Essential Mathematics. Book 1 (Rayner D.)

44 Part 2

6. Draw a sketch of an isosceles triangle ABC with AB = AC.
Point D lies on AC so that ADB = 90°.
If BAC = 40° calculate the size of CBD.

Triangle PQT is drawn inside square PQRS.
PTS = 2 x SPT and RQT = 22·9°
Calculate the size of angle x.

sT A

8. Draw a sketch of a triangle KLM with KL = KM and
KML = 78°. Point N lies on KM and line LN bisects KLM.
Calculate the size of angle LNM.

9. In~the diagram AC bisects
BAD and DE = AE.
Find the angles a and x .

B cD E

10. D r - - - - - - - - - - , c ABE is an equilateral triangle drawn inside square
ABCD.
E Calculate the size of angle DEC.

Decimals 45

2.2 Decimals

It is important to understand decimal numbers.
Decimals are another way of expressing fractions.

(a) 0·1 = -Ia, 0·3 = ?0, 0·02 = 1 ~0 ,

0-41 = 1~0 , 0·30 = ?go = ?o so 0-30 = 0·3,

(b) Which is larger: 0·8 or 0·72?
0·8 = 0·80 = 180~
0·72 = 17io
So 0·8 is larger than 0·72.

(c) Write in order of size, smallest first: 0·62, 0·26, 0·602, 0·3
Write each number with 3 figures after the decimal point.
i.e 0·620, 0·260, 0·602, 0·300
In order of size the numbers are 0·260, 0·300, 0·602, 0·620.

Exercise 7 t')j ~:: ;:•'P'• ~i},)
;t
1. There are 100 small squares in the large square. ·~ ~: '(.(

25 squares are shaded to represent 0·25 F.:-1 ;z~m
;'ij
Draw your own numbers (like 17 or 33) and shade in the ~ (;~; 1,>) :;::.
correct number of squares to represent the equivalent
decimal. Draw your numbers the same size.

In Questions 2 to 5 write down the line which is correct.

2. (a) 0·07 is equal to 0·7 3. (a) 0·23 is equal to 0·32
(b) 0·07 is greater than 0· 7 (b) 0·23 is greater than 0·32
(c) 0·07 is less than 0·7 (c) 0·23 is less than 0·32

4. (a) 0·03 is equal to 0·030 5. (a) 0·09 is equal to 0·1
(b) 0·03 is greater than 0·030 (b) 0·09 is greater than 0·1
(c) 0·03 is less than 0·030 (c) 0·09 is less than 0·1

In Questions 6 to 14 answer 'true' or 'false'.

[ > means 'is greater than', < means 'is less than']

t6. 0·2 = 7. 6·2 < 6·02 8. 6 = 6·00
10. 0·88 > 0·088
9. £3-40 = £3-4 11. 0·001 < 0·0001
13. lOp = £0·1
12. 0·6 > 0·59 t14. 0·75 =

In Questions 15 to 20 arrange the numbers in order of size, smallest first.

15. 0·79, 0·791, 0·709, 0·97 16. 0·3, 0·33, 0·303, 0·033

17. 1, 0·99, 0·989, 0·09 18. 1·2, 0·12, 0·21, 1·12

19. 0·08, 0·096, 1' 0-4 20. 0·008, 0·09, 0·091, 0·0075

46 Part 2

Find the number which the arrow is pointing to on each of the scales.

~(a) 4 6 (b) 8.1 8.2

IIIIIIIIII +

The middle number is 5. Each division is 0·02.
The arrow points to 8·16.
Each division is 0·2.

The arrow points to 4-4. ciiD

@ ~8.1 8.2
8.12 8.14 8.18
4~ 5 6

II I

Exercise 8

Work out the value indicated by the arrow.

1. 40 50 2. 2 3 3. 14 ~ 16

I II

4. 15 15.5 5. 0 ~ 6. 0 ~6

I II I II II I I I II I II

7. 0 10 8. 5 ~ 6 9 . 3.1 3.2

10. 0 ~ 0.1 11· 1.04 12· 0 ~ 1.5

II II I II II I II I I I I

13· 0.2 ~ 0.3 14· 100 ~ 200 15· 2

IIII IIIIII

16. 0 ~ 60 17. 60 ~ 100 18. 4 ~ 4.5

II I I I I I I I I I I I I I I ~I~I~~~~--L-~1~I II II

19. 2.4 ~ 2.8 20. 3.3 3~ 21. 0 ~ Q3

II II II II I II II I II I I I I

22. 70 80 23. 18 19 24· 3.1 3 . 15

II II II

Decimals 47

Basic arithmetic

In many of the questions in this book you are expected to use a
calculator to solve the problem when you have decided which

operation (+, -, x, -;-) is required. It is, however, sometimes still

necessary for you to work out the answers by traditional 'pencil
and paper~ methods.

(a) 5·6 + 19 + 0·24 (b) 8·54-;- 0·2 (c) 5·2 x 0·6

5·6 Multiply both numbers 5·2
by 10 so that you can
1'1 ·o divide by a whole ")(0·6
0·'-ft.. number.
3· \2..
21,-·81,. So work out 85-4-;- 2
There are two figures after
Line up the decimal 4.2·7 the decimal points in the
points as shown. 2.) 85 ·4- numbers being multiplied.
There are two figures after
(d) 743 X 52 the point in the answer.
We work out
(e) 578-;- 23 • 23 into 57 goes 2 times
(743 X 50) + (743 X 2) • 2x23=46
2.5 • 57-46=11
74.3 • 'bringdown' 8
23) 578 • 23 into 118 goes 5 times
)(':)~ • Sx23=115
L,.b~ • 118-115 = 3
37/50 • Answer is 25 remainder 3
/4- "?:,b II8
or 25-i;-
38636 I I5

3

Exercise 9

Work out, without a calculator

1. 19·2- 5·8 2. 11 + 5·2 3. 173 X 8 4. 868-;- 7
5. 98·2-;- 5 7. 73·2-;- 100 8. 5·1 X 100
6. 0·32- 0·025 11. 2900- 1573 12. 0·95 X 9
9. 42 + 0·72 + 5·3 15. 5·24 X 0·5 16. 8·52-;- 0-4
10. 5·48 -;- 4 19. 0·612-;- 0·03 20. 5·2 X 2000
13. 14 490 -;- 6 23. 0·3 X 0·3 24. 5·7-;- 100
14. 4000 - 264
17. 0·52 X 0·04 28. 315 X 72
18. 234 + 23-4 32. 0·714 X 0·34
21. 0·0924-;- 0· 7 36. 27 945 -;- 45
22. 0·72 - 0·065 40. 6071 -;- 28

The next 16 questions involve long multiplication and long division.

25. 213 X 23 26. 325 X 41 27. 612 X 37
29. 2·5 X 3·1 30. 3·12 X 5·3 31. 4·6 X 0·26
33. 2108-;- 17 34. 5170 -;- 22 35. 12 376-;- 34
37. 16 328-;- 52 38. 4824 -;- 18 39. 5295-;- 67

48 Part 2

Know your tables 7 3 4 8 5 6 9 12 2 II

You will (hopefully) know your multiplication tables up to 7
12 x 12. Here is a test square to check both the speed and
accuracy of your memory. you should try to make a copy of 3 12
the square with all 100 answers correct in about five or six 4 36
minutes. 8
5
Two numbers have been put in to show how the square works. 6
9
12

2
II

Operator squares
Each empty square contains either a number or an operation [+, -,
x, ...;-]. Copy each square and fill in the missing details. The arrows
are equals signs.

1. 2. 3.
97 + 56 -+ 73 X 8
574 -+ 1532

X + ·:·

9 X 25 -+

~ ~ [¥; ~ ' ¥0

:

485 + 112 -+ 234 + -+

4. 5. 19.6 7 -+ 6. 0.2 -+
8.42
X
X X !l
II
0.01 X 0.3 -+
~
~ :~ .,"~ l~7J:;~l

I!! -+
+

7. 8. 9. + -+ 906

1.22 X 4.6 -+

+ ,·· '·'- :. -
.'
'
-+ 20 X 52 -+
[¥::, K ~ 1¥ 11 ~
~ •.;,; 1, s~
~
.:y' '
5
440 X -+ 2.8 -+ 7.8 + 526 -+

Decimals 49

10. 11. 12.
4.6 X 5.3 -+ + 1.5 -+ 0.34 -+ 2.1

I

14 . ~ X ,,t'i:;

0.1 X 0.22 -+

t· ,l l ll'l ..

