Essential Mathematics. Book 1 (Rayner D.)

144 Part 5

Bar charts and bar-line graphs

(a) The scores of 35 golfers competing (b) A tally chart/frequency table is made for the
in a tournament were scores.

68 74 71 72 71 68 70 score tally frequency
74 69 71 70 67 73 71
70 74 69 72 73 74 71 67 II 2
72 74 71 72 72 70 73 68 Ill 3
67 68 72 73 72 71 71 69 II 2
70 II II 4
71 ®Ill 8
72 ®II 7
73 II II 4

74 mt 5

(c) This data can be displayed on either a bar chart or on a bar-line
graph. The '-}\-' shows that a section on the horizontal axis
has been cut out.

Frequency Bar chart Frequency Bar-line graph

88

66

44

22

0 72 73 74 0 67 68 69 70 71 72 73 74

Score Score

Exercise 10 Attitude to smoking:
which statement describes you best?
1. In a survey in 1993 almost 30 000
young people were questioned about Percentage responses D Boys
smoking. 100
(a) What percentage of girls aged • Girls
13-14 said they 'don't smoke 80
and never will'?
(b) Is there a connection between 60
sections A and C?
(c) Estimate what percentage of 40
boys aged 15-16 smoke and
don't want to give up. 20

0 ~ ¥~ ~ ~ ~ ~ ¥~
' ' ' ' ' ' ' ' '~ ~ ¥~ ~
$ $ $
~
Don't smoke Smoke, but like Percentage of

and never will to give up smokers

ABc

Charts and Graphs 145

2. The yearly sales figures in the U.K. for Top Sweets Top Snacks
sweets and snacks are shown.
(a) Draw a bar chart or bar line graph for 1 KitKat £118m
each set of figures.
(b) Which products have a particularly large 2 Mars Bars £100m
share of the market?
(c) Is it easier to answer part (b) using just 3 Cadbury Dairy
the figures or by using your charts?
(d) Write down your own order of Milk £65m 1 Walkers
preference for sweets and snacks from 1 Crisps
to 10. You may include other brands not 4 Roses £60m £205m
listed.
5 Twix £60m 2 Golden Wonder

Crisps £65m

3. Pragya thinks she can tell from which newspaper an article is 3 Hula Hoops £55m
taken by counting the number of letters in the words used.
Below are extracts from two newspapers: the first from the 4 Quavers £35m
'Sun'; the second from the 'Independent'.
5 Pringles £30m

[Count numbers like words with digits number of tally
as 'letters'.] letters

Make tally charts for the number of 1
letters in the words in each article. 2
3

Display the results in either one or two bar charts, plotting
frequency on the vertical axis.

You can obtain a very clear comparison of the two articles by
superimposing the two charts on one graph, using different
colours for each newspaper.

Is there a significant difference in the length of words used in the
two articles?

146 Part 5

4. The bar charts show the sales of various goods over a year. D
Unfortunately the labels on the charts have been lost. Decide
which of the charts A, B, C, D shows sales of:
(a) fir trees; (b) crisps; (c) flower seeds;
(d) greetings cards [including Christmas, Valentine's Day etc.]

A8C

JFMAMJJASOND JFMAMJJASOND JFMAMJJASOND

5. The managing director of McDonalds has Comparable chain - sales
just received the sales figures for the year.
He wants to display the results so that £million
McDonalds' sales look as impressive as
possible. 600 Menu
A bar chart of results is shown. Burgers
Work out, as accurately as you can, the
angles on a pie chart which would show the 500 Pizza
same information. Draw the pie chart. Chicken
Which looks more impressive from
McDonalds' point of view: the bar chart or 400 Steaks
the pie chart? Salads

300 Chips

200

100

6. The b;u chart shows the results of a survey in which 2000 people

in each of 20 countries were asked if they had reported that their

car had been stolen in the previous 12 months. Car theft around the world

(a) In which country or countries was the Percentage of motorists who reported that their car had been
reported rate of theft worst? stolen in the previous 12 months

(b) Of the 2000 people questioned in the England & Wales >
United States, how many had reported a Italy
car stolen? New Zealand "(:'

(c) What does the chart show for Norway :J
Switzerland? Scotland (J)

(d) Roughly how many times more likely ••••Canada u."§'
were people in England and Wales to
report a car theft compared to people in -Germany '"c
Germany?
ISwitzerland ·;0::;
(e) Why do you think Scotland's rate of
theft is so much lower than that in "E'
England?
_"s'

;,;

~
:J
0

(J)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Charts and Graphs 147

Data in groups

When a large number of results are to be displayed it is often more
convenient to put the data into groups. Suppose we wanted to
draw a frequency diagram to show the heights of 25 children given
below .

130·2, 145-4, 132·2, 140·9, 150·0, 152·7, 132·1, 141·8, 144·9,
142·0, 137·2, 146·0, 134·6, 143 ·3, 138·9, 144·5, 137·0, 136·2,
139·0, 142·7, 148·1 , 147·2, 136·6, 143·9, 153·8. [In em]

The heights are put into groups as shown and the number of people
in each group is found . A frequency diagram can then be drawn.

Frequency

class interval frequency 8
6
130 ~ h < 135 4 4
135 ~ h < 140 6 2
140 ~ h < 145
145 ~ h < 150 8
150 ~ h < 155
h =height (em) 4
3

130 140 150 160

Heights (em)

• The group 150 ~ h < 155 means 'heights greater than or equal
to 150 and less than 155.
So a height of 150 goes into this group.

• A height of 149·5 goes into the group 145 ~ h < 150.
• All heights are possible (not just certain values) so there are no

gaps between the bars on the frequency diagram.

Exercise 11

1. At a medical inspection the 11 / 12 year-olds in a school have
their heights measured. The results are shown.

136·8, 146·2, 141 ·2, 147·2, 151 ·3, 145·0, 155·0,
149·9, 138·0, 146·8', 157-4, 143·1, 143·5, 147·2,
147·5, 158·6, 154·7, 144·6, 152-4, 144·0, 151 ·0.

(a) Put the heights into groups (b) Draw a frequency
class interval frequency diagram

Frequency

135 ~ h < 140
140 ~ h < 145
145 ~ h < 150

135 height

148 Part 5

2. A group of 7 year-olds were each accompanied by one of their
parents on a coach trip to a zoo. Each person on the coach
was weighed in kg. Here are the weights:

21·1, 45·7, 22·3, 26·3, 50·1, 24·3, 44·2,
54·3, 53·2, 46·0, 51·0, 24·2, 56-4, 20·6,
25·5, 22·8, 52·0, 26·5, 41 ·8, 27·5, 29·7,
55·1, 30·7, 47-4, 23 ·5, 59·8, 49·3, 23-4,
21 ·7, 57·6, 22·6, 58·7, 28·6, 54·1.

(a) Put the weights into groups. class interval frequency

20 ::o:; w < 25
25 ::o:; w < 30
30 ::o:; w < 35

(b) Draw a frequency diagram.
(c) Why is the shape of the frequency diagram different to the

diagram you drew in Question 1?
(d) What shape of frequency diagram would you expect to

obtain if you drew a diagram to show the heights of pupils
in your class?

3. A drug company claims that its new nutrient pill helps people to
improve their memory.
As an experiment two randomly selected groups of people were
given the same memory test. Group A took the new pills for a
month while group B took no pills. Here are the results of the
tests: (A high score indicates a good memory).

Frequency Frequency

)()() 100
50 50

10 20 30 40 50 10 20 30 40 50

Test score Test score

Does it appear that the new pills did in fact help to improve

memory?

Charts and Graphs 149

4. A teacher has a theory that pupils' test results are affected by
the amount of T.V. watched at home.
With the willing cooperation of the children's parents, the pupils
were split into two groups:

Group X watched at least two hours of T.V. per day.
Group Y watched a maximum of half an hour per day.

The pupils were given two tests: one at the start of the
experiment and another test six months later. Here are the
results:

Group X before Group Y before Group X after Group Y after

ff f f

;-- ;-- - ;--
-
- - -- ;-- '--
- - -;-- -

f-- -
f-- r--

----/\, 80 ........~\, 80 ........~\, 80 ../\ 80

30 30 30 30

Test marks Test marks Test marks Test marks

Look carefully at the frequency diagrams.

