Essential Mathematics. Book 1 (Rayner D.)

94 Part 3

±,4. It is possible to do some 'easy' percentages in your head.
Remember 10% = 1~, 25% = 33t% = t etc.

Work these out in your head: (b) 25% of £880
(a) 10% of £230 (d) 33t% of £120
(c) 80% of £5000 (f) 75% of £12
(e) 5% of £2000

5. A joint of meat originally weighs 3-4 kg.
During cooking the weight goes down by
15%.
What is the loss of weight of the meat?

6. The number of children having school dinners is 640. When
chips are not on the menu the number goes down by 5%. How
many fewer children have school dinners?

7. A worm weighs 24 g. After providing lunch for a starling the
weight of the worm is decreased by 8%. By how much does the
weight of the worm decrease?

8. In a restaurant a service charge of 10% is added to the price of a
meal. What is the service charge on a meal costing £28·50?

9. At a garage 140 cars were given a safety test and
65% of the cars passed the test.
(a) How many passed the test?
(b) How many failed the test?

10. Of the 980 children at a school 45% cycle to school, 15% go by
bus and the rest walk.
(a) How many cycle to school?
(b) How many walk to school?

11. A lottery prize of £65 000 is divided between Steve, Pete and Phil
so that Steve receives 22%, Pete receives 32% and Phil the rest.

How much money does Phil receive?

Percentages 95

To encourage sales the price of a car is reduced from £9640
by 4%. What is the new price?

4% of £9640

_ __i_ X 9640
- 100 I

= £385·60

:. New price of car = £9640 - £385·60

= £9254-40

Exercise 11
1. An electrical store increases all its prices by 4%. What are the

new prices of the items below?

£115 £275

2. The petrol consumption of a car is 35 miles per gallon. After a
service the car does 6% more miles per gallon. What is the new
consumption?

3. A dog normally weighs 28 kg. After being put on a diet for three
months its weight is reduced by 35%. How much does it weigh
now?

4. The length of a new washing line is 21m. After being used it
stretches by 3%. Find the new length.

5. A marathon runner weighs 55 kg at the start of a
race. During the race his weight is reduced by 4%.
How much does he weigh at the end of the race?

6. The insurance premium for Wendy's motor cycle is
normally £95. With a 'no-claim bonus' the premium
is reduced by 45%. What is the reduced premium?

96 Part 3

7. A hen weighs 2·7 kg. After laying an egg her weight is reduced
by 1%. How much does she weigh now?

8. A lizard weighs 630 g. While escaping from a predator it loses its
tail and its weight is reduced by 4%. How much does it weigh
now?

9. An electric drill is supposed to work at a speed of 750
revolutions per minute. Due to a fault its speed is reduced by
2% . At what speed does it work?

10. (a) In 1995 a club has 95 members who each pay £40 annual
subscription. What is the total income from subscriptions?

(b) In 1996 the subscription is increased by 15% and the
membership increases by 20%. What is the 1996
subscription? What is the total income from subscriptions
in 1996?

11. Estimate the following (do NOT use a calculator)

(a) 10% of £24·95 (b) 20% of £494

(c) 75% of £398 ·75 (d) 33% of £239·99

(e) 2% of 40 105 kg (f) 9·7% of £68 400.

12. Mrs Adie works in a garden centre which sells most
of its plants to companies. The prices displayed do
not include VAT. Ordinary members of the public
have to pay the marked price plus the VAT at
17-t%.
Mrs Adie often can't find her calculator but she has
a clever way of working out the VAT.
On a tree marked at £58 she works out 17t% of £58
as follows:

10% of £58 = 5·80
5% of £58 = 2·90
2t% of £58 = 1-45

So l7t% of £58 = 10·15

Use Mrs Adie's method to work out the VAT on
(a) a Japanese maple priced at £42
(b) an Acer Griseum tree priced at £85 ·60.

13. A printing firm decided to give its employees the choice of one
of two pay increases.
A: an increase of £450 per year
or B: 3·2% of present salary.
(a) Ben's salary is £18 840 per year at present. Which pay
increase would be better for him?
(b) Leah's pay is £212-45 per week. Which pay increase would
be better for her?
(c) Above what salary would it be better for someone to choose

the 3-2% increase rather than the flat rate of £450.

Percentages 97

Converting fractions to percentages

To change a fraction to a percentage multiply by 100.

(a) In a test Don scored 21 out of 30. (c) In an election
What was his mark as a percentage? 343 voted Tory
217 voted Labour
As a fraction Don's mark is ~6 420 voted LibjDem
Multiply by 100.
ll_ X 100 = 2100 What percentage of the electors
voted Tory?
30 30
mTotal number of voters = 980
= 70%
Fraction who voted Tory=
i(b) Change to a percentage
Percentage who voted Tory = ~~5 x 100
=.l X 1QQ. 300
8I 8 = 35%

-- 37_!2_%

Exercise 12

1. Change the following test results to percentages
g(a) (b) ~~
(c) 7 out of 40 (d) 17 out of 20.

2. In different tests James scored 33 out of 40 and Steven scored 41
out of 50. Who had the higher percentage mark?

3. Work out (a) 13 kg as a percentage of 20 kg
(b) 7 km as a percentage of 25 km
(c) 25p as a percentage of £2

4. In a school 123 children have school dinners and 27 children do
not have school dinners . What percentage of the children at the
school have school dinners?

5. In a survey in North America television
viewers stated that their favourite sports
were as follows:

Ice Hockey 328
Basketball 263
Football 209

What percentage of those questioned chose
Ice Hockey?

6. In 1991 the population of the U .K. consisted
of 29,516,000 females and 28,132,000 males
(to the nearest 1000). What percentage of the
total population was female?

98 Part 3

7. A rectangular area of land is owned by
three brothers
Paul,
Quentin,
Rick.
(a) Find the percentage of the total land
area owned by each brother.
(b) Rick decided to sell 15% of his land
to Paul.
What percentage of the total area is
now owned by Paul?

8. The Government of Russia called a referendum on proposed
new powers for the President. For the proposal to become law
70% of the adult population had to vote and, of those who
voted, 55% had to vote 'Yes'.

Here are the 'Yes' 35412419
figures: 'No' 25 854 743
Adult population 84 740 196

Did the proposal become law?

Two-way tables can read Boys Girls
cannot read
A school inspector is collecting data about 464 682
the reading ability of six year-olds. Children 317 388
are given a short passage and they 'pass' if
they can read over three-quarters of the
words. The results are given in the two-way
table.

Useful information can be found from the can read Boys Girls Total
table. Begin by working out the totals for cannot read
each row and column. 464 682 1146
Find also the 'grand total' by adding Total 317 388 705
together the totals in either the rows or
columns. 781 1070 1851

i

'Grand total'

Percentages 99

(a) What percentage of the boys can read?

Out of 781 boys, 464 can read.

: . Percentage of boys who can read= j~i x 100%
= 59-4% (1 d.p.)

(b) Similarly, the percentage of girls who can read= t~7~ x 100%
= 63-7% (1 d.p.)

So a slightly higher percentage of the girls can read.

(c) What percentage of those who failed the test were girls?
Out of 705 who failed, 388 were girls.
As a percentage this is ~~~ x 100% = 55·0% (1 d.p.)

Notice that this statistic could be misleading! There were more girls
than boys in the test so more girls had the chance to fail.

Exercise 13 can cycle Girls Boys Total
cannot cycle
Give percentages correct to one decimal place. 95 82 215
Total 476
1. This incomplete two-way table shows
details of seven year-olds who can/ Total
cannot ride a bicycle without stabilisers.
(a) Work out the missing numbers.
(b) What percentage of the girls can
cycle?
(c) What percentage of the boys can
cycle?

2. Mrs Kotecha collected the data below to help assess the risk of
various drivers who apply for car insurance with her company.
She needs to know if men drivers are more or less likely than
women drivers to be involved in motor accidents.