~

-+ 0.345 - -+

Designing squares -+
• Make up your own operator squares starting
-+
from a blank grid like the one shown. Try to
make your square difficult to solve, but give l
enough information so that it can be done.
-+
l

• The grid shown opposite is much more difficult -+
to fill (as you will discover!)
Try to make up one of these 'super operator' -+
squares.
ll
l
-+

Here is one that works. 8 -+ 80

10 X

2 X 2 -+ 4

lll

5 X 4 -+ 20

50 Part 2

Proportion

(a) If 11 litres of petrol costs £5· 72, (b) A farmer has enough hay to feed 5
find the cost of 27 litres. horses for 6 days. How long would
the hay last for 3 horses?
The cost of petrol is directly
proportional to the quantity The length of time for which the horses
bought. can be fed is inversely proportional to
the number of horses to be fed.
11 litres costs £5·72
5 horses can be fed for 6 days
1 litre costs £5·72 ...;- 11 = £0·52 1 horse can be fed for 30 days.
3 horses can be fed for 10 days.
27 litres costs £0·52 x 27
= £14·04

In the first example above it was helpful to work out the cost of one
litre of petrol.
In the second example we found the time for which one horse could
be fed.

To do these questions you need to think logically.

• If five men can paint a tower in 10 days, how long would it take
one man?

• If 33 books cost £280·50, how much will one book cost?

Exercise 10

1. If 7 packets of coffee cost £8·54, find the cost of 3 packets.

2. Find the cost of 8 bottles of wine, given that 5 bottles cost
£11·90.

3. If 7 cartons of milk hold 14 litres, find how much milk there is
in 6 cartons.

4. On an army exercise, 5 soldiers took 6 hours to dig a trench.
How long would it have taken 3 soldiers to dig an identical
trench?

5. A party of I0 people exploring a desert
took enough water to last 5 days.
How long would the water have lasted if
there had been only 5 people in the
party?

\..,_

\.

Decimals 51

6. A worker takes 8 minutes to make 2 circuit boards. How long
would it take to make 7 circuit boards?

7. On a rose bush there are enough greenfly to last 9 ladybirds
4 hours. How long would the greenfly last if there were only
6 ladybirds.

8. The total weight of 8 tiles is 1720 g. How much do 17 tiles
weigh?

9. A machine can fill 3000 bottles in 15 minutes. How many bottles
will it fill in 2 minutes?

10. A train travels 40 km in 12 minutes.
How long will it take to travel 55 km
at the same speed?

11. If 4 grapefruit can be bought for £2·96, how many can be
bought for £8·14?

12. £15 can be exchanged for 126 francs. How many francs can be
exchanged for £37·50?

13. Usually it takes 10 hours for 4 men to build a wall. How many
men are needed to build the same wall in 8 hours?

14. A car travels 280 km on 35 litres of petrol. How much petrol is
needed for a journey of 440 km?

15. 10 bags of corn will feed 60 hens for 3 days. Copy and complete
the following:

(a) 30 bags of corn will feed * hens for 3 days.
(b) 10 bags of corn will feed 20 hens for * days.
(c) 10 bags of corn will feed * hens for 18 days.
(d) 30 bags of corn will feed 90 hens for * days.

16. 4 machines produce 5000 batteries in 10 hours. How many
batteries would 6 machines produce in 8 hours?

17. Newtonian spiders can spin webbs in
straight lines.
If 15 spiders can spin a webb of
length 75 em in 30 minutes, how long
will it take 8 spiders to spin a webb
of length 200 em?

18. In the army all holes are dug 4 feet
deep. It takes 8 soldiers 36 minutes
to dig a hole 18 feet long by 10 feet
wide. How long will it take 5 soldiers
to dig a hole 20 feet by 15 feet?

19. It takes b beavers n hours to build a dam. How long will it take

b + 5 beavers to build the same size dam?

52 Part 2
16.6 em
2.3 Mixed problems

Exercise 11
1. Mark is paid a basic weekly wage of £65 and then a further 30p
for each item completed. How many items must be completed in
a week when he earns a total of £171 ·50?

2. What number, when divided by 7
and then multiplied by 12, gives an
answer of 144?

3. A lOp coin is 2mm thick. Alex has a pile of
1Op coins which is 16·6 em tall. What is the
value of the money in Alex's pile of coins?

4. An Air France Concorde leaves Paris at 07 00 and arnves m
New York at 10 20.
A PanAm 747 leaves Paris at 0710 and flies at half the speed of
the Concorde. When should it arrive in New York?

5. Percy's garden is 48 m long and 10m wide
and he wants to cover it with peat which
comes in 60 kg sacks.
10 kg of peat covers an area of 20m2.
How many sacks of peat are needed for
the whole garden?

6. Find two numbers which multiply together to give 60 and which
add up to 19.

0 0 0X = 60, + C) = 19

7. A shopkeeper buys coffee beans at £4·20 per kg and sells them
at 95p per 100 g. How much profit does he make per kg?

8. A Jaguar XJ6 uses 8 litres of petrol for every 50 km travelled.
Petrol costs 56p per litre. Calculate the cost in £'s of travelling
600km.

9. A school play was attended by 226 adults, each paying £1 ·50,
and 188 children, each paying 80p. How much in £'s was paid
altogether by the people attending the play?

Mixed problems 53

10. The exchange rate in Germany is 2-85 Marks to the pound. In a
German shop a television is priced at 598·50 Marks. What is the
equivalent cost in £'s?

Exercise 12
1. A man smokes 50 cigarettes a day and a packet of 20 costs
£2-58.
How much does he spend on cigarettes in six days?
2. As an incentive to tidy her bedroom, a girl is
given lp on the first day, 2p on the second day,
4p on the third day and so on, doubling the
amount each day.
How much has she been given after 10 days?

3. A shopkeeper has a till containing a large number of the
following coins:
£1; SOp; 20p; lOp; 5p; 2p; lp.
He needs to give a customer 57p in change. List all the different
ways in which he can do this using no more than six coins.

4. Place the following numbers in order of size, smallest first:
0-34; 0-334; 0-032; 0-04; 0-4.

5. A book has pages numbered 1 to 300 and the thickness of the
book, without the covers, is 12 mm. How thick is each page?

6. In an election 7144 votes were cast for the two
candidates. Mr Dewey won by 424 votes. How
many people voted for Dewey?

7. The tenth number in the sequence 1, 4, 16, 64 ...... is 262 144.
What is (a) the ninth number,
(b) the twelfth number?

8. Two fifths of the children in a swimming pool are boys. There
are 72 girls in the pool. How many boys are there?

54 Part 2

9. Two weights m and n are placed on scales and m is found to be
more than 11 g and n is less than 7 g. Arrange the weights 8·5 g,
m and n in order, lightest first.

10. A satellite link between Britain and Australia can be hired at a
cost of £250 per minute from 0600 to 1400 and at £180 per
minute after 1400.
The link is used to televise a football match which starts at 13 30
and ends at 15 22.
How much does it cost?

Exercise 13

1. Lisa is 12 years old and her father is 37 years older than her.
Lisa's mother is 3 years younger than her father. How old is
Lisa's mother?

2. 36 small cubes are stuck together to make the block

shown and the block is then painted on the outside.

How many of the small cubes are painted on:

(a) 1 face (b) 2 faces

(c) 3 faces (d) 0 faces?

3. Find the letters in these additions.

(a) 8 7 A (b) A 2 4 5

3B5 5B8 4

+c 4 2 +1 4 c 6

D8 4 E0 5 2 D

±4. Grass seed should be sown at the rate of of an ounce per

square yard. One packet of seed contains 3lb of seed. How
many packets of seed are needed for a rectangular garden

measuring 60 feet by 36 feet? [3 feet= 1 yard, 16 ounces= 1lb]

5. Work out, without a calculator (c) 0·94 + 5·6

(a) 100-;- 10000 (b) 0·1 x 0·4 (f) 4318-;-17

(d) 4·32-;- 0·3 (e) 246 x 32

6. Petrol costs 56·2p per litre. How many litres can be bought for
£17? Give your answer to the nearest litre.

7. A flight on Concorde takes 2 h 36 min. How long would the
same flight take on a plane travelling at half the speed of

Concorde?

Mixed problems 55

8. Seven oak trees were planted in Windsor when Queen
Victoria was born. She died in 1901 aged 82. How old
were the trees in 1993?