What conclusions can you draw? Was the teacher's theory
correct?
Give details of how the pupils m group X and in group Y
performed in the two tests.

5. A car insurance company carried out
some light-hearted but interesting
research. They recorded the cost of
accident repairs from hundreds of
insurance claims and also recorded the
driver's star sign. As an example
suppose the average repair claim for
all drivers is £500. Any average driver
with star sign Sagittarius has a claim
19·3% higher than £500 [ie. £596·50].
In the questions below take the average
repair claim to be £500.
(a) What is the average repair claim
for drivers who are Taurus?
(b) Work out the average repair claim
for drivers of your star sign.
(c) Does the chart provide data
which reinforces your opinion of
the car driving ability of your
mother and/or father?(!)

150 Part 5

Line graphs

Information is sometimes given in the form of a line graph. Line
graphs are particularly useful when quantities vary continuously
over a period of time.

Exercise 12

1. The temperature in a centrally heated house is recorded every time of day
hour from 12 00 till 24 00; the results are shown below.

house temperature in oc

20 .

18
16 .
14
12 .
10
8
6
4
2+··i·····i····+··
0+-'-----+----'---

12.00 13.0014.0015.0016.0017.0018.0019.0020.0021.0022.0023.0024.00

(a) What was the temperature at 20 00? (d) When do you think the central heating
(b) Estimate the temperature at 16 30. was switched on?
(c) Estimate the two times when the
(e) When do you think the central heating
temperature was 18°C. was switched off?

2. A man climbing a mountain measures his height above sea level
after every 30 minutes; the results are shown below.

height above sea level (m)

2400 ++ ; + + +i + +· + + :::;
2200 + + i + + ++ i :A""T·····

2000+++ ++<c~+ ++

1800++ + ++Y

1600 +.;....; +·+;f!· ..

1400 + ;; + ····+····:, ;.......

1200 + 1 +:/+·!
1000 + i j( ++!

800 . / ! !··+···:···+ ··+· +···+ ··!··· +···!····...! · +······i···· +··+· ··+. +\l+···i
600~······>+ , , , , , , , , , , + + + ; ; ; ; t i i

400+!+ +++!+ +++! ·· + +++'+ +++i!
2001 ... ;· ; ; ;;l+·······>····· t : ; · ; ; ; ; ; i + , , , ,

OL-~-L~~~~~_L~_L~_L~_L~+-~

09.00 10.00 11.00 12.00 13.00 14.00 timeofday

(a) At what height was he at 10 00? (e) How high was the mountain? (He got to
(b) At what height was he at 13 30? the top!)
(c) Estimate his height above sea level at
(f) How long did he rest at the summit?
0945. (g) How long did he take to reach the
(d) At what two times was he 2200m
summit?
above sea level?

Charts and Graphs 151

3. The cost of making a telephone call depends on the duration of
the call as shown below.

cost of call
in pence

20p
15p

5p

234567 8 duration of call
in minutes
(a) How much is a call lasting 1 minute?
(b) How much is a call lasting 1 minute (d) How much is a call lasting 4 minutes 27
seconds?
30 seconds?
(c) How much is a call lasting 6 minutes (e) What is the minimum charge for a call?
(f) A call costing 15p is between _ minutes
30 seconds?
and _ minutes in length. Fill in the
spaces.

4. A car went on a five hour journey starting at 1200 with a full
tank of petrol. The volume of petrol in the tank was measured
after every hour; the results are shown below.

volume of petrol in tank (litres)

50

40

30

20

10

0 13.00 14.00 15.00 16.00 17.00 time of day
12.00

(a) How much petrol was in the tank at (d) What happened at 15 00?
1300? (e) What do you think happened between

(b) At what time was there 5 litres in the 15 00 and 16 00?
tank? (f) How much petrol was used between

(c) How much petrol was used in the first 12 00 and 17 00?
hour of the journey?

152 Part 5

5. The number of children inside a school is counted every ten
minutes from 7.30 a.m. until 9.00 a.m. , when the bell rings; the
results are shown below.

1000

0 900
0
800
..u0
VJ
700
.5
c
600
~
"]" 500
..u.. 400
s.0..
300
"'
200
:c::> 100

0 time of day (a.m.)
7.30 7.40 7.50 8.00 8.10 8.20 8.30 8.40 8.50 9.00

(a) How many children were inside the (c) Estimate when the first children arrived.
(d) How many children arrived during the last
school at
10 minutes before the bell rang at
(i) 8.00 a.m.? (ii) 8.35 a.m.? 9 . 0 0 a .m.?
(e) At what time were there 250 children in
(iii) 8.55 a.m.? school?

(b) How many children arrived between

7.30a.m. and 8.30a.m.?

6. A scientist records the height of a growing plant every day for
20 days. The results are shown below.

5 10 15 20 number of days

(a) What was the height of the plant after 5 days?
(b) After how many days was the height

(i) 70cm (ii) 105cm?
(c) What was the greatest increase in height in one day?

(d) What was the full-grown height of the plant?

Cbarts and Grapbs 153

7. The number of people sitting down in a cinema was recorded
every quarter of an hour; the results are shown below.

.,
E
.s(!) 200
u 180

.5 160
140
.BOs:J:)
·v;
~ 120
0..
0 100
8.
80
4-<
~0.... 60
40
:s:::l 20

0

19.00 20.00 21.00 22.00 23.00 time

(a) How many people were sitting down at (d) When do you think the first film ended?

2000? (e) How long was the interval between the

(b) How many people were sitting down at two films?

2115? (f) Which film was more popular?

(c) When do you think the first film started?

8. The petrol consumption of a car depends on the speed, as shown
below.

petrol consumption
(km per litre)

16

10

8

20 40 60 80 100 120 140 160 180 speed of car (kph)

(a) What is the petrol consumption at a speed of
(i) 30 km per hour (ii) 100 km per hour (iii) 180 km per hour?

(b) At what speed is the petrol consumption
(i) 8 km per litre (ii) 12 km per litre (iii) 9 km per litre?

(c) At what speed should the car be driven in order to use the least amount of petrol?
(d) A car is driven at 160 km per hour. How far can it travel on 20 litres of petrol?

154 Part 5

5.4 Puzzles and games 5

Printing a book

When a book is made giant sheets of paper containing several
pages of the book are printed. These large sheets are then folded
to make either 16 or 32 page 'signatures'. The signatures are
then bound together to make the whole book and then the edges
of the pages are trimmed.

The printed pages are spread all over the large sheet and are
upside down in many cases.
Take a sheet of A3 paper (twice the size of A4, 'normal' school
paper) and fold it to make a 16 page signature. Work out where
pages 1 to 16 would be and also which way up they should be
printed.
Now either make drawings, stick in pictures or write a short story
on both sides of your A3 sheet and number the pages. Finally fold
and cut your booklet and fix the pages together with staples.

Diagonals

Look at the squares below.

4 X4 5 X5
I8
9 squares along
27 the diagonals
63
54
8 squares along
the diagonals

12 squares along
the diagonals

(a) Draw a similar diagram for a 7 x 7 square and count the

squares along the two diagonals.

(b) How many squares are there along the diagonals of

(i) a 10 x 10 square? (ii) a 15 x 15 square?

(c) A square wall is covered with square tiles. There are 73 tiles

altogether along the two diagonals. How many tiles are there

on the whole wall?

(d) A square wall is covered by 3364 square tiles. How many tiles

are there along the two diagonals?

Puzzles and Games 5 155

Find the hidden treasure

This is a game for two players: one player hides the treasure and
the other player tries to find it.

(a) Player A draws a grid with x (b) Player B draws his own grid
and y from -6 to 6. He puts and makes his first guess.
the treasure at any point with Say (1, 1).
whole number coordinates.
Say (4, -2).

ii!

......l..........l..........l
B's grid

······~··········~····· ·· ···r

(c) Player A tells player B how far away he is by adding the
horizontal and vertical distances from his guess to the treasure.
So the point (1, 1) is a distance 6 away.