Men Women

had accident 75 88
had no accident 507 820

Total

(a) Work out the totals for the rows and columns
(b) What percentage of the men had accidents?
(c) What percentage of the women had accidents?
(d) Of those drivers who had accidents what percentage were

women?

100 Part 3

3. A psychologist has a theory that women who are over 30 when
their children are born produce more intelligent children. He
measures the I.Q. (intelligence quotient) of hundreds of 12 year-
olds and records the ages of their mothers when the children
were born.

Child's I.Q. Total

80-105 106-135

Mother's 18-30 yrs 414 478
age at 31--46 yrs 297 340
child's birth
Total

(a) What percentage of all the children in the survey had an I.Q.
below 106?

(b) What percentage of the children with a mother in the 18-30
age group had an I.Q. over 105?

(c) What percentage of the children with a mother in the 31--46
age group had an I.Q. over 105?

(d) Do these results support the psychologists's theory?

4. A medical researcher in Smokers Non- Total
Bristol collected data about smokers
the smoking habits and the age 1-25 25
ages of people when they at 26-60 217 32
died. death 61-100 348 195
614
Total

(a) What percentage of those who died aged 26--60 were
smokers?

(b) What percentage of those who died aged 61-100 were non-
smokers?

(c) What percentage of the smokers died before the age of 61?

(d) What conclusions, if any, can you draw?

Percentages 101

5. An agricultural college conducted an experiment on a type of
tomato plant which sometimes failed to produce ripe fruit when
grown in the north of England. Some plants were treated with
a new fertilizer called 'Grosoft' while other plants were given
an ordinary fertilizer.

with 'Grosoft' without 'Grosoft' Total

produced 515 473
ripe fruit 398 714
produced
unripe fruit

Total

(a) What percentage of the plants given 'Grosoft' produced ripe
fruit?

(b) What percentage of the plants not given 'Grosoft' produced
ripe fruit?

Unfortunately 'Grosoft' had to be withdrawn because of a
health scare and it was replaced by the more environmentally
friendly 'Grosmooth'. These are the new results.

with 'Grosmooth' without 'Grosmooth' Total

produced 475 461
ripe fruit 451 703
produced
unripe fruit

Total

(c) What percentage of the plants given 'Grosmooth' produced
ripe fruit?

(d) Was 'Grosmooth' more or less effective than 'Grosoft' in
helping plants to produce ripe fruit.

102 Part 3

3.4 Symmetry

Paper folding

• Take a piece of paper, fold it once and
then cut out a shape across the fold.

cut along r ' ~
the broken line ____,
'
• Fold another piece of paper twice so that the second fold is at
right angles to the first fold. ''' ,-
Again cut along the fold to see what shapes you can make.
See what happens if the second fold is not at right angles to the ''' ',
first fold.

• Fold the paper three times and cut.

• Below are three shapes obtained by folding and cutting as above.
Try to make similar shapes yourself.
Stick the best shapes into your exercise book.

Symmetry 103

• More interesing shapes can be obtained as follows:
(a) Cut out a circle and fold it in half.

(b) Fold about the broken line so that sectors
A and B are equal.

(c) Fold sector B behind sector A. Now
cut out a section and see what you
obtain.

(d) Even more complicated shapes can be
obtained by folding once again down the
middle of the sector.

Here are two shapes obtained by this method of folding.

104 Part 3

Line symmetry

When a shape can be folded like shape A on page 102, it has one
line of symmetry. Shape B has two lines of symmetry and shape C
has four lines of symmetry.

Shade in as many squares as necessary so that the final pattern has lines of symmetry
shown by the broken lines.
(a)

3 squares have been added

(b)

iIt ······-···· +·-+---l·········t'; ·

....... .............1.~- - - - - - · - · l - - - ---- - i - - - - ·- · !
~.- ~!-
;

---·-·-·t ---·-·"''

;
;

.......:..................~...... f--+--f ············: : ............ :········· ··r---+--1 .... !

'! ········t ······

;

9 squares have been added

Exercise 14

Copy each diagram and, using a different colour, shade in as many
squares as necessary so that the final pattern has lines of symmetry
shown by the broken lines. For each question write down how
many new squares were shaded in.

2. ....;.... 1---+---i-+--+-....;_-i + 3.

lj ! i __.. ,··········' ..........;........ ...... ........ ......... ;.......

··· tj ----0--i

"''''!··········

!... . ...........+
.........
.l

l .........•''''"''''t···· .......;..1.......

Symmetry 105

4. 5. 6.

+ ·•.lr-+_.1,.:-.~! ,' '

;,

,, !..:-. :

. .................... ········~ c·>' ;'·,;..,..... ----

; 1···· ,..... ....... .............. """"'~'f"" .........................

......; ; .. ......;.............,............ ,.......+ .. . ..,,, ~ '

~ ' IJ

·····t· ij

7. 8.

Exercise 15

1. You have 3 square black tiles and 2 square white tiles, which can

r--{Jbe joined together along whole sides.
CD L__QSo this
is allowed but this is not allowed.

Draw as many diagrams as possible with the 5 tiles joined Fig. 3
together so that the diagram has line symmetry.
For example fig. 1 and fig. 2 have line symmetry but fig. 3 does
not have line symmetry so fig. 3 is not acceptable.

-+----l----1-+·1-----l···+ Fig. 1

-1·····]·····)-·--·1·····1-----1- Fig. 2

2. Now you have 2 black tiles and 2 white tiles. Draw as many
diagrams as possible with these tiles joined together so that the
diagram has line symmetry.

106 Part 3

3. Finally with 3 black tiles and 3 white tiles draw as many
diagrams as possible which have line symmetry.

Here is one diagram

r-1--,----r.--. -.· ·'which has line

symmetry.

D4. Shape A is a Shape B consists

single square. A of four squares.

Draw three diagrams in which shapes A and B are joined
together along a whole edge so that the final shape has line
symmetry.

5. Shape C is a Shape D consists
single square of five squares.

Draw four diagrams in which shapes C and D are joined •
together along a whole edge so that the final shape has line
symmetry. ••t

Rotational symmetry A

We have already seen that many shapes have one or more lines of •
symmetry. An object like the playing card shown has no lines of
symmetry but it does have rotational symmetry.

Place a sheet of tracing paper over the drawing of the four of hearts
and trace the main features of the card. Now put the tip of your
pencil on the tracing paper exactly in the middle of the card. Turn
the tracing paper until the tracing and the card underneath coincide
just as they did to start with.

The four of hearts has rotational symmetry because the pattern fits
onto itself before a complete turn is made. The shape A does not
have rotational symmetry because it does not fit onto itself before a
complete turn is made. [Any shape will always fit onto itself when a
complete turn is made.]

Symmetry 107

Order of rotational symmetry

The shape B fits onto itself three times when rotated through a
complete turn. It has rotational symmetry of order three.

The shape C fits onto itself six times when rotated through a
complete turn. It has rotational symmetry of order six.

Exercise 16

For each diagram decide whether or not the shape has rotational

symmetry. For those diagrams that do have rotational symmetry
I.Dstate the order.
2. •3. 3• 3• •4. 3• 3•

• •
•c• c• •c• c•

5. A A 6. K K 7. 8.
•
+ + + +

• J•l •

v+ v+ J•l

9. 10. 11. 12.

13.~ 14. 15.~ 16.~

108 Part 3

3.5 Puzzles and games 3

Clocks

1. On how many occasions is a figure 5 used in writing down the
numbers from 1 to 100?

2. On a 24 hour digital clock display (as shown), how many different
times are there with at least one 5 in the display?

3. For how long in a period of 24 hours does a figure 5 appear?

<1. For how long during the day will the display show four digits all
the same?

5. Which digit(s) appears for the longest time in a period of 24
hours?
For how long does that digit appear?

Puzzles

1. What is the largest possible number of people in a room if no two
people have a birthday in the same month?

2 . The letters A, B, C, D, E appear once in every D B
row, every column and each main diagonal of E
the square. Copy the square and fill in the
missing letters

AD

3. Two different numbers on this section of a till etapes at £f.99 :£87.89
receipt are obscured by food stains. What are
the two numbers?