9. A mixed school has a total of 876 pupils. There
are 48 more boys than girls. How many boys
are there?

10. Four 4's can be used to make 12: 44+4
4
(a) Use three 6's to make 2
(b) Use three 7's to make 7
(c) Use three 9's to make 11
(d) Use four 4's to make 9
(e) Use four 4's to make 3

11. A Rover 216XL travels 7-4 miles on a litre of petrol and petrol
costs 53p per litre. In six months the car is driven a total of
4750 miles. Find the cost of the petrol to the nearest pound.

12. In a code the 25 letters from A to Y are obtained from the
square using a 2 digit grid reference similar to coordinates.
So letter 'U' is 42 and 'L' is 54.
The missing letter 'Z' has code 10.

5G A p c Q

~ 40 F R H L

"So
:a
K B M y
""0 3 N
<=
u0
s uQ)

~2 D I E

J T v wX

234 5
[First digit]

Decode the following messages:

(a) 41, 52 (b) 44, 25, 31, 52 (c) 22, 35, 42, 34, 22
25, 34, 52
13, 52, 52, 12 25 34, 42, 33, 33, 32, 22, 44

43, 14, 34, 52 13, 32, 45, 52

22, 42, 43, 22 12, 25, 53

In part (d) each pair of brackets gives one letter

(d) (tof140),(72 +5),(7x8-4),(42 + 3 2 , (!of110), (2671)
)

(3 X 7 + 1), (83 - 31 ), (2 X 2 X 2 X 2 + 5)
(100- 57), (42 - 2), (17 X 2), (151- 99)

(2 X 2 X 2 X 5 + 1), ( t of 56), (2 X 3 X 2 X 3 - 2), (52 - 2).

(e) Write your own message in code and ask a friend to decode
it.

56 Part 2

Exercise 14
1. Unifix cubes can be joined together to make different sized
cuboids.

If the smaller cuboid weighs 96 g, how much does the large
cuboid weigh?

2. A lorry is travelling at a steady speed of 56 m.p.h. How far does
the lorry travel between 10.50 a.m. and 11.05 a.m.?

3. A map has a scale of 1 to 100 000. Calculate the actual length of
a lake which is 8 em long on the map.

4. In a 'magic' square the sum of the numbers in any row, column
or main diagonal is the same.

(a) 3 (b) 14 72

84 X 12

7X 5 9 16

15 3

5. This table shows the approximate weights of coins

lp 2p 5p lOp 20p
3·6 g 7·2 g 3·2 g 6·5 g 5·0 g

(a) What is the lightest weight with a value of 12p made from
these coins?

(b) A group of mixed coins weighs 228 g, of which 48 g is the
silver coins.
What is the value of the bronze coins?

6. The test results of 50 students are shown below.

Mark 5 6 7 8 9 10

Frequency 0 2 12 17 10 9

What percentage of the students scored 8 marks or more?

Mixed problems 57

7. On the 30th June 1994 the day was extended by 1 second to
allow for the irregularity in the speed of rotation of the Earth.
A newspaper carried an article stating that people in Britain eat
54 digestive biscuits every second. How many digestive biscuits
are eaten in a normal day?

8. Which of the shapes below can be drawn without taking the pen

from the paper and without going over any line twice? ~

w~

9. A corn field is a rectangle measuring 300m by 600 m. One
hectare is 10000m2 and each hectare produces 3·2 tonnes of
corn. How much corn is produced in this field?

10. Four and a half dozen eggs weigh 2970 g. How much would six
dozen eggs weigh?

11. The numbers 1 to 12 are arranged on the star
so that the sum of the numbers along each line
is the same.

Copy and complete the star.

12. A jar with 8 chocolates in it weighs 160 g.
The same jar with 20 chocolates in it
weighs 304 g. How much does the jar
weigh on its own?

Exercise 15

1. A floor measuring 5 m by 3·6 m is to be covered with square tiles
of side 10cm. A packet of 20 tiles costs £6·95. How much will it
cost to tile the floor?

2. A small boat travels 350 km on 125 litres of fuel. How much fuel
is needed for a journey of 630 km?

3. Work out (a) 11 + 22 + 33 + 44

(b) 31 X 42 X 53 X ...... X I9T X 1120·

4. A shop keeper bought 30 books at £3-40 each and a number of
C.D.'s costing £8-40 each. In all he spent £312. How many
C.D.'s did he buy?

5. It costs 18p per minute to hire a tool. How much will it cost to
hire the tool from 0850 to 1115?

58 Part 2

6. In a restaurant six glasses of wine cost £7·50.
How many glasses of wine could be bought for
£22?

7. When a car journey starts, the mileometer
reads 23 715 miles. After half an hour the
mileometer reads 23 747 miles. What is the
average speed of the car?

8. A rectangular box, without a lid, is to be made
from cardboard. The cardboard costs 2 pence
per cm2•
Find the cost of the cardboard for the box.

9. The numbers '7' and '3' multiply to give 21 and add up to 10.
Find two numbers which:
(a) multiply to give 48 and add up to 19.
(b) multiply to give 180 and add up to 27.

10. The words for the numbers from one to ten are written in a list
in alphabetical order. What number will be third in the list?

11. The diagram shows a corner torn from a sheet of graph
paper measuring 18 em by 28 em.
Calculate the total length of all the lines drawn on the
whole sheet of graph paper.

12. Sima has the same number of 1Op and SOp coins. The total value

is £9. How many of each coin does she have?

Mixed problems 59

Exercise 16

1. How m,any 26p stamps can be bought for £10 and how much
change will there be?

2. If £1 is equivalent to $1-42,
(a) how many dollars are equivalent to £600,
(b) how much British money is equivalent to $400?

3. A horse starts a journey at 06 10 and runs
50 km at an average speed of 20 km/h. At
what time will he finish?

4. A suitcase is packed with 24 books and 85 magazines. The total
weight of the suitcase and its contents is 8·85 kg. The empty
suitcase weighs 880 g and each book weighs 70 g. Find the
weight of each magazine.

5. Cake mix is sold in two shops in different sized boxes. Which is
the better value for money:
a 1·2kg box for £1·02 or a 2kg box for £1·78?

6. I think of a number. If I add 5 and then multiply the result by
10 the answer is 82. What number was I thinking of?

7. A boat sails 2-4 km in 30 minutes. How long will it take to sail
one km?

8. Show how the 3 by 8 rectangle can be
cut into two identical pieces and joined
togethPT to make a 2 by 12 rectangle.

111111111

9. What is the smaller angle between the hands of a clock at
(a) Half past two
(b) (harder) Twenty past six.

10. The 31st of March 1993 is written 31- 3- 93. This a special date
because 31 x 3 = 93.
(a) How many such dates are there in 1995?
(b) How many such dates are there in 1996?
Give the actual dates in each case.

60 Part 2

2.4 Area

We use area to describe how much surface a shape has.

B contains 10 squares. C has an area of 121- squares.
B has an area of 10 squares. D has an area of 12 squares.

Rectangles

A 2 em by 3 em rectangle may be split into 6 squares as shown. .. I em ..

The area of each square is one square centimetre (1 cm2), so t 1
the area of the rectangle is 6 cm2.
I em 2cm
This result can be obtained by multiplying the length by
+ j

the width.

Area of rectangle = (3 x 2) cm2 3cm

= 6cm2

Exercise 17

Calculate the total area of each shape. The lengths are in em. ,2 4. :~!
1. -5-

'I)3. 8. 3

5 ~6

5.2hl 6. 2~
:o IT.'',L;J3 7. 8

7

9. 10. 11. Find the shaded 12. 3 3
area
53 6 5

8 lD 9 7 6
2 3 2Q] 2

10

\2..

Area 61

Triangles

(a) This triangle has base b, height h and a right angle at A.
Area of rectangle ABCD = b x h

C D By symmetry,
area of triangle ABD = area of triangle CDB.

Area of triangle ABD = b x h
2

(b) Here triangle PQR is drawn Area of triangle PQR =area CD +area Q)
inside a rectangle.

Area of rectangle = (2 x area (D) + (2 x area Q))

:. Area of . PQR = bxh
tnang1e --

2

Notice that this is the same formula as above.

(c) In the triangle below the height Area CD = Area of rectangle - [area Q) + area G)]
(3 units) is measured 'outside' =21-[3~3+3~7]
the triangle

= 21 - [15]
= 6 square units.

We see again that the formula b x h gives the area of
2

the triangle since

bxh 4x3 .
- - = - - = 6 square umts.
22

For any tn.angle, area = ( base x height )
2

base base base

62 Part 2

Exercise 18 4.
Find the area of each shape. In questions 1 to 8 give the answer in
square units.

1. 2. 3.

5. 6. 7. 8.

In questions 9 to 15 the lengths are in em. 11. 4
9.