(d) Player B has another guess and player A gives the distance
from the new point to the treasure.

(e) Play continues until player B finds the treasure.
(f) Roles are then reversed so that B hides a new treasure and A

tries to find it in as few goes ,as possible.

• After several games you may realise that you can improve your
chances by using a 'mathematical strategy'.

Part 6

6.1 Probability

Chance

(a) What are the chances of England winning
the next soccer World Cup?

(b) What are the chances of your maths teacher
being away tomorrow?

(c) What are the chances that the day after
Wednesday will be Thursday?

(d) What are the chances that you will live to be
1000 years old?

Some events are certain: (c) above.
Some events are impossible: (d) above.
Some events are 'in between' certain and
impossible: (a) and (b) above.
The probability of an event is a measure of the chance of it
happening.
Probability is measured on a scale from 0 to 1. An event which is
impossible has a probability of 0.
An event which is certain has a probability of 1.
Events which are neither impossible nor certain have a probability of
between 0 and 1. For example, the probability of obtaining a 'head'

when tossing a coin is 1"·

Exercise 1 (For discussion)
What is the probability of the following events?

1. The day after Monday will be Tuesday.
2. The day after Saturday will be Tuesday.
3. When a pound is equally divided between ten people, each

person will receive 1Op.
4. When a coin is tossed, the result will be a 'head' or a 'tail'.
5. When an ordinary dice is thrown, the number showing will be

more than 8.

Probability 157

6. The letter 'a' appears somewhere on the next page of this book.
7. Liverpool will score over 100 000 goals in their next game.
8. Obtaining a total of one when two dice are thrown together.
9. Someone in the class is more than 3m tall.
10. There will be at least one baby born somewhere in Great Britain

on the first of April this year.

Experimental probability

The chance of certain events occurring can easily be predicted.
For example the chance of tossing a head with an ordinary coin.
Many events, however, cannot be so easily predicted.
For example what is the chance of a drawing pin landing point
upwards when dropped onto a hard surface?

tYou may argue that the chance is because it can only land 'point

t-up' or 'point down' so one out of two is

But a drawing pin is not symmetrical like a coin or a dice so you
cannot use that argument.

Suppose the bag shown below contains an
unknown mixture of marbles coloured red,
blue and green.

One marble is selected, its colour noted, and it
is then replaced in the bag. This procedure is
repeated 30 times and on 7 occasions a red
marble was selected.

The experimental probability of selecting a red

marble from the bag is 7 •
30

We call the selection of a marble from the bag
a 'trial'.

Expen.menta1 probab.11t. t yN=um-be-r -of-tri-al-s i-n -wh-ic-h -ev-en-t o-cc-ur-s
Total number of trials made

In gambling casinos, where large bets are involved, all the dice used
must by law be tested to make sure that they are fair. Much of the
early work on the theory of probability was carried out by
mathematicians who were interested in gambling.

158 Part 6

Exercise 2
Carry out experiments to work out the experimental probability of
some of the following events.

Toss two coins. What is
the chance of tossing
two heads?

Dice Pontoon
This is a game for two players using a dice.

Version 1

• Player A throws a dice as many times as he likes and keeps a
running total for his 'score'.
So if he throws 2, 3, 5, 1 his score is 11.

• If he throws the same number twice consecutively his score
returns to zero and his turn is finished.
So if he throws 2, 3, 5, 1, 1 his score is 0.

• He can decide to stop throwing the dice at any time to avoid the
risk of losing his score.

• Player B then has his turn and throws the dice following the rules
above.

• The winner is the player with the higher score.

Version 2

• The players take turns to throw the dice.
• If a player throws the same number that his opponent has just

thrown then his score goes to zero.
For example a game could go like this:

A3 AS A3
B4 Bl B3

Now B automatically loses because his score is zero.

Probability 159

• A player can decide to stop throwing at any time. If he does stop
his opponent may continue throwing on his own until someone
wins.

Here is another example of a game: B3 B4 B5 B5

A5 A6 A6
B2 Bl B2

So A is the winner because B's score is zero after throwing two
consecutive five's.

Notice that B would have won if he had thrown any number
apart from 5 on his last throw.

Think about the rules of the two versions of the game and decide
which version you think will give both players the most even
chance of winning.

Which version do you think involves more skill?

It is not easy to decide which version will give both players the
most even chance of winning.
One way to find out is to do an experiment.

Play each version of the game several times and make a Winner
tally chart, recording which player (first or second) won. First player Second player
Work out the experimental probability of each player
winning for each version. llit II II

For example, if you played the game 14 times and the
first player won 9 times, the experimental probability of
the first player winning would be (4 .

Write a couple of sentences to say what the results of the
experiments showed.

Try to think of any further changes to the rules which would make
the game more fair or perhaps more interesting. Play the game
using ycmr changes and decide if it really does give a better game.

160 Part 6

Expected probability

For simple events, like throwing a dice or tossing a coin, symmetry
can be used to work out the expected probability of an event
occurnng.

For a fair dice the expected probability of throwing a '3' is 1;.

For a normal coin the expected probability of tossing a 'head' is f

Expected probabt.1t.ty = -th-e-n-u-m-b-er-o-f'w'-ay-s-t-he--ev-e-nt..ca:n:.h.a:p.p.en-

the number of possible outcomes

Ten identical discs numbered 1, 2, 3, 4,
5, 6, 7, 8, 9, 10 are put into a bag. One
disc is selected at random.

In this example there are 10 possible
equally likely outcomes of a trial.

(a) The probability of selecting a '2' = /0

This may be written p(selecting a '2') = 1
10

(b) p (selecting an even number) = 5
10
=t

(c) p (selecting a number greater than 7) = ?o

Exercise 3

1. One card is picked at random from a pack of 52.
Find the probability that it is
(a) a Jack
(b) the King of hearts
(c) a diamond.

2. Ten discs numbered 1, 1, 2, 2, 4, 5, 6, 7, 8, 8 are placed in a bag.
One disc is selected at random.
Find the probability that it is
(a) an even number
(b) a six
(c) less than 3.

3. A bag contains the balls shown. One ball is

taken out at random. Find the probability

that it is R =red
W =white
(a) red (b) white (c) blue B =blue

One more red ball and one more blue ball

are added to the bag.

(d) Find the new probability of selecting a

red ball from the bag.

Probability 161

4. A dice has its faces numbered 2, 3, 3, 3, 4, 7.
Find the probability of rolling
(a) a '7'
(b) an even number
(c) a prime number.

5. One card is selected at random from the
nine cards shown.

Find the probability of selecting

(a) the ace of diamonds (b) a king

(c) an ace (d) a red card

6. If Mala throws a 3 or a 5 on her next throw when playing
'Snakes and Ladders' she will slide down a snake on the board.
What is the probability that she will avoid a snake on her next
throw?

7. The 26 letters of the alphabet are written on discs. The five discs
with vowels are put in bag A and the other discs are put in bag B.

Find the probability of selecting ( ) ( )

(a) an 'o' from bag A

(b) a 'z' from bag B AB

(c) a 'w' from bag A

vowels consonants

8. A golfer is practising shots from six feet. He In Miss
47
records how many either go in or miss the hole. 33
Estimate the probability that the next shot he

hits will go in.

9. A shopkeeper is keen to sell his stock of left-handed scissors. He
has read that 9% of the population is left-handed. What is the
probability that the next person to enter his shop is left-handed?

10. Helen played a game of cards with Michelle. The cards were
dealt so that both players received two cards. Helen's cards
were a seven and a four. Michelle's first card was a 10.

Find the probability that Michelle's second card was
(a) a picture card [a King, Queen or Jack]
(b) a seven.

162 Part 6

11. Each of the 11 letters of the word 'CALCULATION' was
written on a small card. One card was selected at random.
Find the probability that the letter drawn was
(a) an 'A'
(b) a letter which appears in the word 'DOG'
(c) a letter which appears in the word 'BEG'

12. A pack of cards is split into two piles. Pile P
contains all the picture cards and pile 0
contains all the other cards.
(a) Find the probability of selecting
(i) the Jack of hearts from pile P
(ii) a seven from pile 0
(b) All the diamonds are now removed from both piles.
Find the probability of selecting
(i) the King of clubs from pile P
(ii) a red card from pile 0.

