b)
3 million
(c)
20 thousand
(d)
16 million
(e)
200 million
(f)
150 thousand (g)
2 ⋅ 4 million
(h)
8 ⋅ 21 million
(i)
sixteen thousand four hundred
(j)
one million one hundred and fifty thousand
Write each of the following numbers in scientific notation.
(a)
0 ⋅ 0043
(b)
0 ⋅ 000021
(c)
0 ⋅ 064
(d)
0 ⋅ 000000765
(e)
0 ⋅ 000502
(f)
0 ⋅ 0309
(g)
0 ⋅ 000009
(h)
0 ⋅ 0008
(i)
0 ⋅ 00005002 (j)
0 ⋅ 02345
(k)
0 ⋅ 65
(l)
0 ⋅ 00000000054
(m)
0 ⋅ 06067
0 ⋅ 0000002
(o)
0 ⋅ 5508
(p)
0 ⋅ 000000000000213
(n)
Write each of the following numbers out in full.
(a)
2 ⋅ 6 × 103
(b)
5 ⋅ 41 × 10 2
(c)
7 ⋅ 45 × 10 4
(d)
2 ⋅ 81 × 10 5
(e)
6 ⋅ 4 × 10 7
(f)
6 ⋅ 2 × 10 0
(g)
2 ⋅ 34 × 101
(h)
5 ⋅ 12 × 10 6
(i)
7 ⋅ 003 × 10 2
(j)
2 ⋅ 3 × 10 9
(k)
6 ⋅ 009 × 10 4
(l)
9 × 1012
Write each of the following as an ordinary number.
(a)
4 ⋅ 3 × 10 −2
(b)
3 ⋅ 57 × 10 −4
(c)
1 ⋅ 9 × 10 −1
(d)
9 ⋅ 8 × 10 −3
(e)
7 ⋅ 04 × 10 −5
(f)
6 × 10 −2
(g)
8 × 10 −7
(h)
5 ⋅ 22 × 10 −1
(i)
2 ⋅ 007 × 10 −5
(j)
7 ⋅ 8 × 10 −9
(k)
14 × 10 −4
(l)
450 × 10 −6