3

78 1
---------'r- ----- --
9
'
J
3

9

13. 14. Find the shaded area. 15. Find the shaded area.

54 7

1

10 4

j 8 18

3 ~t19.
14cm
In Questions 16 to 23 the area is written inside the shape. Calculate
the length of the side marked x .
16. 17. 18.

12 cm2 5 em

X
X

Area 63

21. ~ 22. X 23.
cmv~r 8~ct20.
X

Irregular shapes

It is not easy to find the exact area of
the triangle shown because we do not
know either the length of the base or
the height.

We could measure both lengths but this
would introduce a small error due to
the inevitable maccuracy of the
measuring.

• A good method is to start by draw-
ing a rectangle around the triangle.
The corners of the triangle lie either
on the sides of the rectangle or at a
corner of the rectangle.

Calculate the area of the rectangle.
In this example: Area of rectangle = 3 x 4

= 12 square units.

• Now find the areas of the three triangles marked A, B and C.
This is easy because the triangles each have a right angle.
Use the symbol '6A' to mean 'triangle A'

Area of 6 A = 4 x 1 = 2 square units
2

Area of /\ B = -2x- 2 = 2 square um.ts

u

2

Area of 6 C = 3 x 2 = 3 square units
2

Now we can find the area of the required triangle by subtracting

the areas of 6 A, 6 B and 6 C from the area of the rectangle.

Area of shaded triangle= 12 - [2 + 2 + 3]

= 5 square units.

64 2. Part 2

Exercise 19 6.
Find the area of each shape.

1.

! 5.
I
I

1: :: :: :: .

····- ~ ----!..---J·-·---·1---- ~---

4.

For Questions 7 to 13 draw axes with x and y from 0 to 10.
Plot the points given, join them up in order

7. (1 , 7), (3 , 2), (7, 6). 8. (4, 2), (6, 8), (l , 3).

9. (1, 2), (3, 6), (7, 5), (5, 3). 10. (3, l), (8, 3), (5, 6), (l, 4)

11. (2, 0), (7' l), (8, 4), (5, 5), (l ' 3) 12. (3, 1), (6, 0), (9, 2), (7, 5), (3, 6), (1, 4)

13. (2, 3), (5, l), (II, 5), (9, 7), (5, 6),
(2, 8), (0, 6), (2, 3).

14. A triangle and a square are drawn on
dotty paper with dots l em apart.
What is the area of the shaded region?

15. A triangle is drawn inside a regular
hexagon. What is the area of the
triangle as a fraction of the area of
the hexagon?

Area 65

Find the connection

You need square 'dotty' paper or ordinary
squared paper.

(a) Draw any triangle ABC and shade it.
(b) Draw a square on each of the sides of

the triangle.

(c) Join PQ to form L:. PQC (marked (D).
Join RS to form L:. RSB (marked Q)).

Join UT to form L:. UTA (marked Q)).

(d) Find the areas of triangles ABC, PQC,

RSB, UTA. You may be able to use the

formula ( base x height ) or you may
2

need to draw construction lines as in the

questions above.

(e) What do you notice? Draw another

triangle and repeat the procedure. Do

you get the same result?

Mixed area problems 15

Exercise 20

1. The diagram shows a picture 20 em by
15 em surrounded by a border 5 em wide.
What is the area of the border?

2. A rectangular lawn 17m by 8 m is
surrounded by a path 1m wide. What is
the area of the path?

3. A wall measuring 5 m by 3·5 m is to be covered by square tiles
measuring 50 em by 50 em. How many tiles are needed?

4. A rectangular area 3m by 1·2 m is to be covered by paving slabs
measuring 16 em by 40 em. What is the least number of slabs
needed?

5. How many panes of glass 35 em by 10 em can be cut from a
sheet llOcm by 110cm? What is the area of glass wasted?

66 Part 2

6. A flag has a sloping strip drawn across.
Calculate the area of the shaded strip.

7. A rectangle has a perimeter of 28m and a length of 6·5 m. What
is its area?

8. The field shown is sold at auction for 380 yards
£55 250. Calculate the price per acre which
was paid.
[1 acre = 4840 square yards]

500 yards

9. A groundsman has enough grass seed to cover three hectares.
[lhectare = 10000m2] . A tennis court measures 15m by 40m.
How many courts can he cover with seed?

10. A rectangular field 350m long has an area of 7 hectares.
Calculate the perimeter of the field.

11. A waterproofing spray is applied to the outside of the 4
walls, including the door, and the roof of the garage
shown.
(a) Calculate the total area to be sprayed.
(b) The spray comes in cans costing £1·95 and each can
is enough to cover 4m2. How much will it cost to
spray this garage? [Assume you have to buy full cans].

12. -+-3m--. A gardener is using moss killer on his lawn. The

2mt Sbed \ instructions say that 4 measures of the mosskiller,
in water, will treat 10m2 of lawn. The box contains
I Lawn 250 measures and costs £12 ·50.
Find the area of the lawn and hence the cost of the
lOrn moss killer required.

+I lol, J I3m

j ll T1 I
•~•~---13m----•._5m-.

Area 67

13. The shaded triangle is drawn inside a rectangle with
longer side 12 em.

(a) If area of triangle @ = 2 x (area of triangle (D),

find the length x .

(b) If area of triangle @ = 3 x (area of triangle (D), find

the length x.

14. In a recent major survey of children's mathematical ability only length
1 in 20 of fifteen year olds gave the correct answer to the
following question: i
'Find the length of the rectangle if the area is tcm2.' Area = f cm2 2 cm

Calculate the length. 5

L....--~

Maximum flow

A hill farmer wants to make a long chute to transfer water from a
mountain stream down to one of his fields . He has a large number
of flat metal sheets which can be bent to form a channel.
The width of each sheet is 20 em and he can
make the channel as long as he likes by
joining several sheets together.

He wants to bend the sheet in such a way that he can obtain the
maximum possible flow of water to his field. He realises that he
requires the area of the cross-section of the channel to be as large
as possible.

He decides to bend the sheets in one of two ways:

(a) as three sides of a rectangle, or (b) as two sides of a triangle

- -- -- __--__--__--__--
-

Your task is to decide how the sheet should be bent so that the area

of the cross-section is as large as possible. Iem 1 em
For example the channel shown on the right has a
cross-sectional area of 42 cm2. 3 3

~------1-4e-m------~

When investigating the 'V' shaped channels you will
need to make accurate drawings with different
angles x between the sides.

68 Part 2

2.5 Sequences

Sequences are very important in mathematics. Scientists carrying out
research will often try to find patterns or rules to describe the results
they obtain from experiments.
Codewriters and codebreakers can use complicated sequences to
work out new codes for transmitting secret information.
When answering questions on sequences try to keep an 'open' mind
so that you can find rules by 'trial and error'.

Here are three sequences. Try to find the next number.

• 5, 8, 12, 17, ?

• 2, 2,4, 12,48, 240J
• 15, 14, 16, 13, 17,?

Exercise 22

1. Find the next number in each sequence

(a) 5, 10, 20, 40, (b) 1, 6, 11, 16,
(c) 8, 9, 11, 14, 18, (d) 780, 78, 7·8,
(e) 54, 52, 49, 45, (f) 54, 18, 6, 2,
(g) 100, 10, 1, 0·1, (h) 1, 2, 6, 24, 120,

2. In the sequence 2, 5, 14, .... . . each new term is found by
multiplying the last term by 3 and then subtracting l . Find the
next term in each of these sequences

(a) 4, 7, 13, 25, 49,
(b) l, 4, 13, 40, 121
(c) l, 5, 13, 29, 61

3. Find the missing numbers. (b) (c)
(a) ~

4. Multiplying by 999, we obtain this sequence

l X 999 = 999
2 X 999 = 1998

3 X 999 = 2997

(a) Write down the next two lines of this sequence.
(b) Write down the answer to 9 x 999 without writing down any

more lines in the sequence.

Sequences 69

5. l N3 N4

14
II 13 15 17

12 16

The numbers Nl, N2, N3, N4 and Ml, M2, M3, M4 form two
sequences.
(a) Find MS, M6, NS, N6
(b) Think of rules and use them to find M15 and N20.

6.

7 13 19
6 8 12 14 18 20
5 9 11 15 17 21
4 lO 16 22

l C1 C2 C4
(a) Find CS, DS
(b) Use a rule to find ClO and D30

7. Bl

4
··············+ .. ···· ! ················· ········-······

3
2

Use a rule to find AlO, B20, A35 .