13. Each letter in the words of the sentence below is written on a
separate card.
'I have told you a million times, don't exaggerate!'
The cards are placed in a bag and one card is selected at
random. Find the probability of selecting
(a) an 'a'
(b) a 't'
(c) a 'b'

14. A box contains 12 balls: 3 red, 2 yellow, 4 green and 3 white.
(a) Find the probability of selecting
(i) a red ball
(ii) a yellow ball
(b) The 3 white balls are replaced by 3 yellow balls. Find the
probability of selecting
(i) a red ball
(ii) a yellow ball.

15. One person is selected at random from the crowd of 14 750
watching a tennis match at Wimbledon. What is the probability
that the person chosen will have his or her birthday that year
on a Sunday?

16. One ball is selected at random from a bag containing x red balls

and y white balls. What is the probability of selecting a red ball?

Probability 163

Number of successes

When an experiment (like rolling a dice or toss'ing a coin) is repeated
several times we can calculate the number of times we expect an
event to occur. Call the event in which we are interested a 'success'.

Then expected number of successes = (probability of a success) x (number of trials)

The spinner shown has nine equal
sectors. How many S's would you
expect in 450 spins?

lProbability of spinning a 5 =

Expected number of S's in 450 spins

= p (spinning a 5) x number of spins

t= X 450

=50

t;A dice is biased so that: the probability of rolling a '6' =
t;the probability of rolling a '1' =

the probability of rolling each of

the numbers '2' '3' '4' or '5' = ...i.
'' 32"
The dice is rolled 160 times.

How many times would you expect to roll (a) a '1'.

(b) a '3'.

(a) expected number of l's = p(l) x 160

t= X 160

= 20 times
(b) expected number of 3's = p(3) x 160

i= 2 X 160

= 25

Exercise 4

1. One ball is selected at random from the bag R =red
shown and then replaced. This procedure is G =green
repeated 360 times. How many times would W =white
you expect to select:
(a) a red ball
(b) a white ball?

2. A fair dice is rolled 480 times. How many times would you
expect to roll:
(a) a 'two'
(b) an odd number?

164 Part 6

3. A spinner, with 12 equal sectors, is spun 600
times. How often would you expect to spin:
(a) a shaded sector
(b) an even number
(c) a vowel
(d) a prime number?

4. A bag contains eight discs numbered 2, 2, 2, 3, 3, 4, 4, 6. One
disc is selected at random and then put back. This procedure is
repeated 200 times.
How many times would you expect to select:
(a) a '2'
(b) an even number
(c) a prime number?

5. A coin is biased so that the probability of tossing a 'head' is
0·58.
(a) How many 'heads' would you expect when the coin is tossed
200 times?
(b) How many 'tails' would you expect when the coin is tossed
1000 times?

±6. For a dice the probability of rolling a '3' is written p (3).
A dice is biased so that: p (6) =
p(l)=/2
ip(2) = p(3) = p(4) = p(5) =
This dice is rolled 300 times. How often would you expect
(a) a '1'
(b) a '6'
(c) an even number
(d) an odd number?

7. The number of pupils attending Faldo College is 973. How
many of these pupils would you expect to celebrate their
birthdays on a Saturday in the year 1999?

8. Five cards are taken from a pack. These cards are shuffled and
one card is selected at random. This procedure is repeated until
30 cards have been selected. Here are the results:

K99K3 K99K9 3K9K3
3K99K 3K99K 9939K

Which do you think were the five cards used?

Explain your answer.

Probability 165

9. Six cards are taken from a pack. These cards are shuffled and
one card is selected at random. This procedure is repeated until
30 cards have been selected. Here are the results:
77337 Q3737 377QQ
77Q7Q Q7Q77 QQ7Q7
Which do you think were the six cards used?
Can you be certain?

10. A dice has its six faces marked
Op, Op, Op, Op, 5p, 20p.
In a game at a school fair players pay 5p to
roll the dice and they win the amount shown
on the dice.
During the afternoon the game is played 540
times.
(a) How much money would be paid by the people playing the
game?
(b) How many times would you expect the dice to show '20p'?
(c) How many times would you expect the dice to show '5p'?
(d) How much profit or loss would you expect the stall to
make?

11. At another stall at the fair players pay 20p
to spin the pointer on the board shown.
Players win the amount shown by the
pointer.
The game is played 800 times.
Work out the expected profit or loss on this
game.

12. A bag contains w white balls, y yellow balls and r red balls. One
ball is selected and replaced 1000 times. How many times would
you expect to select a white ball?

13. A coin is biased so that the probability of tossing a 'head' is x.
[x is a fraction]
(a) How many 'heads' would you expect when the coin is tossed
50 times?
(b) How many 'tails' would you expect when the coin is tossed
300 times?

166 Part 6

Listing possible outcomes

• When a 1Op coin and a 20p coin are 10p 20p
tossed together there are four possible
outcomes. head head
head tail
±·So, for example, the probability of tossing tail head
tail tail
two tails =

• Suppose a red dice and a blue dice are rolled together. This time
there are many possible outcomes. With the red dice first the out-
comes can be listed systematically:
(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)
(2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)
(3, 1) (3, 2)

(6,' 5) (6,' 6)

There are 36 equally likely outcomes

A neat way of listing the
outcomes is on a grid.

The point marked • shows a blue
four on the red dice and a five dice
on the blue dice.

The probability of rolling a

four on the red dice and a five 2 + ············· +·············+·············+·············+·············+

on the blue dice is 1 •
36

Exercise 5 23456
red dice

1. Roll a pair of dice many times and in a Total Frequency
tally chart record the frequency of
obtaining the totals from 2 to 12. 2
3

12

2. (a) Work out the expected probability of getting a total of 5
when two dice are rolled together.
Compare your answer with the experimental probability of
getting a total of 5 obtained in the experiment in Question 1.

(b) Work out the expected probability of other totals and
compare them with the experimental results.

3. A red dice is thrown first and then a blue dice is thrown.
(a) Find the probability that the score on the blue dice is the
same as the score on the red dice.
(b) Find the probability that the score on the blue dice is one

more than the score on the red dice.

Probability 167

4. Two dice are rolled together and the 5
difference is found.
In the grid the point X has a difference of 3 4
obtained by rolling a 2 and a 5. blue

Find the expected probability of obtaining a dice 3 ····+ •m+············+
difference of (a) 3
2 ..
(b) 0

23456
red dice

5. The spinner shown has six
equal sections on the outside
and three equal sections in the
middle.
The spinner shows a '5' and
an 'A'.

Find the probability of
spmnmg
(a) a 'C'
(b) a '7'
(c) a '6' and an 'F' at the same time

6. A bag contains a 2p coin, a 5p coin and a 1Op coin. Two coins
are selected at random.
(a) List all the possible combinations of two coins which can be
selected from the bag.
(b) Find the probability that the total value of the two coins
selected is
(i) 15p
(ii) 7p
(iii) 20p

7. The four cards shown are shuffled and placed face down on a
table.

••3
•• •~:

Two cards are selected at random.
(a) List all the possible pairs of cards which could be selected.
(b) Find the probability that the total of the two cards is

(i) 5
(ii) 9

168 Part 6

6.2 Problem solving 2

Three friends Alan, Ben and Chris have three
cars they can use: a Ford, a Honda and a Rover.
Alan likes the Ford and the Rover,
Ben likes the Ford,
Chris likes all three.
How can we arrange it so that everyone takes a
car they like?

One way is to draw a network diagram Ae eF
Be eH
(a) Draw 3 dots on the left for Alan, Ben, Chris. c• eR
Draw 3 dots on the right for Ford, Honda, Rover.

(b) Draw lines to show who likes each car.
So join A to F and R, B to F and C to F, Hand R.

(c) Now think of a solution
Show it with thick lines (or use colour)

The solution is for Alan to take the Rover,
Ben to take the Ford,
Chris to take the Honda.