• • •<1. Draw four straight lines which pass through all
9 points, without taking your pen from the

paper and without going over any line twice. •••

[Hint: Lines can extend beyond the square).

•••

Puzzles and games 3 109

• • •5. Draw six straight lines to pass through all 16 •
points, subject to the same conditions as •
• • •above. •
•
•••

•••

In Questions 6, 7. 8 each letter stands for a different single digit from
0 to 9. The first digit of a number is never zero.

6. Work out what digit A must be in this addition. c A N+
[Remember you can have 'carry's').
ADD

7. Find one solution for this addition. s uMs

A R E+

EAs y

8. Explain why a solution cannot be found for the
addition shown.

L u V+
y0u

[Work out what the U in LUV would have to be).

9. Three friends are trying to weigh themselves but the weighing
machine only works for weights over 90 kg and they all weigh
less than that. They decide to go on the machine two at a time.

John and Steve together weigh 119 kg.
John and Mark together weigh 115 kg.
Steve and Mark together weigh 107 kg.

How much does John weigh?

Part 4

4.1 Approximating

Decimal places

• Suppose a group of 7 people share a prize of £20. Using a calcula-
tor, each person should receive £2·8571428.
Since the penny is the smallest unit of money most of the figures
in this answer are meaningless for everyday use.

So how much does each person receive? 2.85 2.8571428
On a number line we can see that the
answer is nearer to 2·86 than to 2·85. t

2.855 2.86
38 .47
We will round off the answer to £2·86 correct to 2 decimal places.

• A prize of £500 divided between 13 people gives £38-461538 each.

On a number line the answer is nearer to

38-46 than to 38-47. t38.46 38.465
So the answer is £38-46, correct to

2 decimal places (2 d.p. for short). 38.461538

• Suppose the calculator shows 1·575. This number is exactly half
way between 1·57 and 1·58. Do we round up or not?
The rule for rounding off to 2 decimal places is:

If the figure in the 3rd decimal place is 5 or more round up.

Otherwise do not. -

Examples: 3·7521 = 3·75 to 2 d.p.

i

141·2974 = 141·30 to 2 d.p. [We need the zero!]

i

0·355124 = 0·36 to 2 d.p.

i

• Rounding off to 1 decimal place: If the figure in the 2nd
decimal place is 5 or more round up. Otherwise do not round
up.

Examples: 15·6234 = 15·6 to 1 d.p.
= 0·8 to 1 d.p.
i

0·7936

i

Approximating 111

Exercise 1

1. Write the following numbers correct to 2 decimal places.
(a) 3·75821 (b) 11 ·64412 (c) 0·38214 (d) 138·2972

2. Write the following numbers correct to 1 decimal place.

(a) 18·7864 (b) 3·55 (c) 17·0946 (d) 0·7624

3. Write the following numbers correct to the number of decimal
places indicated.
(a) 8-4165 (3 d.p.) (b) 0·7446 (2 d.p.) (c) 18·2149 (3 d.p.)
(d) 18·0612 (1 d.p.) (e) 0·07451 (3 d.p.) (f) 0·0312 (2 d.p.)
(g) 1·355 (2 d.p.) (h) 9·974 (1 d.p.) (i) 0-45555 (3 d .p.)

4. Work out the following on a calculator and write the answer
correct to 2 decimal places.
(a) 11 7 7 (b) 213 7 11 (c) 1-4 7 6 (d) 29713
(e) 1·3 X 0·95 (f) 1·23 X 3·71 (g) 97 7 1·3 (h) 0·95 X 8·3

Significant figures

Sometimes numbers are rounded off to a certain number of
significant figures rather than decimal places.

For decimal places we started counting from the decimal point.
For significant figures we approach from the left and start counting
as soon as we come to the first figure which is not zero. Once we
have started counting we count any figure, zeros included.

(a) 52·7211 = 52·7 to 3 significant figures. (3 s.f.)

t

[Count 3 figures. The 'next' figure is 2, which is less than 5] .

(b) 7·0264 = 7·03 to 3 significant figures.
t

(c) 0·0237538 = 0·0238 to 3 significant figures.
t

(d) 2475·6 = 2500 to 2 significant figures.

Notice that we need the two noughts after the '5' as the
original number is approximately 2500.

Exercise 2

1. Write the following numbers correct to 3 significant figures

(a) 1·0765 (b) 24-897 (c) 195·12 (d) 0·7648

(e) 17·482 (f) 0·07666 (g) 28774 (h) 2391·2

2. Write the following numbers to the degree of accuracy indicated
(a) 19·72 (2 s.f.) (b) 8·314 (l s.f.) (c) 0·71551 (3 s.f.)
(d) 1824·7 (3 s.f.) (e) 23 666 (2 s.f.) (f) 0·03476 (2 s.f.)

3. Work out the following on a calculator and write the answer
correct to 3 significant figures.
(a) 1773·1 (b) 0·13 x 0·11 (c) 270·11 (d) 87719
(e) 1·7 x 8·32 (f) 570·753 (g) 1970·021 (h) 170·7

112 Part 4

4.2 Using a calculator T -\:.~ov~'v-...t

Order of operations co..\c:.u.\o.Tor<;. MO~IZ.

Here is an apparently simple calculation ·,\... eo.s'::JJ
4·52 + 3·5 X 6·3 .

Working left to right on a calculator press

Depending on what make of calculator you use, Brackets
you may get 50·526 or you may get 26 ·57. Divide
Which is correct? Multiply
Add
Where there is a mixture of operations to be Subtract
performed to avoid uncertainty you must follow
these rules: (a) work out brackets first

(b) work out 7 , x before +, -.

Some people use the word 'BODMAS'
to help them remember the correct
order of operations.

Here are four examples

• 8+676=8+1=9
e 20 - 8 X 2 = 20 - 16 = 4
• (13 - 7) 7 (6- 4) = 6 7 2 = 3
• 20 - 8 7 (5 + 3) = 20- 8 7 8 = 19

Exercise 3

In Questions 1 to 20 do not use a calculator to work out the answer.

l. 13+973 2. 40 - 5 X 7 3. 8 X 3- 14 4. (8 + 3) X 6
5. 15- (7 + 5) 7. 17-12...;-4 8. 24 7 (1 + 7)
6. 8 + 3 X 2 11. 3 + 20 7 2
9. 3 X 4 + 5 X 2 15. 3 X 5 - 12 7 2 12. (1 0 7 2) 7 4
13. 16 - (8 + 3 X 2) 10. 30 7 3 + 5 X 4 16. 8 X 2 - 4 7 2
14. (9 7 1 - 2) X 4 19. 36-1272
17.~ 20. (6 + 5) X 2
1 810.+- -3 -X -2 5+ 1
10 - 8 12 - 57 5
872

In Questions 21 to 32 use a calculator and give the answer correct to two decimal places.

21. 8·51 + 8·7 7 3·2 22. 14·82 + 1·24 X 1·1 23. 7·3 X 1·9- 5·04
24. 9·764 + (8·2 7 2·93) 25. 11·7379·2-0·971
27. (111 ·774·5-6)x 1·9 28. 12·2 + 3·7 X 3·7 26. (8·779)+1·9

30. 1·234 +-8·5 31. 8·96 + 2·761 29. 9·25 7 (1-4- 0·07)
9·2
4·02 32. 83 - 2·04

-4

3·3

Using a Calculator 113

Using the memory

IWe will use the following memory keys: Min I Puts a number into the memory.
IMR I Recalls a number from the memory.

[2]To clear the memory we will press IMin 1.

Many calculators will keep a number stored in the memory even
when they are switched off. A letter 'M' on the display shows that
the memory does contain a number.

The IMin I key is very useful because it automatically clears any
number already in the memory when it puts in the new number.

So if you pressed ~ IMin II]}] IMin I, the number in the

memory would be 6·5. The 13·2 is effectively 'lost'.