70 Part 2

8. The odd numbers can be added in groups to give an interesting
sequence

1 1 (1 X 1 X 1]
3+5 8 (2 X 2 X 2]
7+9+11 27 (3 X 3 X 3]

The numbers 1, 8, 27 are called cube numbers. Another cube
number is 53 [we say ' 5 cubed']

53 = 5 X 5 X 5 = 125

(a) Write down the next three rows of the sequence to see if the

sum of each row always gives a cube number.

(b) Now look at a similar sequence using the even numbers.
2 2 [Hint: 13 +1]

4+6 10

8 + 10 + 12 30

Write down the next three lines of the sequence and work
out the sum of each row.
Give a rule which describes the numbers you obtain.

9. A large 'T' can be drawn inside the number square so that all 5
numbers in the T are inside the square.

2345678

9 10 II 12 13 14 15 16

The T can be moved around but it must stay upright.
The 'T-number' is the number in
the middle of the top row. So
this is Tl8.

17 18 19

26

34 I

(a) What is the smallest possible T-number?

(b) Work out the total of the numbers in T21

Seuences 71

(c) Work out, as quickly as you can
(Total of numbers in T37)- (Total of numbers in T36)

(d) Fill in the numbers for T75

(e) Use x's to write the numbers for Tx

10. A famous sequence in mathematics is Pascal's triangle.

A

I/ B

2 /1 /

33 I

/
464
/
s / 10 10 s

6 IS 20 IS 6

(a) Look at the 'hockey stick' shapes. or

Write down what you notice.
Does it make any difference where
the hockey stick is drawn?

(b) Look at the diagonal marked A.
Predict the next three numbers in the sequence 1, 3, 6, 10, 15, ....

(c) Look at the diagonal marked B.
Predict the next three numbers on that diagonal.

11. (a) Work out the sum of the numbers in each row of Pascal's
triangle. What do you notice?

(b) Without writing down all the numbers, work out the sum of
the numbers in the lOth row of the triangle.

12. (a) What is the sum of all the numbers in the triangle down to
and including (i) the 3rd row,
(ii) the 6th row?

(b) Predict the sum of all the numbers in the triangle down to
and including the lOth row.

72 Part 2

13. [For enthusiasts] Draw Pascal's triangle down to the 15th row.
Use a computer to do this if possible.
Colour in all the even numbers. What do you observe?

Sequences in coordinates
In the next exercise sequences of triangles or squares are drawn and
you are asked to work out the coordinates of various points.
Look at the sequence of the coordinates for the first few shapes.
For example, if a sequence started (1, 5), (2, 10), (3, 15)

the next term will be (4, 20)
and the 50th term will be (50, 250).

Exercise 23

1. Here is a sequence of touching triangles.

y ..,.. ........Ll ..li. Find the coordinates of:
(a) the top of triangle 5
l d3)4 ! <4b) ! <6b) I (b) the top of triangle 50
3 =~~~f (c) the bottom right corner of triangle 50
2 (d) the bottom right corner of triangle 100.

2 3 4 5 6 7 8x

2. Write down the coordinates of the centres of y !•••••••••••• iw••••••••••••+ •!••••••••••• • + ••••••• •••••+ '··•••••••
squares 1, 2 and 3.
Find the coordinates of: 8~• ••••••••••••••••+
(a) the centre of square 4
(b) the centre of square 40 6
(c) the top right corner of square 4 5
(d) the top right corner of square 40. 4
3
2

2 3 4 5 6 7 8x

Sequences 73

3. y Find the coordinates of
the top vertex of:
5 (a) triangle 4
4 (b) triangle 20
3 (c) triangle 2000.

2 3 4 5 6 7 8 9 10 II 12 X

4. Write down the coordinates of the centres of y
squares 1, 2 and 3. 7 -+ ·············-+ ····· ··+···············-+··············~
Find the coordinates of:
(a) the centre of square 4 6
(b) the centre of square 10
(c) the top vertex of square 70. 5

4

2 3 4 5 6X

5. Write down the coordinates of the centres of
the first six squares.
Find the coordinates of:
(a) the centre of square 60
(b) the centre of square 73
(c) the top left corner of square 90
(d) the top left corner of square I01

y
5

4 - ····

3

2

III
2 3 4 5 6 7 8 9 10 II 12 13 X

74 Part 2

Count the crossovers ><
Two straight lines have a maximum of one crossover

Three straight lines have a maximum of three crossovers. A

Notice that you can have less than three crossovers if the *
lines all go through one point. Or the lines could be
parallel.
In this work we are interested only in the maximum
number of crossovers.

Four lines have a maximum of six crossovers.

~ Draw five lines and find the maximum number of crossovers.

~ Does there appear to be any sort of sequence in your results?
If you can find a sequence, use it to predict the maximum number
of crossovers with six lines.

~ Now draw six lines and count the crossovers to see if your
prediction was correct.
[Remember not to draw three lines through one point.]

~ Predict the number of crossovers for seven lines and then check if
your prediction is correct by drawing a diagram.

~ Write your results in a table: Number of lines Number of crossovers

(a) Predict the number of crossovers for 20 2 1
lines. 3 3
4 6
(b) [Harder] Predict the number of 5
crossovers for 2000 lines. 6

Puzzles and Games 2 75

2.6 Puzzles and games 2

Puzzles

1. The totals for the rows and columns are given. Unfortunately some
of the totals are hidden by ink blots. Find the values of the letters.

(a) (b)

A A A A 28 A B A B B 18

A B c A 27 B B E c D 21
A c D B 30
A B B A B 18
DBBB
c B c B c 19

• 25 30 24 E B D E D 26

27 10 25 23 17

The next two are more difficult. (d)

(c) A B B A 22

A A A A 24 A A B B 22
A B A B 22
c A c D 13
B B A B 17
A B B A 18

B B D c 12

• 18 15 18 27 17 22 17

2. In these triangle puzzles the numbers For example:
a, b, c, d are connected as follows: (c)
a x b=c
c X b= d

Copy and complete the following triangles:
(a) (b)

(d) (e) (f)

76 Part 2

In Questions 3 to 6 each letter stands for a different single digit from
0 to 9. The first digit of a number is never zero.

3. (a) M E (b) K L M
M E+ LM
AM
M L M+
[Find two solutions] L MM

(c) N A v E Work out E, V, A and S. You will find that
wA v E
vR A N, W and R can each have three different

E+ values. [SEVE is a shortened version of

s EvE 'SEVERIANO'].

4. Some of these questions have more than one solution.

(a) 0 N E (b) c u T (c) s 0 N
c0 T su N
0 N E+ F+ I s+
T w0
0 wN
0 AT

(d) T 0 u R (e) F 0 u R
F v E+
s0 uR

R 0 A R+ N NE

p ERR

5. These two look similar but the second question is more difficult

(a) H I T (b) H T
T H E+
T H E+
BAL L
BEL L

6. This question has only one solution.

B0 0 M

B 0 0 M+
sM
LE

D7. The box represents a mathematical rule.

DFor example if 3 5 = 11

Dand 4 1 = 9,

Dthe rule for is 'twice the first number plus the second number'

Find the rules in (a) and (b) below

D(a) 3 2= 7 D(b) 4 1= 6

D5 1 = 14 D5 4= 13
D6 3 = 15 D2 6 = 14

Puzzles and Games 2 77

Happy numbers 1

• (a) Take any number, say 23.

(b) Square the digits and add: 22 + 3 2 = 4 + 9 = 13
(c) Repeat (b) for the answer: 12 + 3 2 = 1 + 9 = 10
(d) Repeat (b) for the answer: 12 + 0 2 = 1

23 is a so-called 'happy' number because it ends in one.

• Take another number, say 7.

Write 7 as 07 to maintain the pattern of squaring and adding the
digits.
Here is the sequence: 07

.j \,

0 + 49 = 49

.j \,

16 + 81 = 97

.j \,

81 + 49 = 130
.!~\.

1+9 + 0 = 10

.j \,
1 +0

So 7 is a happy number also.

With practice you may be able to do the arithmetic in your head
and write: 07 -+ 49 -+ 97 -+ 130 -+ 10 -+ 1.

You may find it helpful to make a list of the square numbers 12, 22,
32, 00.92.

• Your task is to find all the happy numbers from 1 2 3 4 5 6 7 8 9 10
1 to 100 and to circle them on a grid like the

one shown. 11 12 13 14 15 16 17 18 19 20

This may appear to be a very time-consuming 21 22 23 24 25 26 27 28 29 30
and rather tedious task! 31 32 33 34 35 36 37 38 39 40
But remember: Good mathematicians always 41 42 43 44 45 46 47 48 49 50
look for short cuts and for ways of reducing the 51 52 53 54 55 56 57 58 59 60
working.