Check that no other solution is possible.

Exercise 6

1. Three sisters are given sweets by their grandparents. There is a
packet of chocolates, a packet of opal fruits and a packet of
toffees.
Denise likes all three kinds of sweets.
Ellie likes chocolates and opal fruits.
Fran likes opal fruits .

Draw a network diagram with points D, E, De e C(Choc.)
F on the left and points C, 0, T on the e O(Opal F.)
right. Ee e T (Toffees)

Draw lines on the diagram to decide how the Fe

sweets should be distributed.

Problem Solving 2 169

2. Here is a network showing four teachers K, Teacher Class
L, M, N and the classes they like to teach K IR
lR, IV, lC, IT.
For example teacher K likes to teach 1V and L IV
lC.
Work out which teacher should take which M IC
class so that each teacher takes one class
only. N IT

3. Four travellers W, X, Y, Z have to make a journey, for which 4
tickets are available: by plane; by train; by car; by boat.

W prefers to go by plane or car
X prefers to go by boat
Y prefers to go by train or boat
Z prefers to go by train, car or boat.

Draw a network diagram and work out who should take each
ticket.

4. A mixed class of 4 boys and 4 girls are going
on a coach trip and their misguided teacher
thinks it would be 'nice' if boys sat next to
girls. The teacher asks the girls A, B, C, D
which of the boys E, F, G, H that they
know. Each girl can sit next to a boy she
knows.

A knows E, F
B knows F, G, H
C knows E, H
D knows F, H

Find two solutions where the girls all sit next
to boys that they know.

5. A class is electing five girls as captains of the following teams;
swimming; gymnastics; netball; athletics; tennis.
It is clearly best if the girl elected is good at the sport for which
she is to be captain.

Vera is good at swimming and gymnastics.
Wilma is good at gymnastics, netball and tennis.
Xenia is good at swimming and athletics.
Yasmin is good at netball.
Zara is good at netball and athletics.

Draw a network and work out the five games captains.

170 Part 6

6. In a sort of 'Blind Date' a computer agency A p
arranges marriages. The only requirement is B
that the intended couple are friends before Q
they marry! c
The diagram shows six men A, B, C, D, E, F R
and six women P, Q, R, S, T, U. D
A line indicates that they are friends. E s
F
Arrange the six marriages. T

u

7. Six actors A, B, C, D, E, Fare trying to decide who should play
the six parts in a play. The parts are:
the hero (H); the princess (P); the villain (V); the reporter (R);
the tree (T); the nurse (N).

A would like to be the princess or the reporter.
B would like to be the hero or the villain
C would like to be the hero or the nurse
D would like to be the villain or the tree
E would like to be the nurse or the tree
F would like to be the villain, the reporter or the tree.

Draw a network and work out who should play each part.
Find two different solutions.

6.3 Mental arithmetic

If you have to work out 49 + 47 in your head there are several ways

of doing it.

50..,...4-7=97 50+-50=100 =4-0+-4-0 =. 80
q+ 7 /6
~7-f-:::- 96 coo- 4- =::. '16 80+ lb =CJb

\

Do this experiment in your class.
Ask everyone to work out the answers to the following three
questions in their heads.
(a) 98 +56
(b) 103- 38

(c) 11 x 36

Mental Arithmetic 171

The interesting part of the experiment is to write down the different methods which the people in
your class have used. You may be surprised at the number of different ways that people choose
for the same calculation.
It does not really matter which method you use as long as you understand it.

There are several sets of mental arithmetic questions in this section. It is intended that a teacher
will read out each question twice, with all pupils' books closed. The answers are written down
without any written working. Each test of 30 questions should take about 20/25 minutes.

Test 1 16. If June 14th is a Tuesday what day of the
week is June 23rd?
1. What are 37 twos?
17. True or false: 1kg is about 2 pounds?
2. What is the smaller angle between the
hands of a clock at 8 o'clock? 18. How many millimetres are there in 3·5
metres?
3. Two angles of a triangle are 55° and 30°.
What is the third angle? 19. A daily newspaper costs 25p from
Monday to Saturday and 45p on Sunday.
4. What is 5% of £44? What is the total cost for the seven days?

5. How many 5p coins are needed to make 20. Work out a half plus a tenth and give the
£10? answer as a decimal.

6. A car costing £8500 is reduced by £120. 21. A clock ticks once every second. How
What is the new price? many times does it tick between stx
o'clock and seven o'clock?
7. What number is twice as big as sixty-nine?
22. Add eleven to nine times eight
8. On a tray fourteen out of fifty peaches are
rotten. What percentage is that? 23. A rectangular piece of wood measures 1·5
metres by lOcm. What is its area?
9. Add together 11, 18 and 9.
24. An egg box holds six eggs. How many
10. A C.D. cost £13·55. Find the change from boxes are needed for 100 eggs?
a £20 note.
25. How many 19p stamps can I buy for a
11. What five coins make 51 p? pound?

12. A train journey of 186 miles took three 26. Work out roughly 0-496 x 602·8.
hours. What was the average speed?
27. A till contains 22 five pence coins. How
13. Write one twentieth as a decimal. much is that?

14. How many minutes are there between 28. How many magazines costing 95p can I
8.15 p.m. and 10.20 p.m.? buy with £10?

15. A pools prize of six million pounds is 29. What is a half of a half of 3?
shared equally between one hundred
people. How much does each person 30. True or false: 3 feet is slightly longer than
receive? 1 metre?

172 Part 6

Test 2 22. Five pounds of carrots cost 75p. How
much do they cost per pound?
1. By how much is three kilos more than 800
grams? 23. Between 6 p.m and midnight the
temperature falls by 15°C. The
2. How many 20p coins do I need to make temperature at 6 p.m is 8°C. What is the
£400? temperature at midnight?

3. How many square centimetres are there in 24. Write down ten thousand pence in
one square metre? pounds.

4. How much more than £108 is £300? 25. What is a quarter of two hundred and
ten?
5. Two angles of a triangle are 44° and 54°.
What is the third angle? 26. Find two ways of making 66p using five
cams.
6. Work out 1% of £50.
27. How long does it take a car to travel
7. My watch reads ten past eight. It is 15 30 km at a speed of 60 km/h?
minutes fast. What is the correct time?
28. What number is exactly mid-way between
8. A SOp coin is 2 mm thick. What is the 12·1 and 12·2?
value of a pile of SOp coins 2 em high?
29. A can of drink costs 45p. How much do
9. Add together £2·35 and £4·55. you pay for six cans?

10. A ship was due at noon on Friday but 30. John weighs 8 stones and Jim weighs
arrived at 8.00 a.m. on Saturday. How 80 kg. Who is heavier?
many hours late was the ship?
Test 3
11. By how much is half a metre longer than
1 millimeter? (answer in mm). 1. What number is 10 less than ninety
thousand?
12. What number is thirty-five more than
eighty? 2. I want to buy 4 records, each costing
£4-49. To the nearest pound, how much
13. How many minutes are there in two and a will my bill be?
half hours?
3. The number of centimetres in one inch is
14. From nine times seven take away fifteen. 2·54. What is 6 inches to the nearest em?

15. A T.V. show lasting 45 minutes starts at 4. What is the total of 57 and 963?
10 minutes to eight. When does it finish?
5. How many quarters are there in one and
16. A train travels at an average speed of a half?
48mph. How far does it travel in 1±
hours? 6. Which of these numbers 2, 4, 5, 6, 8 is not
a factor of 560?
17. What is the perimeter of a square of area
144cm2? 7. A triangle has a base 2 em and an area of
10 cm2. What is its height?
18. A string of length 390 em is cut in half.
How long is each piece? 8. What number is exactly mid-way between
3·07 and 3·08?
19. A half is a quarter of a certain number.
What is the number? 9. Work out four squared plus five squared.

20. A man died in 1993 aged 58. In what year 10. The pupils in Darren's class are given
was he born? lockers numbered from 32 to 54. How

21. Roughly how many millimetres are there many pupils are there in Darren's class?

in one foot?