Work out, correct to 2 decimal places. (b) 8·51- ( 3"24)
1·73
(a) 8·97
1·6- 0·973

Work out the bottom line first. Work out the brackets first.

G[8] I0·9731 BIMini 13·24IEJ~BIMinl

B18·971 IMRI B [ill[jiMRIB

Answer= 14·31 to 2 d.p. Answer= 6·64 to 2 d.p.

EJA very common error occurs when people forget to press the button at the end of

the calculation.

Exercise 4

Work out and give the answer correct to 2 decimal places.

1. 5·63 11·5 8·27
2·8- 1·71 2. 3.

4. 3·7- 2-41 5·24 + 1·57 2·9X1·35
1·9 + 0·72 8·5 + 9·3. 6. 0·97 X 3·85
5.
12·9- 8·72 1·24 + 4·63
9. 11·626-·3-
7. 14·5- c-·9) 8. 8-41- 3·2 X 1·76
0·7 9·8
11. 5·62 + 1·98 +1-·2 12. 8 · 58-·9-
9·84 X 0·751 4·5
10. 11·6
14. 8-43 + 1·99 1 517·.6 - -1·9-2
6·3 X 0·95 9·6- 1·73
6·3 8·2 8-4 8-41
13.-+- 17. (9·8- 4-43)2 18. 18·7- 2·332
4·2 11·9 20. 9·232 - 7-422
16. 25·1 - 4·22 21. 16· 1 - 1·1 2

19. 8·21 2 + 1·672

114 Part 4

22. 16·1 - 1·82 23. (17·2- 1·2)2 24. 9·9- 8·3 X 0·075
4·7 9·8
2 711.·-7-- 3-·7-3
25. 1·21- - 9- 26. 3·72 + ll-4 2-452
142
1·7 3 08-.1P1-6+·-73-2
8·94 2
2 9 .3-·2-1 - - 33. 8 · 31+·9 - -1·-7
2 84·.8-+-1-·7-2 8·2-4·11 8-4 6·5
1
8·2 4·1 32. 8·5 - (1·62 + 1·92)
31. 8·7+-+- 36. (8·2- - -) X 8·2
2 8·2
9·7 5·6 353.--4+ (2-•1 )

( 14)2 1·6 1·3

34. 3·2 + 3·2 +5

4.3 Travel graphs

On a travel graph the motion of a moving object is shown, with time
usually plotted on the horizontal axis and distance from a point
plotted on the vertical axis.

• The graph shows the journey of a car from

Amble to Cabley via Boldon. The vertical axis

shows the distance of the car from Amble 100

between 1400 and 1700.

(a) At 1530 the car is 60 km from Amble. 80
(b) The car stopped at Boldon for 45

minutes. The graph is horizontal from

1430 until 1515 which shows that the car 60 -! ·········!··············

does not move. Boldon -1 ., ·wt--~+-

(c) The car takes+ hour to travel 50 km from
Amble to Boldon. Thus the speed of the 40

car is 100 km/h.

(d) The speed of the car from Boldon to 20

Cabley is 40 km/h.

distance travelled 50] Amble -+----'-----'--'--+---'---'-----'--t---'----'----'---+_.
t[Speed =
=- 1400 1500 1600 1700

time taken 1 Time

• This graph shows the details of a cycle ride

that Jim took starting from his home.

(a) In the first hour Jim went 30 km so his

speed was 30 km/h. 20

!(b) He stopped for hour at a place 30 km

from his home.

(c) From 0930 until 1100 he cycled back

home. We know that he cycled back home 10

because the distance from his home at

1100 is Okm.

(d) The speed at which he cycled home was 0800 0900 1000 1100

20 km/h.

Travel Graphs 115

Exercise 5

1. The graph shows a car journey from
A to C via B.
(a) How far is it from A to C?
(b) For how long does the car stop
at B?
(c) When is the car half way
between B and C?
(d) What is the speed of the car
(i) between A and B?
(ii) between B and C?

0600 0700 0800 0900

2. The graph shows the motion of a
train as it accelerates away from
Troon.
(a) How far from Troon is the train
at 0845?
(b) When is the train half way
between R and S?
(c) Find the speed of the train
(i) from R to S

(ii) from Q to R

(iii) (harder) from P to Q.

0700 0800 0900 1000

3. The graph shows a car journey from
Lemsford.
(a) For how long did the car stop at
Mabley?
(b) When did the car arrive back at
Lemsford?
(c) When did the car leave Mabley
after stopping?
(d) Find the speed of the car
(i) from Lemsford to Mabley
(ii) from Nixon back to
Lemsford.

1400 1500 1600 1700

116 Part 4

4. This graph shows a car journey from London to Stevenage and
back.

0900 1000 1100 1200 1300 1400

(a) For how long in the whole journey was the car at rest?
(b) At what time was the car half way to Stevenage on the

outward journey?
(c) Between which two times was the car travelling at its highest

speed?

5. The graph shows the journey of a coach and a lorry along the
same road between Newcastle and Carlisle.

0800 0900 1000 1100 1200 1300

(a) How far apart were the two vehicles at 0915?
(b) At what time did the vehicles meet for the first time?
(c) At what speed did the coach return to Newcastle?
(d) What was the highest speed of the lorry during its journey?

Travel Graphs 117

6. The graph shows the motion of
three cars A, B and C during a
race of length 140 km.
(a) What was the order of the
cars after 40 minutes?
(b) Which car won the race?
(c) At approximately what time
did C overtake B?
(d) At what speed did car B finish
the race?
(e) Describe what happened to
car C during the race.

20 40 60 80 100 120 140

7. Explain why the two graphs below cannot be travel graphs.

distance

time time Distance from Dover
B
8. The diagram shows the travel graphs of five
objects.
Which graph shows:
(a) A car ferry from Dover to Calais
(b) A hovercraft from Dover to Calais
(c) A car ferry from Calais to Dover
(d) A buoy outside Dover harbour
(e) A cross channel swimmer from Dover?

Time

Solving problems with travel graphs

Exercise 6 , , , , , , ,.,.• ............. .............. ........••.... .............. ............. ............ ............. ............

In Questions 1 to 6 use the same scales as in question 5 of the last • Distance from Lisa' s home (km) !
exercise . '.

±1. At 17 00 Lisa leaves her home and cycles at 20 km/h for 1 hour. 20
She stops for hour and then continues her journey at a speed
1 tof 40 km/h for the next hour. She then stops for hour. -+ -10 ;········•''············! · ; ; ;· + :
Finally she returns home at a speed of 40 km/h.

Draw a travel graph to show Lisa's journey. 1700 1800 Time
When did she arrive home?

118 Part 4

2. Suzy leaves home at 1300 on her horse and
rides at a speed of 20 km/h for one hour.
Suzy and her horse then rest for 45 minutes
and afterwards continue their journey at a
speed of 15 km/h for another one hour.
At what time do they finish the journey?

3. As Mrs Sadler leaves home in her car at
13 00 she encounters heavy traffic and

travels at only 20 km/h for the first 1- hour.

In the second half hour she increases her
speed to 30 km/h and after that she travels
along the main road at 40 km/h for ~h. She

stops at her destination for 1- hour and then

returns home at a steady speed of 40 km/h.
Draw a graph to find when she returns
home .

4. At 12 00 Mr Dean leaves home and drives at a speed of 30 km/
h. At 12 30 he increases his speed to 50 km/h and continues to

his destination which is 65 km from home. He stops for 1- hour

and then returns home at a speed of 65 km/h.
Use a graph to find the time at which he arrives home.

5. At 08 00 Chew Ling leaves home and cycles towards a railway
station which is 65 km away. She cycles at a speed of 30 km/h

until 09 30 at which time she stops to rest for 1- hour. She then

completes the journey at a speed of 20 km/h.
At 09 45 Chew Ling's father leaves their home in his car and
drives towards the station at 60 km/h.
(a) At what time does Chew Ling arrive at the station?
(b) When is Chew Ling overtaken by her father?