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 . 79 80

So think about what you are· doing and good luck! 81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
As a final check you should find that there are 20
happy numbers from 1 to 100.

Part 3

3.1 Using algebra

Many problems in mathematics are made easier to solve when letters
[x, y, a, n etc.] are used in place of numbers. This is using algebra.
Computer programs rely heavily on algebra for instructions which
are clear and unambiguous.
Mathematicians in countries throughout the world use the same
rules of algebra and they find it easy to communicate even if they
cannot speak the same language.

This section begins with the 'number walls' investigation where
algebra can be used at the end to find rules which are difficult to
find using only numbers.

The second section concentrates on some of the basic rules of
algebra which must be understood before harder problems can be
solved.

Number walls 379
Here we have three bricks with a number written inside each
one.

A wall is built by putting more bricks on top to form a sort
of pyramid.

The number in each of the new bricks is found by adding
together the numbers in the two bricks below like this:

Here is another wall.

Using Algebra 79

A • If you re-arrange the numbers at the bottom, does it affect the
total at the top?

• What is the largest total you can get using the same numbers?
• What is the smallest total?
• How do you get the largest total?

B • What happens if the bottom numbers are
(a) the same? [e.g. S, S, S, S]
(b) consecutive? [e.g. 2, 3, 4, S]

• Write down any patterns or rules that you notice.

C • What happens if you use different numbers at random? [eg 7,
3, S, 11]

• Given 4 numbers at the bottom, can you find a way to predict
the top number without finding all the bricks in between?

D • Can you find a rule with 3 bricks at the bottom, or 4 bricks?
Can algebra help? Hint:

Simplifying expressions

The expression Sa + 2a can be simplified to 7a. This is because
Sa+ 2a means five a's plus two a's, which is equivalent to seven a's.
(It can also be remembered as'S apples+ 2 apples= 7 apples'.)
Similarly, the expression 7c- 3c can be simplified to 4c.

The expression lOx+ x can be thought of as lOx+ lx which can be
simplified to llx.
Similarly, 9a- a can be thought of as 9a- la which can be
simplified to 8a.
Notice that when we simplify an expression we are not finding its
value when a is a particular number. We are just rewriting the
expression in a simpler form.
Some expressions cannot be simplified.
The expression 7x + 2x consists of two terms, 7x and 2x.
The expression Sx + 3y consists of two terms, Sx and 3y.

7x and 2x are called like terms.
Sx and 3y are called unlike terms.
An expression which is the sum or difference of two terms can only
be simplified if the terms are like terms.

80 Part 3

Exercise 1

Simplify as many of the following expressions as possible. This exercise could be done orally.

1. 7x- 2x 2. 4a +Sa 3. 3a + 2b 4. 9y- 2y
6. Sc- 2d
5. Sx + 4x 7. 4x+ 3 8. 9d+ d
10. 6d- 4
9. 13y- y 14. 13x- 9x 11. 6x + 3y 12. 4h + 2h
18. 7d- 3d 16. 3a + b
13. 7y- Sy 15. 7a +a
22. 9c + Sc 19. lOa- 4a 20. 17t- 2t
17. 4- 2x 24. Sc- S
26. 9a + 9 23. Sc- c 28. llb- b
21. 19b + 3b
27. llb- 11
25. 9a +a

Collecting like terms

The expression 12x + 3y- 4x + 9y cannot be simplified down to a

single term, but we can simplify the two 'x' terms and the two 'y'
terms separately.

This process is called 'collecting like terms'.

The sign in front of the term is part of the term. If we change the
order of terms the sign in front of each term stays with the term.
We can emphasise this fact by drawing loops around each term to
include the sign.

(a) Simplify 7x + Sy + 2x- 3y

@@@@=@@@@

i = 9x+ 2y

no sign ]

[ means+

(b) Simplify Sa - 2x - a + 2x

®@9@=®9@@

=4a

Exercise 2

Simplify the following expressions as far as possible by collecting like terms.

1. 7x + 3y + 2x + Sy 2. 9x + 2y- 3x + y 3. Sa+ 6y- 2a- 4y

4. lit+ 7- t- 4 5. 8y- 3 + y + 7 6. 9x + 2b- 8x- 7b

7. 6a+ 10-2a-2 8. 6h-2y+3h+9y 9. 8y-3+y+9
10. 4x- 10- x + 3 12. 7y + S + 7y- 4
11. x + l2y + 3x- 2y

13. 3a- 2c + Sc- 2a 14. Sx + 2y- 7y + Sx 15. 7d- 4 + 10- 6d
16. Sa+ 2c- 2a- Sd 17. lOx+ 7 - 4x + x
20. a- 4c + 2a + lOc 18. 6x- 2y + x + 4
19. 1ly- 3 + 2y- 2
21. 8d- S- 7d + 9

22. 4a- 11 + 2 + 6a 23. 14a + 13c- 2a- 8c 24. 2 + 3y + 7- 2y

25. 4y-2x+8y+5x 26. 6c+l3d-7d+4c 27. 8a+5y+2a-y

28. 9a+c-8a+c 29. Sx+lly-2y+9 30. 6a+3x-2a+10a

Using Algebra 81

Using algebra to explain connections

(a) Draw a 3 (b) Add the 3

triangle and numbers lying

write any three at either end

numbers at the of each side

corners. and write the

answer in the

6 7 middle of the 6 7

side.

(c) (i) Add the three corner numbers: (d) Here is 5
another 19
3 + 6 + 7 = 16 triangle where
the side
(ii) Add the three side numbers: numbers are
found in the
9 + 10 + 13 = 32 same way.

8 II

corner numbers: 5 + 8 + 11 = 24
side numbers: 13 + 16 + 19 = 48

(e) In both triangles it looks as if the sum of the side numbers is
twice the sum of the corner numbers. To show that this
connection is always true it is not enough to simply try lots of

examples with random numbers. It might not work with
fractions or decimals or negative numbers.

(f) Use algebra by calling the corner numbers c
a, b, c (or any other letters).

(g) Find the side numbers as before. ab
c

a a +b b

(h) Add the three corner numbers: a+ b + c
Add the three side numbers: a + c + a + c + c + b

So we can see that the sum of the side numbers is always twice
the sum of the corner numbers for any values of a, b and c.

82 Part 3

Exercise 3

1. (a) Write any 3 2 (b) Add 4 to 8 and write 2

numbers at the the answer here. ---.__

corners of a Add 2 to 8 ~

triangle and write and write the 12 , ',8 13
another number
inside the triangle. answer here.~:----
Add 5 to 8 as : ',,

5 4 shown. 5 10 4

(c) Add the numbers along each side: (d) Repeat the 3

5+10+4=19 calculations for

2+12+5=19 a triangle with

2+13+4=19 different

This suggests that the sum of the numbers numbers.

along each side is the same.

79

(e) Now use letters a, b, c, d to represent any numbers. Will a
the sum of the numbers on each side always be the
same? [Do not write 'yes' or 'no'. Write a short sentence
to say what you find].

b c

2. (a) Draw a square and write any four 3.----------,5
numbers at the corners.

9 13

Add the corner numbers and write 3 .-----22-----, 5
the answer in the middle of the line
opposite as shown. 18 12

9 8 13

(b) Add the four corner numbers: 3 + 5 + 13 + 9 = 30
Add the four side numbers: 18 + 22 + 12 + 8 = 60
This suggests a rule that the sum of the side numbers is twice

the sum of the corner numbers.

Using Algebra 83

(c) Show that the rule works for a square with numbers 3, 7, 11, 15
at the corners.
Show that the rule works for a square with numbers a, b, c, d at
the corners.

3. (a) Draw a square and 3 7 (b) Add these two 3 7
write any two
numbers at the top numbers and put
corners.
the answer in the

middle of the 10
square.

(c) Add 3 to 10 and 3 7 (d) Add the top corner numbers:

write the answer 3 + 7 = 10

opposite the 3. Add the bottom corner numbers:

Add 7 to 10 and 10 17 + 13 = 30
write the answer

opposite the 7. This suggest that the sum of the bottom

corner numbers is three times the sum of

17 13 the top corner numbers.

(e) Show that the rule works for a square with numbers 5 and 8
at the top corners.
Show that the rule works for a square with numbers x and y
at the top corners.

4. (a) Draw a rectangle and write any two numbers at 3 r - - - - - - - 2_ _ _ _ _...,4
the top corners and one in the middle.

(b) Add these three numbers and write the answer in 3 ,...,....._ _ _ _.....;2....__ _ _ _...., 4
the middle of the rectangle.