Mental Arithmetic 173

11. Write 7 divided by 100 as a decimal. 2. What number is three times as big as 55?

12. Jane is 35 em taller than William, who is 3. A bird remains airborn for five days. How
1·34 metres tall. How tall is Jane? many hours is that?

13. A toy train travels 6 metres in 3 seconds. 4. Liz buys three items at 12p each and pays
How far will it go-in one minute? with a £10 note. What change does she
receive?
14. Which is larger: 2 cubed or 3 squared?
5. The coordinates of 3 corners of a square
15. What number is next in the series 3, 9, are (0, 0), (0, 3) and (3, 0). What are the
27,0 0.? coordinates of the fourth corner?

16. To the nearest metre, what is 11·95 m 6. How many grams are there in 12-4
divided by 2? kilograms?

17. Joe borrowed £4·68 from his father. He 7. If 15 out of 60 children in a school can
paid him back with a £10 note. How swim, what percentage of the children
much change did he receive? cannot swim?

18. What is a tenth of 2-4? 8. What is the next number in the sequence
8, 16, 32?
19. I think of a number and subtract 6. The
result is equal to 7 times 3. What is the 9. Work out six squared times ten squared.
number?
10. One video lasts an hour and 22 minutes
20. A newspaper said that there are 1600 and a second video lasts two hours and
children in the local school. If this figure 45 minutes. What is the total time for the
is correct to the nearest hundred, what is two videos?
the least number of children that there
could be in the school? 11. If one angle in an isosceles triangle is 161 o
find the other two angles.
21. How much longer is 7·5 metres than 725
centimetres? t12. How many degrees are there in 1 full turns?

22. The mean weight of 3 apples is 120 grams. 13. Roughly how many 24p stamps can I buy
What is their total weight? for £5?

23. If p = 6 and q = 9, what is p + q divided 14. How many centimetres are there in 40 metres?

by 2? 15. What is the smallest number which can be
added to 34 to make the answer exactly
24. If one kilogram= 2·2 pounds, how many divisible by 8?
pounds does a 4 kg bag of flour weigh?
16. How many 50ml glasses can be filled
25. I think of a number, multiply it by 3 and from a one litre bottle?
add 4. The result is 10. What is the
number? 17. Find the difference between 7t and 20.

26. How many lines of symmetry does a 18. How many 2p coins are worth the same
square have? as twenty 5p coins?

27. What is a quarter of a half? 19. In a test Kate got 16 out of 20. What
percentage is that?
28. Answer true or false: 1km is longer than
1 mile 20. Work out one fifth plus a quarter and
give the answer as a decimal.
29. Work out 200 times 300.
21. I bought a magazine for 79p and paid
30. Write ten million pence in pounds. with a £1 coin. My change consisted of
five coins. What were they?
Test 4

1. A pile of 5p coins has a value of 80p.
How many coins are there?

174 Part 6

22. How many days are there m thirteen 12. What is the angle between the hands of a
weeks? clock at half past seven?

23. True or false: an ordinary family car 13. A drink costs 65p. I buy two drinks and
weighs about 20 000 grams? pay with a £10 note. Find the change.

24. I start with a half. If I add one and then 14. Find two ways of making 27p with five
double the answer what do I get? coins.

25. Add together 999 and 999. 15. If the first of May is a Monday, what day
of the week is the twenty-ninth of May?
26. I bought 2 kilograms of sugar and used
350 grams of it. How many grams of 16. Find the area of a stamp which measures
sugar do I have left? 18mm by 3cm.

27. A toddler weighs 2 stones 1 pound. How 17. How many spots are there on an ordinary
many pounds is that? dice?

28. How much more than £74 is £180? 18. What number do you need to add to 675
to make 999?
29. If cable costs 8 pence for 50 em, how
much will 6 metres cost? 19. If one angle of a parallelogram is 33°,
what are the other three angles?
30. A triangle has base 5 em and height 9 em.
What is its area? 20. How many numbers between 1 and 20
inclusive are square numbers?
Test 5
21. A metal pipe of length 15·2cm is cut
1. Work out 15% of £60. exactly in half. How long is each piece?

2. How many centimetres are there m 22. At what point does the line y = 7 cut the
10km? line y = x?

3. What is a half of a half of 0·2? 23. True or false: 8 miles is about 5 km?

4. A pie chart has a sector representing 10% 24. Between 6 pm and 11 pm the temperature
of the whole chart. What is the angle of falls by 10°C. The temperature at 11 pm
the sector? is -13°C. What was the temperature at
6pm?
5. A peach costs 37 pence. What is the total
cost of four peaches? 25. Work out a fifth plus a quarter and give
the answer as a decimal.
6. Telephone charges are increased by 25%.
What is the new charge for a call which 26. A rectangular card is 8 inches wide and
previously cost 60p? one foot long. Find the area of the card
in square inches.
7. Write down the next prime number after
32. 27. I am thinking of a number. When I
double the number and then add 11 the
8. How many edges does a cube have? answer 1s 19. What number am I
thinking of?
9. A boat sails at a speed of 16 knots for five
hours. How far does it go? 28. How many Sp coins are worth the same
as a hundred 2p coins?
st10. Multiply by 100.
29. Add together £2-40, 25p and 36p.
11. What is 0·1 divided by one tenth?
30. A pile of 1000 sheets of paper is 50 em

deep. How thick is each sheet in mm?

Puzzles and Games 6 175

6.4 Puzzles and games 6

Puzzles

1. King Henry has 9 coins which look identical
but in fact one of them is an underweight fake.
Describe how he could discover the fake using
just two weighings on an ordinary balance.

2. Write the digits 1 to 9 so that all the answers 3.,_------------------.

are correct. 1. I +- I :: .2.
2 • .2 ~ 2.. := l.,..
0-0=0
3. 3 y... 3 :=. b
X 4. 4- -:- 4- ::. 8

070=0 5. Exactly 2 statements
II in this box are true

0+0=0 How many statements in the
box are true?

4. Starting at the 5 and going around the spiral,
the spaces can be filled in with signs (+, -, x , 2 3 7
f7) so that you obtain the correct answer. 1 r-2--+!_,_! -4-t 1 ~

5 t_! 5 2

t

Check that the signs shown opposite do give 2 + !3 !- 7
the correct answer. X 2 = !4

5 5 !+ 2

t

Copy the following spirals and work out the missing signs. Work in
pencil so that you can rub out mistakes. There may be more than
one solution to each question but you only have to find one.

372 ~
10 II
lstit±J
534

tt

5. A polygon has exactly 20 diagonals. How many sides has it?

176 Part 6

Boxes: a game for two players • Draw around the border of a 10 x 10 square
on dotty paper (you can also use squared
~ .I paper).

0L. • Two players take turns to draw horizontal or
.I J vertical lines between any two dots on the
grid.
I~
• A player wins a square (and writes his initial
inside the square) when he draws the fourth
side of a square.

• After winning a square a player has one
extra turn.

• The winner is the player who has most
squares at the end.

In the game above J has
two squares so far and
R has one square.

Break the codes

1. The symbols y, j , !, -e-, l_ each stand for one of the digits 1, 2, 3,

5 or 9 but not in that order. Use the clues below to work out

what number each symbol stands for.

(a) i x i = l_

(b) B-x j=j

(c) -e- + -e- = y
(d) y + i =!

2. The ten symbols below each stand for one of the digits
0, 1, 2, 3, 4, 5, 6, 7, 8 or 9.

!::,. * 0d'0 CDWV Y ~

Use the clues below to work out what number each symbol

stands for.

d' +(a) !::,. = 0 cJ(f) X !::,.= 0
(b) d' X d' =!::,.
(g) y X y = V
*(c) + D = D
(h) y + CJ = CD
(d) W x D = D
(i) y +!::,. = ~
(e) d' X d' X d' = 0

3. The ten symbols used in part 2 are used again but with different

values.

The clues are more difficult to work out.