6. Kate lives 80 km from Kevin. One day at 1200 Kate cycles
towards Kevin's home at 25 km/h. At the same time Kevin
cycles at 30 km/h towards Kate's home

Draw a travel graph with 'Distance from Kate's home' on the
vertical axis.
Approximately when and where do they meet?

Travel Graphs 119

In questions 7 and 8 use a scale of 2 squares to 15 minutes across
the page and I square to I 0 km up the page.
7. At 0100 a bank robber leaves a bank as the alarm sounds and

sets off along a motorway at 80 km/h towards his hideout
which is 150 km from the bank.

150km

Police station Robber' s
hideout

As soon as the alarm goes off a police car leaves the police
station, which is 40 km from the bank, and drives at 80 km/h to
the bank. After stopping at the bank for 15 minutes, the police
car chases after the robber at a speed of 160 km/h.

Draw a travel graph with 'Distance from police station' on the
vertical axis.
(a) Find out if the police caught the robber before the robber

reached his hide out.
(b) If the robber was caught, say when. If he was not caught say

how far behind him the police were when he reached his
hideout.

8. The diagram shows three towns A, B and C. The distance from
A to B is 50 km and the distance from B to C is 110 km.
At the same moment 3 cars leave A, B and C at the speeds
shown and in the directions shown.

50km l!Okm
A

The cars from A and C are trying to intercept the car from B as
quickly as possible.
(a) Which car intercepts the car from B first?
(b) After how many minutes does the car from A catch the car

from B?

120 Part 4

4.4 Problem solving 1

In this section we will investigate networks and then use them to
solve some practical problems.
Networks can be used to give a simplified plan of roads, railway
lines or airline routes.
A well known example of a network is the London Underground
Map.
Stations are shown in the correct order but,
although distances between stations may look
the same, in reality they are not.

Traversable networks

A network is traversable if it can be drawn without going over any
line twice and without removing the pen from the paper.
• It is easy to see that this network is traversable.

• This network is traversable but it is not so easy to show it. C~D
See if you can do it.
Describe the route you took using the letters shown . B~A
[Ask your teacher if you cannot do it.]

Exercise 7 4.

Decide whether or not each network ts traversable. Keep your
answers for the next exercise.

1. 2.

5. 6. 7.

Problem Solving 1 121
9.
CYJ f ! ) ~10. II. 12.

Nodes

A node is a point where lines meet or

Y kAn odd number of lines meet at an odd node. e.g.

XAn even number of lines meet at an even node. e.g. or)
0w~0This network E
E has 2 odd nodes(o) and 3 even nodes (E).

Exercise 8

1. For each network in the last exercise count the number of odd
and even nodes. Record your results in a table as below.

Network Traversable? odd nodes even nodes
5
1 ...; 0
2

12

Try to think of a rule which will tell you whether or not a
network is traversable.
Don't look for a terribly complicated rule!
[Hint: The rule mentions either odd nodes or even nodes, but

not both]

2. Here is a complicated network of roads.
Use your rule from above to predict whether
or not the network is traversable.
Check by drawing to see if your prediction
was correct.

122 Part 4

3. (a) Use your rule to predict whether or not
this network is traversable.

(b) What difference (if any) will it make if
the wavy line from A to B is removed?
Is the network now traversable?

(c) Go back to the network in figure 1.
Will the network be traversable if the
wavy line from E to F is removed?

4. Count the odd and even nodes in this network. A
Point A is an odd node since one line goes to it. Point B is an
even node and point C is an odd node.
So we have 2 odd nodes and 1 even node.

You have seen that you can draw networks with an even number
of odd nodes but can you design a network, either traversable or
not, with an odd number of odd nodes? Try it.

Shortest route

The network shows towns P, Q, R, S, T, U, V
joined by roads, with distance in miles.
Find the shortest route from P to V.

To keep track, label the routes you
have tried with the letters

PTUV = 16 miles
PQTUV = 18 miles
PQRSV = 15 miles

PSV = 17 miles

No other route is shorter so the shortest route
is 15 miles via Q, Rand S.

Problem Solving 1 123

Exercise 9
1. Villages A, B, C, D, E, F, G, H, I are joined by a network of
roads. The lengths of the roads are in miles.

6

(a) Find the shortest route from A to I.
(b) Find the shortest route from B to E.
(c) A family on a sight-seeing tour wish to visit all of the

villages as they go from A to I. Find the shortest route
passing through all the villages.

2. This network shows the roads joining towns
A, B, ... L.
(a) Find the shortest route from B to K.
(b) Find the shortest route from G to L.
(c) Find the shortest route for a waste
disposal lorry which has to visit every
town as it travels from A to L.

3. The networks below show streets in a town. A postman has to
go along each street at least once, starting and finishing at
point A.
Work out the shortest route that he can take for each network.
[The unit of distance is lOOm].

(a) B (b)

A

c

B

(c) (d)

124 Part 4

4. The owner of a firm operating delivery vans knows that his
network has no odd nodes so that it should be traversable from
any place. To cut operating costs the owner wants to build his
depot at a point from which the network is traversable.

(a) Show that the network is traversable, starting and finishing at A.
(b) For this traversable route, what is the distance travelled by a

van travelling along all the roads on the circuit?
(c) The owner decides that the long route between C and D (115

miles) is no longer required. Is the reduced network traversable,
starting and finishing at the same point, either A, B or C?

4.5 Puzzles and games 4

Triangles and quadrilaterals

1. On a square grid of 9 dots it is possible to draw several

different triangles with vertices on dots.

V: • • • •
••
• • •••
A and B are different but C is the same as A.

(a) Copy A and B above and then draw as many different

triangles as you can. Check carefully that you have not

repeated the same triangle.
If a triangle could be cut out and placed exactly over

another triangle then the two triangles are the same.

[We say the two triangles are congruent]. •••
(b) Go through your triangles and draw a line
'21.. :
of symmetry with a broken line in those
triangles with line symmetry.

Puzzles and Games 4 125

2. On a grid of 9 dots it is also possible to draw several different
quadrilaterals. Here are three:

CJ

••• •••

(a) Copy the three shapes above and then draw as many other
different quadrilaterals as possible. You are doing well if
you can find 12 shapes but there are a few more!

(b) Draw a line (or lines) of symmetry on all the quadrilaterals
which have line symmetry.

Crossnumbers 2

All the numbers given have to fit inside the cross number grid.
Work your way through the puzzle logically, crossing out the
numbers as you put them into the grid.

1. 248714
432562
4 452344
456144

236 13 34
237 18 34
246 25 46
251 26 56

2. 3745124 4483 182 436 53

4253464 4488 324 573 63

8253364 6515 327 683 64

8764364 337 875

375615

126 Part 4

3. 1258315 231 682 25

1275465 241 742 36

1275525 242 42

582 73

45137

46147

7150
7240
7250

4. 8 215613 24561
246391 24681
I 246813 34581

7 1349 5321 121 285 21 55

2457 5351 136 335 22 56

I 2458 5557 146 473 22 58
3
3864 8241 165 563 23 61

4351 8251 216 917 36 82

4462 9512 217 53 83

91

I

5. This question is more difficult. Several numbers have been
given to help

8 3 32276 7245 164 274 22 54
5 32473 7246 173 281 22 62

32476 7248 225 284 23 63

32486 7346 243 285 42 74

32576 8222 271 327 45 87

33576 273 449 46

3 52476 692

52487

7 1717
5 1978
2 5245
6366
7242

Puzzles and Games 4 127

6. This one looks virtually impossible, but it can be done! Think
ahead and do not make guesses.

3891234980 888881
4814567384 888882
4844567284 888883
4891234987 888884

9991
9992
9993
9994

598 21 45
598 24 46
698 36 47
698 40 48
892 41 48
892 42 51
893 43 84
893 44 84

Biggest number: a game for the whole class

(a) Draw a rectangle like this with 4 boxes

(b) Your teacher will throw a dice and call out the number which is

showing. (eg 'four') I4 I
(c) Write this number in one of the boxes.