(c) Add 3 to 9, 2 to 9 and 4 to 9 and write the
answers as shown.

13 11 12

(d) Add the numbers across the top: 3+2+4=9

Add the numbers across the bottom: 13 + 11 + 12 = 36.

(e) Draw a rectangle with letters a, b, c across the top. Work
out the other four numbers as above.
Write down the connection between the sum of the top
numbers and the sum of the bottom numbers.

84 Part 3

5. Any three numbers [e.g. 2, 3, 7] are written at the corners of a 2 7
triangle.
Each corner number is increased by 4 and the answer is written 36
on the side opposite.
/
Sum of corner numbers= 2 + 3 + 7 = 12
Sum of side numbers = 11 + 6 + 7 = 24 (2 + 4)

What do you notice?

Draw a triangle with letters a, b, c at the corners and repeat the
calculations.
What do you notice? Is the sum of the side numbers always
twice the sum of the corner numbers?

6. (a) Draw a square and write four numbers at the
corners and a fifth number in the middle. Four
triangles are formed.

7

8

(b) Add the numbers at the corners of each triangle and
write the answer inside the triangle.

(c) Add the numbers inside opposite triangles: 15 + 16 = 31
18+13=31.

(d) Is the sum of the numbers inside opposite triangles always

the same?
Draw and label a square with letters a, b, c, d, e.

Solving Equations 85

7. Draw a square with diagonals and write any numbers at the top 3 .,---------- 2
corners and one in the middle as shown. 71
7
(a) Add the numbers at the corners of the top triangle and
write the answer inside the triangle and at the bottom 21
right corner of the square.

(b) Add the numbers at the corners of the right hand
triangle and write the answer inside the triangle and at
the bottom left corner of the square.

(c) Add the numbers at the corners of the bottom triangle
and write the answer inside the bottom triangle. Add
the numbers at the corners of the left hand triangle and
write the answer inside the left hand triangle.

(d) Compare the sums of the numbers inside opposite

triangles [12 + 40 =52; 31 + 21 =52]

(e) Repeat the calculations using letters a, b, c where 3, 2
and 7 were above.
Is the sum of the numbers inside opposite triangles
always the same?

3.2 Solving equations

An equation is a mathematical statement which has an equals sign.

Examples: 7x- 3 = 4; 5y- 3 = 2y + 9

Equations are like weighing scales which are balanced. The scales
remain balanced if the same weight is added or taken away from
both sides.

• On the left pan is an unknown weight

x plus two 1kg weights. On the right , 0 II II 11 \

mffbI
pan are four 1kg weights. ~

If two 1kg weights are taken from ,0 I \ nlJl
~
each pan the scales are still balanced.
So the weight x is 2 kg.

• Here is another set of balanced scales

\~where the weights are again in kg.

Take an x weight from both sides. ,@0
I

Take a 2 kg weight from both sides. \CXXX1
Two x weights are 8 kg.
So each x weight is 4 kg.

86 Part 3

Exercise 4
Find the weight x by removing weights 'in your head' [or by putting
a finger over them to show they are removed]. Weights are in kg.

1. 2.

01

3, \(XXX) [5] I 4. ,cflb [5]
I
I
6. ,cffi-2
\ffiffiffiS, \ 0I [iT2] I
8.
II \ (XXY) [2] I

?. \ffiffi tB I Rffi,@0~1 I
II
10.
. JPlb9. [!] I
\~1 12.
I

11.

Exercise 5

In this exercise the symbols D, 6, 0 and * represent weights which

are always balanced. Use logic to answer the questions.

1. (a) (b) 0 * (c)

000 !':,.!':,.!':,. 0?

~ ~
How many 6's?

2. (a) D*O 66 (b) 0 (c)

~ *D* 66 *?

~ ~

How many D's?

Solving Equations 87

3. (a) *0 =DODD 4. (a) 0 = !'::.!'::.
(b) 0 =DOD (b)
(c) (c) *=DO
* * * = How many D's? (d) 0 = 0!'::.!'::.
* = How many f'::.'s?

5. (a) * = !'::.00 6. (a) *D = !'::.0
(b) 0 = 00 (b) DO=!'::.
(c) *0 = 0!'::.!'::. (c)
(d) 0 0 = How many f'::.'s? (d) 0 = 00
* = How many O's

7. (a) !'::.!'::.=DO 8. (a) *D = 00
(b) 0!'::. = 0
(b) Of'::.=* (c) *0 =DODD
(d) D D = How many f'::.'s?
(c) !'::.0 = *D

(d) !'::.!'::.!'::.!'::.=How many D's

9. (a) * = !'::.0 10. (a) 000 = *0
(b) !'::.=ODD (b) ** = 000
(c) (c) D* = 00
(d) *0 = !'::.!'::.0 (d) * = How many D's?
0 = How many D's?

11. (a) *!'::. = 0
(b) * = !'::.0
(c)
(d) **!'::.=000
!'::. = How many O's?

Rules for solving equations

Equations are solved in the same way as we solve the weighing scale problems. The main rule
when solving equations is: 'do the same thing to both sides'.
You may add the same thing to both sides.
You may subtract the same thing from both sides.
You may multiply both sides by the same thing.
You may divide both sides by the same thing.

Solve the equations (b) 7x + 3 = 9
7x + 3- 3 = 9- 3 [subtract 3 from
(a) 4x- 1 = 8
7x = 6 both sides]
4x- 1 + 1 = 8 + 1 [Add 1 to both
4x = 9 sides]

4 = 2_ [Divide both sides by 4] 7x = ~[Divide both sides by 7]
x
44 77

X -- 214. 6

X=-

7

Exercise 6 2. 3x + 4 = 16 3. 5x- 4 = 6

Solve the equations for x. 5. 3x+ 5 = 32 6. 6x- 11 = 1
8. 9x = 2
1. 2x- 3 = 1 9. 4 + 3x = 5
4. 2x+ 1 = 2 11. 10 + 20x = 11
7. 6x = 5 12. 8 +x = 19

10. 3 + 7x = 31

88 Part 3

In Questions 13 to 24 solve the equations to find a.

13. 3a- 1 = 2 14. 5a+ 7 = 9 15. Sa+ 5 =53
17. 3 + 6a = 4
16. 25a - 7 = liS 18. S + 5a = 10

19. 13a- 7 = 32 20. 3 + 5a = 3 21. 5a- 11 = 7

22. 9a - 20 = 16 23. Sa+ 9 = 9 24. 7a- 2 = 1

----· --·-
Solve the equations where the 'x' terms are on the right hand side.

(a) 25 = 7x- 10 (b) S = 6 + 3x
25+ 10 = 7x-10+ 10 [Add 10 to S - 6 = 6 + 3x - 6 [Subtract 6 from
both sides]
35 = 7x both sides]

2 = 3x

35 = ?x [Divide both sides by 7] ~ = 3x [Divide both sides by 3]

77 33
l3 -- x
5= X

Notice that x is written on the right hand side throughout.
------- -· -·-·-- --.
.

In Questions 25 to 36 solve the equations to find x.

25. 4 = 3x+ 1 26. 9 = 5x- 1 27. 15 = 6x- 3
29. 0 = 5x- 6
28. 20 = 4x- S 30. 17 = 13 + 3x
32. 3 = 2 +4x 33. 4 + 9x = 4
31. S = 5 + 2x 36. 71 = 3x + 69
35.llx+ll=l2
34. 35 = 25 +40x

In Questions 37 to 48 find the value of the letter in each question.

37. 3y- 7 = 1 38. 20 = 4c + 10 39. 4 = 4 + 5t
41. S+ 5m = 90S
40. 6p -7 = 17 44. S = 1OOb + 7 42. 2x- 3 = 0

43. 7 = 5u- 73 47. 3x - 1- = 11- 45. 66 = 6 + 6n

46. 2x +1- = 1 t48. 2t + = 1-

Equations with the unknown on both sides

(a) 5x + 1 = 2x + 7 (b) 4x- 3 = 2x + 9 [A]
4x - 2x = 9 + 3
5x + 1 - 1 = 2x + 7- 1 [Subtract 1]
2x = 12
5x = 2x + 6
5x- 2x = 2x + 6- 2x [Subtract 2x] X= 6

t3x = 6 In this solution we have performed
x = [Divide by 3] two operations at the same time.
In line [A] we have added 3 to both
X= 2 sides and also subtracted 2x from
both sides.

In both equations the method was to put all the 'x' terms on one side of the equation and
all the number terms on the other side.