(a) * -!::,. = y Y_t(f) y + y +J' = w
(g) u - D = d
(b) W+ cJ = CD * -(h) w - (y + d') =!::,.
(i) v - 0 = 6.
+ +(c) !::,. D = y ~

(d) 0 + 0 = w

d(e) x CD = CD

Part 7

7.1 Multiple choice papers

Test 1 A4 8. Find the area A 96cm2
of triangle ABC. B 60cm2
1. Solve the equation B -4 All lengths are
4x- l = 0 cI in em. c 48cm2

D 41 A D None of
-4
9. ! of 14 = the above

2. A pile of 250 cards is A 0-4cm 10. £1600 decreased by 4%
1m deep. How thick is
each card? B 0-4m IS

c 2·5cm 11. In the triangle the size A 3
of the largest angle is 70
D 0·25cm

3. Without using a A £8-80 B 8·2
calculator, work out c 8-4
2·5% of £220 B £88
D 84
c £5 -50
A £1596
D £55 B £64

4. Find the size of angle a A 40° c £1200

B 50° D £1536

c 60° A 70°

D 70° B 80°

5. A train travels 20 km in A 6-4min c 90°
8 minutes. How long
will it take to travel B 9min D 100°
25 km at the same
speed? c lOmin

D 12·4min

6. The line y = 3 cuts the 12. What is the value of A 3
line y = -2 at the point 1 - 0·15 as a fraction? 20
A (3, -2)
with coordinates: B (-2, 3) 13. A car travels for 20 cB 17
minutes at a speed of 20
c (0, 3) 66 km/h. How far does
the car travel? 15
D None of 100

the above D Impossible

to find

7. Find the value of x A5 A l3·2km
3 x 5t-4=x-t
B6 B l320km

c 13 c 33km

D 14 D 22km

178 Part 7

14. Which of the A 1 only 22. The number midway A 3·105
statements between 3·1 and 3·11 is
is(are) true? B 1 and 2 B 3·111

1. a= b c 1 and 3 c 3·15
2. c =a
3. c = d D 1, 2 and 3 D 3·5

23. A quarter share of a A £40
fifth share of 10% of
£8000 is B £80

c £400

D £36

15. Which point does not lie A (1, 1) 24. The number of seconds A 9000
on the line y = x? in a day is about
B (0, 0) B 90000

c (-3, -3) c 30000

D (-2, 2) D 300000

16. A metal ingot weighing A 3·2kg 25. A quarter of A 2·2
64 kg is made into (8·8--;-- 0·01) is
20 000 buttons. What is B 3·2g Do not use a calculator. B 22
the weight of one
button? c 312·5g c 220

D 312·5 kg D 88

Test 2

17. The number of letters in A1 1. A floor measures 6 m by A 54
the word ROUTE that 41 m and is covered by
have line symmetry is B2 square carpet tiles B 108
measuring 50 em by
c3 50 em. How many c 120
carpet tiles are needed?
D4 D 216

18. What fraction of the A I
area of the rectangle is
the area of the triangle? 4

B I 2. Work out%+ 0·15 A 18

8 3. Solve the equation "ii4
cI 8=9-2x
16 B 45
400
D I
c9
32 10

D 0·09

A I

19. How many prime A2 2
numbers are there
between 10 and 20? B3 B -2I

c9 c 8_!_
2
D None of
D1
the above
4. Which of the following A and 2
1 and 3
20. The probability of A I statements are true? B 2 and 3
selecting an Ace at 52 c 1, 2 and 3
random from a pack of
52 cards is B I 1. 1% of 10 = 0·1

13 2. 12--;-- 5 = 2-4 D

c I 3. 7--;-- 8 = 0·875

4

D 3

52 A (-3, -3)
B (3, 3)
21. What is the area, in m2, A 0·0009 5. At what point will the
of a square with sides of line y = x cut the line c (-3,0)
length 0·03 m? B 0·009 X= -3?
D Impossible
c 0·06
to say
D 0·09

Multiple Choice Papers 179

6. A picture measuring A 200cm2 Use the graph below for questions 12 to 15.
30cm by 20cm is
surrounded by a border B 225cm2
5 em wide. What is the C 400cm2
area of the border? D 600cm2

7. Find the area of triangle A7
OAB in square units
B8
4 -+ · ···'·····=--,;.,---······+············+·· ·+····
c9
3 -1 +·I + !""""'++
D 10
2

8. A man's heart beats at A 6000
70 beats/min. How
many times will his B 8400
heart beat between 0330
and 2330 on the same c 72000
day?
D 84000

0900 1000 1100 Time

9. Which is the best A 19m 12. When does the car A 0900
estimate when 20 yards B 21m arrive in Harrogate?
is converted into metres. C 210m B 0930
D 1·9m
c 1030

D 1115

10. The pie chart shows the A 10·28° 13. When does the bus A 0900
nationalities of people leave Harrogate?
on a ferry. What angle B 35° B 0945
should be drawn for the
UK sector? c 126° c 1000

D 132° D 1126

14. At what speed does the A 30km/h
car travel on the return B 40km/h
journey to York? C 50km/h
D 100km/h

11. In how many ways can A1 15. How far apart are the A 2·5km
you join the square X to B2 car and the bus at 1015? B 5km
shape Y along an edge C3 C 10km
so that the final shape D4 D 22·5km
has line symmetry?
16. A roll of wallpaper is A2
10m long and 50cm
wide. How many rolls B3
of wallpaper are needed
to cover a wall 2·5 m c4
high and 8 m long?
D5

Xy

180 Part 7

17. What fraction of the A I 25. Without a calculator, A 0·0856
area of triangle ABC is work out B 0·7956
the area of triangle 4 3·72 X 0·23
c 0·8556
ADE? c B 3
D 8·556
4 c 10

D I
4
2

3

5

A 2B Test 3 A £177
B £155
18. The value of A 30 1. Car Hire
11 X 22 X 33 is c £77·13
B 36 £13 per day plus
14p per km D £771·30
c 72
Mr. Hasam hired a car A n=4
D 108 for 6 days and travelled B n = 36
550 km. How much did
19. What is the next A 76 it cost? c n=8
number in the sequence B 84
6, 13, 27, 55? 2. Find n if D n=9
c 111 72-;-n=9

D None of

the above

20. Which of the following A 1 only 3. Anna has £4·80. She A 84p
B 94p
statements are true? B 2 only t tspends on presents,
c £1 ·04
1. 200mm =2m C 3 only 1on a magazine and on
D 2 and 3 D £1·14
t2. %is greater than sweets. How much is

3. All rectangles have left?

four lines of symmetry.

21. Draw triangle ABC in A 53° 4. 1 kg= 2·2 pounds. A 3kg
which AB = AC. If Which is closest to
ACB = 53°, calculate B 64° 7 pounds? B 4kg
the size of ABC.
c 74° c 14kg

D 127° D 15kg

22. Four 1's are written in D 5. Which is the odd one Ab
the square so that no out? Bd
two 1's are on the same c Cf
row, column or Dg
diagonal. Where must B
the ' 1' go in the third I A
column?

23. How many wine glasses A3 6. Paul's car averages 35 A £342 ·50
of capacity 30 ml can be miles to 1 gallon of
filled from a barrel B 125 petrol. Petrol costs B £4195·62
containing 240 litres? £2-40 a gallon. If Paul
c 800 drives 28 770 miles in c £822
one year, how much
D 8000 D £1972·80
does his petrol cost?
24. A car travels half a mile A 1-mph
in a minute. What is its B 30mph
speed? C 60mph

D 120mph

Multiple Choice Papers 181

7. How many t's A2 14. There were 4 candidates At
Bt
@are there in B5 in a class election. Mary ct
Dt
l4?. c6 !got of the votes,
t,George got Henry got
D7 t. What fraction did

Sheena, the 4th

candidate get?