(d) Your teacher will throw the dice again. (eg 'two') I4 I I2 I
Write the number in another box

(e) Your teacher will throw the dice two more times (eg 'three' and
then 'two') and again you write the numbers in the boxes.

l2l4l3l2l

(f) The object of the game is to get the biggest possible four figure
number. The skill (or luck!) is in deciding which box to use for
each number.
You score one point if you have written down the largest four
digit number which can be made from the digits thrown on the

4322dice. In the example above you score a point if you have

and no points for any other number.
The game can also be played with 5 boxes or 6 boxes for
variety.

Part 5

5.1 Metric and Imperial units

Originally measurements were made by .using appropriately sized
bits of human being. The inch was measured using the thumb,
(hence we still sometimes say 'rule of thumb' when we mean rough
measurement), the foot by using the foot. As people did not usually
travel far, and used only local materials, it did not matter that
units were not the same everywhere. After all, if it took a day to
travel to the next town, it made little difference if it was 25 or 26
miles, or that the church clocks told different times. With the
advent of railways it became more important, particularly with
time.

Other measurements were based on actions. The Mile was taken
from the Latin Milia passuus, meaning a thousand passing-paces (the
distance covered from one foot touching the ground to the same
foot touching the ground again). As the Roman soldiers were no
doubt well drilled, this distance might well be fairly standard. A
Fathom (6 feet), was used for finding the depth of water, and was
measured using a weighted string, dropped at the bow of the ship,
each fathom being measured by stretching the string between
outstretched arms.

A Furlong (probably from furrow-long) was the distance a horse
could pull a plough without having to stop for a rest, and this gave
the lengths of fields. The chain, which was a tenth of a furlong, was
the width of the field. The product of these two gave the basic
measurement of land measurement, the Acre. However, no doubt
because of the differing types of soil, this varied all over the country.
As late as 1889 the acre was 2400 square feet in Devon, while in
Cheshire it was 3840 square feet!

Even today many people, when asked their height or weight, will

give the answer as '5 feet 3' or '9t stone' rather than 'I metre 59' or

60 kg. -

After the French Revolution in 1789 the standard unit of length
became the metre and the unit of mass became the kilogram. All the
smaller and larger units are obtained by dividing or multiplying by
ten, a hundred, a thousand and so on.

Metric and Imperial Units 129

Here is a table with details of the most commonly used units for
length, mass and volume.

Metric units Imperial units

Length lOmm = 1cm 12 inches = 1 foot
lOOcm =1m 3 feet = I yard
IOOOm = 1 km
1760 yards = 1 mile
Mass 1000mg = 1g
1000g = 1kg I6 ounces = 1 pound
14 pounds = 1 stone
1000 kg = 1 tonne 2240 pounds = I ton

Volume 1000ml = 1 litre 8 pints = I gallon
1ml = 1cm3

Exercise 1

Copy and complete

1. 57 em= m 2. 1·3 km = m 3. 0·24kg = g 4. 600g = kg
5. 17mm = em 6. 300kg = t 7. 0·6m = em 8. l-4mm = em
9. 2000ml = £ 10. 305 g = kg 11. 80cm = m 12. 200mm = m
13. 2·5t= kg 14. 2-4m= mm 15. 20 g = kg 16. 4-5£ = ml
17. 2£= cm3
18. 5·5m = em 19. 56mm = m 20. 7 g = kg

Questions 21 to 30 involve imperial units

21. 3 feet = inches 22. 5 yards = feet
23. 2 pounds = ounces 24. 9 stones = pounds
25. 24 inches = feet
26. 1- pound = ounces
27. 2 feet 6 inches= inches
29. 8 stones 4 pounds = pounds 28. 1 ton = pounds
30. 5 feet 2 inches = inches

Questions 31 to 45 involve a mixture of metric and imperial units.

31. 0·032 kg = g 32. 6 feet= yards 33. 8 ounces = pound
34. 1 mile= feet 36. 0-42 t = kg
37. 11·1 em= m 35. 235mm = em 39. 7 litres = ml
40. 4 yards= feet 42. 2 gallons = pints
43. 400m = km t38. pound= ounces 45. 10 miles = yards

41. 7mm = em

44. 5 gallons = pints

Converting between metric and imperial units

It is sometimes necessary to convert imperial units into metric units
and vice versa.
Try to remember the following approximate equivalents:

1 inch ~ 2·5 em 1 kg ~ 2·2 pounds
1 foot ~ 30cm 1 gallon ~ 4·5 litres

t1 km ~ mile

[The ·~· sign means 'is approximately equal to'.]

130 Part 5

(a) Change 45 kg into pounds (b) Change 6 feet into metres

1kg ~ 2·2 pounds 1 foot~ 30cm
45 kg ~ 45 x 2·2 pound 6 feet ~ 6 x 30 em

~ 99 pounds ~ 180cm
6 feet ~ 1·8m

(c) Who is heavier: John who weighs 52 kg or Jim who weighs 9 stones 6 pounds?

52 kg ~ 52 x 2·2 pounds
~ 114-4 pounds

9 stones 6 pounds = (9 x 14) + 6 pounds

= 132 pounds
So Jim is heavier.

Exercise 2

Copy and complete using the approximate conversions given above.

1. 3 kg ~ pounds 2. 10 inches ~ em 3. 4 gallons~ litres
4. 24 km ~ miles 5. 5 feet~ em 6. 80 kg ~ pounds
7. 4 inches~ em 8. 10 gallons ~ litres 9. 6 feet 2 inches ~ em

10. All the teachers at Gibson Academy must be at least 5 feet
8 inches tall. Mr Swan is 1·66 m tall. Is he tall enough to teach at
Gibson Academy?

11. A boxer must weigh no more than 10 stones just
before his fight. With two days to go he weighs
65kg.
How much weight in pounds does he have to lose
to get down to the 10 stone limit?

12. A car manual states that 2! gallons of oil must be
put into the engine before it is started. Roughly

how much will it cost if oil costs £1·20 per litre?

13. If the speed limit on a road in Holland is 80 km/h,
what is the equivalent speed limit in m.p.h.?

14. The perimeter of a farm is about 36 km. What is the
approximate perimeter of the farm in miles?

±15. A carpenter requires a 5 mm drill for a certain job but he has

only the imperial sizes t, ?6 and inch.

Which of these drills is the closest in size to 5 mm?

16. At a charity cake sale all the proceeds were collected in 1Op
coins and then the coins were arranged in a long straight line
for a newspaper photo.
If the diameter of a lOp coin is just under one inch and the line
of coins was 50 metres long, roughly how much money was

raised?

M,etric and Imperial Units 131

i

Exercise 3

Some of the old units had interesting names. Here is a selection of
the units taken from the National Curriculum in 1863.

LONG MEASURE ALE MEASURE MONEY
3 Barleycorns make l Inch 8 Pints make l Gallon 12 pence make 1 shilling
9 Gallons make l Firkin 5 shillings make 1 crown
4 Inches make l Hand 2 Firkins make l Kilderkin 20 shillings make 1 pound
12 Inches make l Foot 2 Kilderkins make l Barrel 21 shillings make 1 guinea
126 Gallons of Claret make 1 Pipe
3 Feet make l Yard METRIC CONVERSION
6 Feet make l Fathom l Pound = 454 g
l Inch = 2·54 em
5t Yards make 1 Rod, Pole or Perch APOTHECARIES WEIGHT l Pint= 0·57 litre
20 Grains make l Scruple
40 Poles make 1 Furlong 480 Grains make l Ounce
8 Furlongs make l Mile

l00 Links make l Chain
10 Chains make l Furlong

l00 Furlongs make l Cable

AVOIRDUPOIS WEIGHT
16 Ounces make l Pound
14 Pounds make 1 Stone
2 Stones make l Quarter

1. Translate the following 'Tale of metrication' into
English.

rk '7 2.54 ~ IIUf UleUf ~ tb ~. '7

~ led IIUf ~ 454 ~ is IIUf ~. '7

«144 ~ IIUf 304·5'HIM ~ ~. 4#14 'J

~ 6a1t Uk 4 6·36~ ~ IIUf 5·03, Ult6 tb

0·91· ~. '71 ~ '7 ~ ~ 4 651119 ~"""
'J ~ ~ 4e#tt 4 15'3 fH., M ef/liUQ.d 4 12· 71 ~
~em 0·042 ~ ~. (), tb ~ 10·16t:M, '7
~ Hilt 1. 5'3, ~ ~ tb pi4ee «144 is
~. 7~ He# <1-tep ~ 1·6hH. ~· tb
0·0025' ~ ~ ~. ~ tU teaa ~ «144 4

576£ 1M 4 0·30, ~. 7~ ~ ~ IIUf 1·30

~ ~ eU '7 ~ (6 ~ IIUf 0·20

~~tkpi4ee.