Solving Equations 89

Exercise 7 2. 4x - 3 = 2x + 5 3. 6x + l = x + 16
5. 6x + 5 = 3x + 7
Solve the equations for x. 6. 4x - 10 = x - 4
8. 7x+9 = 3x+9
1. 5x + 2 = 3x + 14 9. 7x- 4 = 3x + 1
4. 4x - 7 = 3x + 8 11. 8 + 3x = 17 + x 12. 9 + 1Ox = 5x + 10

7. llx- 6 = 5x

10. 5x + 3 = 2x + 13

In Questions 13 to 21 solve the equation to find n.

13. 7n - 5 = 6n - 4 14. 8 + 9n = 6n + 9 15. 7 + 4n = 10 + n

16. l2n + 100 = 2n + 200 17. 3n- 8 = 2n- 8 ±18. 3 + lln = 8n + 33

19. 5n + t = 3n + 1 t20. 4n- = n t21. 6n + = 5n +

(c) 2x + 14 = 6x - 2 (d) 5- 4x = 1 + 3x
14 + 2 = 6x- 2x
16 = 4x 5-1 = 3x +4x

4 = 7x

4 =X .7± = x

In both equations we have taken the x terms to the right hand side because there were

more x terms on the right hand side at the start. This method avoids the need to have

negative x terms which are a common source of error.

In Questions 22 to 36 solve the equation to find x.

22. 4x + 3 = 8x + 1 23. 3x + 1 = 5x - 7 24. 4x + 1 = 5x - 2
25. 2 + 2x = 1Ox 27. 18- 2x = 9 + x
28. 11 - 3x = 2x + 1 26. 6 + 3x = 5 + 9x 30. 10 - 5x = 7 + 2x
31. 8x + 4 = 3x + 5 33. 13 + 2x = 9 + 4x
34. 6 - 3x = 5 + 2x 29. 8 - x = 6 + 2x 36. 8 - 3x = 4x + 8
32. 1Ox + 1 = 8 - x

35. 2x - 7 = 7 - 5x

Using equations to solve problems

Philip is thinking of a number. He tells us that when he
doubles it and adds 7, the answer is 18. What number is
Philip thinking of?

Suppose that Philip is thinking of the number x

He tells us that 2x + 7 = 18

Subtract 7 from both sides: 2x = 11

Divide both sides by 2 x -- l l2

x = 5t

tSo Philip is thinking of the number 5

90 Part 3

Exercise 8

In each question I am thinking of a number. Use the information to
form an equation and then solve it to find the number.

1. If we multiply the number by 3 and then add 2, the answer is 13.

2. If we multiply the number by 5 and then subtract 3, the answer
is 9.

3. If we multiply the number by 6 and then add 11, the
answer is 16.

4. If we multiply the number by 11 and then subtract 4,
the answer is 7.

5. If we double the number and add 10, the answer is 30.

6. If we multiply the number by 10 and then subtract 4,
the answer we get is the same as when we multiply the
number by 7 and then add 2.

7. If we multiply the number by 6 and subtract 1, the answer we
get is the same as when we double the number and add 5.

8. If we multiply the number by 7 and add 3, the answer we get is
the same as when we multiply the number by 2 and add 5.

9. If we treble the number and add 10, we get the same answer as
when we multiply the number by 9 and add 8.

10. If we double the number and subtract from 7 we get the same
answer as when we treble the number and add 2.

11. If we double the number, subtract 11 and then add the original
number we get the same answer as when we subtract the
number from 9.

Harder problems

The diagram shows a square.
Find the length of each side of the square
and hence find the area of the square.

A square has equal sides. Sx- 8

So: 5x - 8 = 3x + 2
5x- 3x = 2 + 8

2x = 10

X =5

The side of the square is 3x + 2
With x = 5, 3x + 2 = 17

The side of the square= 17 units

Finally the area of the square = 17 x 17
= 289 square units.

Solving Equations 91

A common difficulty with problem solving is knowing how to get
started. If the problem does not include a letter 'x' (or any other
letter) then a general rule is to Jet x be the quantity you are asked
to find, not forgetting to state the units.
In the first example on the previous page x is given in the question but
in the second example we Jet x stand for the quantity to be found.

The total mass of three coins A, B and C is 33 grams. Coin B
is twice as heavy as coin A and coin C is 3 grams heavier
than coin B. Find the mass of coin A.

Let tl;le mass of coin A be x grams.
Draw and label a simple diagram to help visualise the
problem.

ABc

c;:;=:;o

X 2x + 3

B is twice as heavy as A. So the mass of coin B is 2x grams.

C is 3 grams heavier than B. So the mass of coin C is 2x + 3

grams.

The total mass of the three coins is 33 grams.

x + 2x + 2x + 3 33

5x+ 3 = 33
5x = 30
X= 6

The mass of coin A is 6 grams.

Exercise 9 5x- I

1. The diagram shows a rectangle. Write an
equation and solve it to find x.

3x + 2

2. In a quadrilateral ABCD, BC is twice as o A
long as AB and AD is three times as
long as AB. Side DC is lOcm long. The c
perimeter of ABCD is 31 em.
Write an equation and solve it to find
the length of AB.

3. The length of a rectangle is three times its width. If the perimeter

of the rectangle is 20 em, find its width. '
/

4. The length of a rect'angle is 3 em more than its width. If the

perimeter of the rectangle is 30 em, find its width.

92 Part 3

5. The angles of a triangle are A, B and C. A
Angle B is twice as big as angle A and
angle C is 10° bigger than angle A.
Find the size of angle A.

BC R

6. The sum of three consecutive whole numbers is 192. Let the first
number be x. Write an equation and solve it to find the three
numbers.

7. The total mass of four parcels A, B, C and D is 94 kg. Parcel C
is twice as heavy as parcel B and parcel D is 25 kg heavier than
parcel C.
Parcel A is 3 kg lighter than parcel B.
Find the mass of each parcel. [Let mass of B be x kg].

8. The total distance from P toT is 181 km. The

distance from Q to R is twice the distance from

S toT.
R is mid-way between Q and S.
The distance from P to Q is 5km less than the
distance from S to T.

Find the distance from S to T.

9. Abila's prize in a competition is a giant box of Smarties. Ahila
decides to share the prize equally with her friend Meera and
they each start filling empty Smarties tubes with their share.
After a while Meera has filled 5 tubes and has 38 Smarties left
over and Abila has filled only 3 tubes and has 152 left over.
How many Smarties go into each full tube?

10. The triangle shown is equilateral. Find x
and hence state the length of each side of
the triangle.

3x- 2

11. Two football teams have the same Won Drawn Lost

number of points. The teams get x 5 2 7
3 10 1
points for a win, 1 point for a draw Man. Utd

and no points for losing. The results Spurs

for the two teams are shown.

Form an equation involving x and

work out how many points are awarded for a win.

12. The width of a rectangle is 2x + 3 and its perimeter is 12x + 6.

(a) Find the length of the rectangle (in terms of x)

(b) Find x if the length of the rectangle is 7em.

Percentages 93

3.3 Percentages

Percentages are fractions with denominator (bottom number) equal
to 100.

So 25% means 12~0 , 31% means ?do and so on.

Percentages are used in a wide variety of situations.

• School test results are best understood by
most people as percentages.

• Shops sell goods at '30% off in sales.
• A Bank might charge interest of 12·3% on

loans.
• The owner of a shop might set his prices so

that he makes a profit of 55% of his
expenditure.

""'"·- -- -- -~ -.-
(a) Work out 16% of £15.
(b) Work out 23% of £350
16% of£15 (Quick way)
23% = 0·23 as a decimal
=...!.§._X _!i
100 I So 23% of £350 = 0·23 x 350

= 240 = £2-40 = £80·50
100

-

Exercise 10 (b) 24% of £44 (c) 19% of £1120
(e) 5% of £12·60 (f) 120% of £400
1. Work out
(a) 55% of £310
(d) 6% of £406

2. Give the correct units in your answers to the following:

(a) 18% of28km (b) 97% of4000kg (c) 35% of400m
(d) 62% of $35 000 (e) 11·2% of 710 km (f) 7·2% of $155

3. Using a calculator we find that 13·2% of £12·65 = £1·6698 .

This answer has to be rounded off to the nearest penny, since
the penny is the smallest unit of currency.
So 13·2% of £12·65 = £1·67 to the nearest penny.

Work out, to the nearest penny:

(a) 8% of £11·64 (b) 37% of £9·65

(c) 3·5% of £13·80 (d) 12% of £24·52

(e) 3t%of£11·11 (f) 115%of£212·14