8. I left the cinema at A 19.50 15. What fraction of the Af
22 25. The film lasted 2 shape below is shaded? Bt
hours and 35 minutes. B 20.10
At what time did it ct
begin? c 19.10
D%
D 19.35

9. Which list is arranged in 16. Which pair of angles is A a and g
ascending order? equal? B band d
A 0·14, 0·05, 0·062, 0·09
B 0·14, 0·09, 0·062, 0·05 C c andf
D band e
c 0·050, 0·062, 0·09, 0·14

D 0·050, 0·090, 0·14, 0·062

10. A greengrocer sells 9 kg A 4·11
of potatoes for £2·52.
How many kg can be B 45 t
bought for £1-47?
c
11. How many lines of
symmetry does the D 5.1.
rectangle below have?
4
D
A1 17. A box has a mass of A 210g
230 g when empty.
B2 When it is full of sugar B 860g
the total mass is 650 g.
c3 What is its mass when it c 440g
is half full?
D4 D 420g

12. A window frame in a A 868cm2 18. Work out on a A 1·73
church measures B 174cm2 calculator, correct
24·3 em by 35·7 em. C 1080cm2 to 3 s.f. B 1·72
80% of the window is D 694cm2
filled with stained glass. -1-4+1-·9 c 1·728
What is the area of
stained glass in the 1·7 2·1 D 1·30
window?
19. 5a+2a-9-3a+4= 15 A a= 5
B a = 2·5
13. One litre of water A 220g C a= 0·5
weighs 1 kg. One litre of D a=2
ice weighs 870 g. What B 780g
is the difference in 20. 999 mm is equal to A 99m
weight between 5 litres c 1100 g B 9·9m
of water and 4 litres of C 0·99m
ice? D 320g D 0·099m

182 Part 7

7.2 Mixed exercises

Revision exercise 1
In the diagram three vertices of a rectangle are
given.
Find the coordinates of the fourth vertex and
write down the equations of any lines of
symmetry.

0 2 4 6x

Put the answers in a table:

Shape Vertices given Other vertex Lines of symmetry
(5, 1)
Rectangle (1' 1), (1' 3), (5, 3) X= 3, y = 2

Draw axes with values of x and y from -10 to +10.

Draw the shapes given and hence copy and complete the table.

Shape Vertices given Other vertex Lines of symmetry
(1, 6), (1, 10), (3, 6) ?
1. Rectangle (4, 3), (4, -1), (10, 3) ? ? '?
(5, -2), (10, -2), (5, -3) ? ? '?
2. Rectangle ? ? '?
3. Rectangle (5, -1 0), (1 0, -8) y = -8 only
4. Isosceles
(2, -4), (0, -8) ? x = 2 only
triangle
(-6, 4), (-8, 7), (-6, 10) ? ? '?
5. Isosceles (3, 3), (3, -3), (-3, -3) ? Give four lines.
(-8, -2), (-7, 1), (-5, 1) ?
triangle (-8, -5), (-4, -5), (-5, -8) x = -6 only
Give three none
6. Rhombus possibilities
7. Square none
8. Trapezium Give three
9. Parallelogram possibilities y = x is one line
Find three more.
10. Parallelogram (-3, 4), (-2, 6), (-1, 6) Give two points

11. Square (6, 6), (6, 9)

Mixed Exercises 183

Revision exercise 2

1. A cake recipe calls for 500 g of flour to mix with 200 g of sugar.
How much sugar should be used if you have only 300 g of flour?

2. Find the total cost:
5 bags of cement at £3·95 per bag
13m of wire at 40p per m.
8 sockets at 65p each.
Add V.A.T. at 17"4-%.

3. In a History test Joe got 27 out of 40.
What was his mark as a percentage?

4. Work out to the nearest penny

(a) 6%of£85·15 (b) 11·5%of£33-40

5. Here is a list of numbers (c) prime numbers.
5, 8, 9, 11, 12, 13, 17, 18, 20.

Write down the numbers which are
(a) factors of 40 (b) multiples of 4

6. Use a calculator to work out the following. Give your answers

correct to 3 significant figures.

(a) 18·3- (1·91 x 2·62) ( b )5-·23- -
9·2- 7-63

(c) ~8·91+ 1·54 2
0·97
(d) (1-4+0·761)

1·76

7. Write down the next term in each sequence.
(a) 75, 69, 63, 57
(b) 1, 4, 10, 22
(c) 8, 9, 11, 14

8. Calculate the shaded area -scm--+

7cm

II em

9. Solve the equations (b) 3x - 8 = x + 1
(a) 3x- 1 = 17

(c) 5x + 1 = 8 - 2x.

10. Draw an accurate copy of each triangle and find the length x
and the angle y.

(a)~ (b)

S.6cm Scm

184 Part 7

11. What fraction of each shape is shaded? (c)
(a) (b)

Revision exercise 3 y
5
1. (a) Draw a pair of axes with values from
-5 to 5. ~------------T------------+~x

(b) Draw and label the following lines -5 5
line A: x = 4
line B: y = -2 -5
line C: y = x

(c) Write down the coordinates of the point
where
(i) line A meets line B
(ii) line A meets line C
(iii) line B meets line C
(iv) line B cuts the y axis.

2. Work out the following, without using a calculator, giving your

answer as a decimal.
(a) ! of 17
(b) 22% of 30 (c) ~~of 40

3. (a) The diagram shows a trapezium. D 8cm c
Calculate the area of triangle ABD and
the area of triangle DCB. Hence write i
down the area of the trapezium ABCD.
Scm
B
l

(b) Calculate the area of each trapezium below.

(i) 7 em (ii) 5cm

+\~)~)-------lo.\

9cm

IOcm

4. Copy and fill in the missing numbers
(a) 2·3 m = em (b) 45 g = kg (c) 2 feet= inches
(d) 260m = km (e) 3 litres = ml (f) 1 pound= ounces

(g) 25 mm = em (h) 1 yard = feet (i) 2·6 kg = g

Mixed Exercises 185

5. A bag contains 6 coloured balls. One ball is selected at random
and then replaced in the bag. This procedure is repeated until

50 selections have been made. Here are the results:

[B = Blue, G = Green, Y = Yellow]

BY BY BY BY B G BY B
G BBBBYG BYBG BY
YBYYBBBG BBYB
BYBYG BY BYBG B

What do you think were the colours of the balls in the bag?
Justify your answer.

6. If 5 melons can be bought for £7, how many can be bought for
£12·60?

7. If 7 bags of flour cost £6-44, find the cost of 3 bags.

8. Find the angles marked with letters

(a) (b)
X

9. An opinion poll was conducted to find out which party people
intended to vote for at the next election. The results were.

Conservative 768
Labour 840
Lib Dem 612
Don't know 180

Work out the angles on a pie chart and draw the chart to
display the results of the poll.

10. A worker takes 8 minutes to make 12 items. How long would it
take to make 15 items?

11. Draw a pair of axes with values from 0 to 8.
Plot two corners of a square at (2, 4) and (6, 4).
Find the coordinates of the six possible positions for the other
corners of the square.

12. The diagram shows a rectangle. 2x+ 6
Work out x and then find the area of the
rectangle.

5
em

6x- 10

186 Part 7

Revision exercise 4

1. Work out the total cost, including V.A.T.
13 kg of sand at 57p per kg.
2 tape measures at £4-20 each
2000 screws at 80p per hundred
250 g of varnish at £6-60 per kg

V.A.T. at 171-% is added to the total.

2. Use a calculator to work out the following and g1ve your
answers correct to 3 significant figures.

(a) 8_62 _1 ·71 (b) 8-02-6·3 5·6 9·7
0·55 (c) 1·71- 11 ·3
1·32 + 4·6

3. A normal pack of 52 playing cards (without jokers) is divided
into two piles

Pile A has Pile B has the
all the picture rest of the pack.
cards (Kings,
Queens, Jacks)

Find the probability of selecting
(a) any 'three' from pile B
(b) the King of hearts from pile A
(c) any red seven from pile B.

4. The numbers A1, A2, A3 and
B1, B2, B3 make two sequences.

Work out (a) A4
(b) B7
(c) A100
(d) B1000

5. A grass playing field in the shape of the D c
quadrilateral shown is to be sprayed
with weedkiller at the rate of 3 grams of lOOm
weedkiller per square metre of grass.
The weedkiller costs £5 ·50 per kg. L-------------------~' B

(a) Draw an accurate scale drawing of 85m
the field using a scale of l em to 10 m.

(b) Find the areas of the triangles ABC
and ADC and hence find the total
area of the playing field in square
metres.

(c) Find the cost of the weedkiller
needed for this field. Give your

answer to the nearest pound.