2. In 1823 two wealthy brothers decided to put
together their savings in order to buy a boat and
sail around the world.
Ebenezer had £12 7s 9d and Charles had
£13 15s 6d.
How much did they have altogether?

3. When Ebenezer returned, after 16 years, he found
that he now had seven grandchildren, including a
pair of twins.
He gave each child a present which cost £1 7s 3d.
How much did the presents cost altogether?

132 Part 5

Exercise 4
Some of the statements below are 'true' but some are false.
(a) Write down the statements which are true.
(b) For the statements which are false suggest a more accurate

number.

;_{' b ook i s
thic k.
<::-~
~~~~""l',0~~&~· ~- - - - - - - - - -
'i"VJ ~ An ordinary biro
0 is about 2~ inches long

::§'

An average 12 year old
boy weighs about six stones

Changing units

When a problem has quantities measured in different units the first

thing you must do is change some of the units so that all quantities

are in the same units. ~ ~80cm

• Find the area of the rectangular table top shown. 1.5 m ~
~
Write 80 em as 0·8 m.

Area of table= 1·5 x 0·8 .
= 1·2m2.

Metric and Imperial Units 133

• A piece of metal weighing 2 kg is 50 geach
melted down and cast into small cubes
each weighing 50 g. (jJ(jJ
How many cubes can be made? (jJ(jJ(jJ

Write 2 kg as 2000 g.

Number of cubes= 2000-7- 50.
= 40.

0• A very common error occurs where the units of an area are chan-
ged.
Hereisa The same square

square of side 1m. Area has sides of 100cm.

= 1 m2 1m Area 100 em

= 10000 cm2

lm !OOcm

So 1m2 = 10000cm2 [NOT 100cm2!].

Exercise 5 (c) 3m
1. Find the area of each shape in m2•
(a) (b) 2m l60cm

2.2m .------'

!Ocm

90cm 80cm

2. An urn contammg 56 litres of water is used to fill cups of 140m!
capacity 140 ml. How many cups can be filled?

3. A shopping bag weighing 65 g cor,ttains the following items:
• 3 bags of flour weighing 1·2 kg each
• 1 tube of toothpaste weighing 200 g
• 15 packets of crisps each weighing 25 g.
Find the total weight of the bag and its contents in kg.

4. A large lump of 'Playdoh' weighing 3·6 kg is cut up into 800
identical pieces. Find the weight of each piece in grams.

5. In an equatorial forest a young tree grows at a constant rate of
18 em per day. How much in mm does it grow in one minute?

6. A solid gold coin weighing 0·84 kg is melted down and cast into
tiny coins each weighing 150 mg.
How many small coins can be made?

134 20cm Part 5
20cm
7. Every face of the rectangular block shown is 60 cm
painted. The tin of paint used contains
enough paint to cover an area of 8m2.
How many blocks can be painted
completely?

8. Water is leaking from a tap at a rate of
· 1·2cm3 per second.
How many litres of water will leak from the
tap in a seven day week?

9. Calculate the area of each shape in cm2. D6mml (c)
(a) (b)
0.8m
1.1 m
!
1.8m 8mm

10. The diagram shows the outline of a strip of farmland. 600m

(a) Calculate the area of the field m hectares.
[1 hectare= 10 000 m2]

(b) The field is sprayed with a pesticide and it takes 3 seconds to
spray 100m2.

How many hours will it take to spray the entire field?

11. The waterfall with the greatest flow of water in the world is the
'Guaira' between Brazil and Paraguay. Its estimated average
flow is 13 000 m3 per second. The dome of St Paul's Cathedral
has a capacity of ?800m3. How long would it take the waterfall
to fill the dome?

12. In the U.K. about 90 000 tonnes of tobacco are smoked as
cigarettes each year.
(a) If one cigarette weighs 0·9 g, work out how many cigarettes
are smoked in the U.K. every year. [You can work out the
answer without a calculator].
(b) If a packet of 20 cigarettes costs £2·50, how much is spent in

the U.K. on cigarettes every year?

Fractions 135

5.2 Fractions IIII

Equivalent fractions

iIn the diagram, of the shape is shaded. If you look

at it a different way you can see that 1- of the shape

is shaded.

iThe two fractions and 1- are equivalent fractions.

In practice we choose the simpler fraction which in

this case is 1-·

Fill in the missing numbers to find the equivalent fractions

(a) .l-_- (b)
2 4 TO

Answer·. .2l -- 14. -- i10 Answer·• 17. -- l4l2. -- 13!5!.

Exercise 6

In questions 1 to 6 write down the fraction shaded then find a
simpler equivalent fraction.

3.

4. I I 6. I I I I I I

r--

....__

LJIJLJ

In Questions 7 to 15 fill in the missing numbers

7· 2__ 9 83__ 98 __
· 4-s-TI · u-6-3
6-3 -
11. ?2 = 24 = l!. 12. ?o = 30 = 1.!
!10. = TO = 20

13• TO- -s3---25 14• 2 1- -37- -1-5 15· g7 ---54 -- -49

In questions 16 to 20, each of the three shapes has a certain fraction
shaded. Find these fractions in their simplest form, and so find
which shape is the odd one out.

16. (a) (b) (c)

136 Part 5

m17. (a) (b) (c)
(b) (c)
18. (a) (c)

19. (a) (b)

20. (a) (b)

Multiplying with fractions

• A prize of £5000 is shared equally between 8 people.

tEach person receives of £5000.
t of £5000 = £5000 -;- 8

= £625

tSo to find of £5000 we divided £5000 by 8.

• In a mixed school with 840 pupils, f of the pupils are girls. How

many girls are there?

tWe need to find of 840.
tNow of 840 = 840-;- 7

= 120

t tSo of 840 = 120 x 3 [Because of 840 is three times
tas many as of 840.]
= 360

There are 360 girls in the school.

Fractions 137

-~

To find %of £95

t.We need to work out% x 9

(a) Multiply the numbers on top (the numerators)
(b) Multiply the numbers underneath (the denominators)
(c) Simplify the answer

Sol. x ~ = 285
5I 5

=£57.

Similarly l8. of 124 = l8. x .ilIl.

_372

8

= 46·5 [Using a calculator]

Exercise 7

Calculate the following

! i1. of 100 tonnes 2. of 3500 kg i3. of £496
?6. 2 of66cm
4. i of £15 5. f of 12 hours
i9. of £8-40
7. ~6 of $4000 8. ?o of £55

!10. On each bounce a ball rises to of its

previous height.
How high will a ball bounce the first
time if it is dropped from a height of
2m?

11. In an examination full marks were 92.
How many marks did Sarah get if she

ihad of full marks?

12. Paul's new jeans are 96 em long when they leave the shop. After
washing they shrink to :~ of their previous length.

What is the new length of the jeans?

13. The petrol tank of a car holds 60 litres. How much more petrol

tcan be put into the tank when it is full?

14. A 'super bouncy' ball rises to 7 of its previous height on each
10
bounce. One of these balls is dropped from a height of 4 m.

(a) How high will it rise after one bounce?

(b) How high will it rise after the second bounce?

15. Work out a half of ninety-nine and a half.

t16. On a calculator = 0·1111111

Without using a calculator, write down 960 as a decimal.

138 Part 5

Fractions of fractions !The black section is t of

The grey shaded strip is of the rectangle.

! of the rectangle ...' ------
1
I
I
I

-------' ------~' ------,'------

-------' ------,' ------,'------

The rectangle on the right is divided into 20 equal parts so the black

section is 2~ of the rectangle.

!So t of of the rectangle = 2~ of the rectangle

Notice that l x l5 = ..l.

4 20

Exercise 8

1. What fractions are these? t(c) of%-

(a) toft (b) toft

2. The diagram shows a square of side 1m divided

into four rectangles A, B, C and D. 2
.
(a) Find the areas of A, B, C and D in m

(b) Check that the sum of your answers is 1m2•

A B

~m c D

3. Nicki says that 0·3 x 0·3 = 0·9

Write 0·3 as a fraction.
Multiply the fraction by itself and write the answer as a decimal.
Was Nicki's answer correct?

4. Draw a copy of the rectangle.
(a) Shade in t of the squares.
(b) Draw crosses in t of the unshaded squares.
(c) How many squares are neither shaded nor
have crosses in them?

Fractions 139

5. Draw a copy of the star
(a) Shade in ± of the triangles.

(b) Draw ticks (J) in 1- of the

unshaded triangles.
(c) How many triangles are neither

shaded nor have ticks in them?

6. Javed is painting all the window frames on his parents' house.

tOn the first day he paints of the frames and on the second

1day he paints of the remainder. On the third day he paints the

rest.
If there are 72 window frames on the house, how many did
Javed paint on
(a) the second day
(b) the third day?

7. Richard has a packet of 32 Polos.

For some unknown reason Richard gives f

of his Polos to his sister Jane.
Generous Jane then gives± of her share to
a friend and eats the rest.

Richard meanwhile eats %of his remaining

Polos.
(a) How many Polos does Richard have

left at the end?
(b) How many Polos does Jane eat?

8. A vet treats 36 sick mice with a new antibiotic.

tAfter 6 hours of the mice have recovered and are runmng

around happily.

tAfter 12 hours of the remaining mice are cured but

unfortunately the others have died .
(a) How many mice eventually recovered?
(b) How many mice died?

9. Two friends Gill and Natasha share a large farm field between
them.

lGil has! of the field and Natasha has
tin her part Gill plants of the land with potatoes and the rest

with sugar beet.
In her part Natasha grows carrots in~ of the land and the rest is
devoteo to potatoes.
(a) What fraction of the total area of the field is planted with

(i) sugar beet
(ii) carrots
(iii) potatoes?
(b) Should your answers above add up to I?
If so, check that they do.

140 Part 5

t10. In her will Granny Sheldrake left of her

money to her sister Emily,% of her money to
her grandson Eric and the rest to her cat,
which was to be looked after by Eric.

tEric immediately spent of his inheritance

on a new car and put the rest in the bank.
One day, while driving his new car, he ran
over the cat and consequently inherited the
eat's share of the money.
Eric put this money in the bank.
If Emily inherited £45 000, work out
(a) how much money was left to the cat.
(b) how much money Eric had in the bank after the eat's

'accident' .

11. A new giant Rolo for the Norwegian market is made from 5 g of
chocolate and 7 g of toffee.
How much toffee is needed to make 984 g of these Rolo's?

12. A jar of marmalade is made from 50 g fruit, 140 g sugar and 10 g
water.
How much fruit is needed to make 2 kg of this marmalade?

Questions 13, 14 and 15 are more difficult.

±13. A cylinder is After 60 ml of
full of water water is added the

1cylinder is full. r------~

Calculate the total volume of the cylinder.

±14. A swimming pool is full. After a further 6000 gallons of water

are pumped in the pool is %full.
(a) What is the total volume of the pool?

(b) How many more gallons are required to fill the pool?

±15. By noon one day of the eggs in

a crocodile's nest have hatched.

By midnight a further 5 baby

crocodiles are walking around,

tleaving only of the eggs still to

hatch.

How many eggs were in the nest

originally?

Charts and Graphs 141

5.3 Charts and graphs c

Information is often much easier to understand B
when it is presented in the form of a chart
rather than as words or numbers. Newspapers A
regularly present the results of surveys or
research in the form of pie charts, bar charts
or bar line graphs. Most computers are
programmed so that they can easily display
information in interesting charts, possibly in
colour or with a '3-D' effect.
It is a good idea to use colours freely in your
own work and sometimes people make charts
more eye-catching by drawing small pictures
to illustrate the sectors.

Pie charts

This pie chart shows the afterschool
activities of 200 pupils.

(a) The number of= f:o x 200 = 20

pupils doing
drama
(b) the number of = ~:6 x 200 = 80
pupils doing
sport

(c) the number of = 3~0 x 200 = 25

pupils doing
computing

A packet of breakfast cereal weighing
480 g contains four ingredients:

Oats 120 g
Barley 80 g
Wheat 60g
Rye 220g

Calculate the angles on a pie chart.

The fraction of the packet which is oats = !~~

The angle on the pie chart for oats = !~~ x 360 = 90°.

1°Similarly the angle for barley = 80 x 360 = 60°

142 Part 5

Exercise 9

1. Lara had £24 to spend on presents. The pie chart

shows how much she spent on each person.

How much did she spend on:

(a) her mum (b) her dad

(c) her brother (d) her grandma

(e) her friend (f) her auntie?

[Make sure that your answers add up to £24.]

2. Six hundred families travelling home on a ferry
were asked to name the country in which they
had spent most of their holiday. The pie chart
represents their answers.
(a) How many stayed longest (i) in Portugal?
(ii) in Spain?
(b) What angle represents holidays in Switzerland?
(c) How many families stayed longest in Switzerland?

3. A group of 72 people were asked which language they spoke at

home. Their replies were:

English 40 Greek 9

Urdu 22 Welsh 1

(a) Work out (i) j~ x 360 (ii) ~~ x 360

(iii) j X 360 (iv) 1 X 360.
2 72

(b) Using a compass and a protractor, draw and label an

accurate pie chart to display the languages spoken.

4. A survey was carried out to discover what pets 400 children

kept.

120 kept cats 30 kept rabbits

160 kept dogs 10 kept snakes

80 kept goldfish

(a) Work out what fraction of the children kept:

(i) cats (ii) dogs (iii) goldfish

(iv) rabbits (v) snakes.

(b) Draw and label an accurate pie chart to display this data.

(c) Illustrate your pie chart with suitable drawings.

5. Opinion pollsters asked over a thousand people which TV

channel they liked best. Their answers were:

lTV 30%

Channel 4 15%

Satellite 20%

BBC1 27%

BBC2 8%

Find the angle on a pie chart representing

(a) Channel 4 (b) BBCl.

Charts and Graphs 143

6. A hidden observer watched Stephen in a 60 Jj JrH f<~k4! .>< . ""~" •

minute maths lesson. This is how he spent The ~n5 w .w ,.., .30

his time:

Looking for a calculator 8 minutes

Sharpening a pencil 7 minutes

Talking 32 minutes

Checking the clock 2 minutes

Working 4 minutes

Packing up 7 minutes

Draw an accurate pie chart to illustrate
Stephen's lesson.

7. The pie chart illustrates the sales of four
brands of petrol.
(a) What percentage of total sales does
BP have?
(b) If Shell accounts for 35% of total
sales, calculate the angles x and y.

8. In a survey 320 on an aircraft and 800 people on a ferry were
asked to state their nationality.

Aircraft Ferry
320 800
people people

(a) Roughly what percentage of the people on the aircraft were
from the U.K.?

(b) Roughly how many people from France were on the ferry?
(c) Jill looked at the charts and said 'There were about the same

number of people from Italy on the aircraft and on the
ferry'. Explain why Jill is wrong.