Numeracy
Outcome 1
[INTERMEDIATE 2]
Introduction
3
1.
Reading Scales
4
2.
Bar Charts (simple)
6
3.
Percentage Bar Charts
10
4.
Compound Bar Charts
12
5.
Component Bar Charts
17
6.
Histograms
20
7.
Pie Charts
24
8.
Line Graphs
29
9.
Distance–Time Graphs
35
10.
Ogives or Cumulative Frequency Curves
38
11.
Scatter Diagrams
44
12.
Dot Plots and Stem–Leaf Charts
47
13.
Tables (general)
51
14.
Flow Charts
54
15.
Miscellaneous Charts
60
Tutor Assignment
68
Answers to SAQs
76
INTRODUCTION
The purpose of this pack is to make you familiar with a wide variety of graphs, charts,
tables and other diagrams which either illustrate or gather together data for the purpose of
analysis.
The following is an extract from the National Unit Specification statement of standards.
Don’t be alarmed by its formal vocabulary and style – this pack contains all you need.
Outcome 1
Interpret graphical information when presented as a number of related but
straightforward forms or in a complex form.
Performance Criteria
(a)
Extract information from tables, graphs, charts or diagrams when presented
as a number of related but straightforward forms or in a complex form.
(b)
Explain extracted information appropriately in terms of the context.
You might well have got a fright when you saw the contents section – a huge number of
different things to learn!
NOT SO!
A lot of the stuff you will have done before, at school or at work, or subconsciously while
reading a newspaper article or watching TV.
It often takes a long time to explain in writing an idea or procedure which can actually be
tackled quickly, so get stuck in and ‘enjoy’ it!
Resources
You’ll need a calculator (a scientific one is best because of the way it works, not because of
how many buttons it has).
You’ll also need a protractor for the pie charts on pages 22–26.
Assessment
Your tutor/college/training centre will keep you right about how and when to sit your
formal assessment to achieve this Outcome.
OUTCOME 1: NUMERACY/INT 2
3
READING SCALES
SECTION 1
The most common types of data illustration you will see are bar charts of various types
and line graphs. Each of these requires you to be able to read a scale, and many
students have difficulty with this. So here is a quick lesson on scale reading.
Example 1a
Here is an example of a vertical scale. What numbers are represented by the letters A
and B?
Solution:
First count how many intervals there are between two convenient numbered lines:
130
A
Next, subtract the two numbers on those same lines:
120
B
V
120 – 110 = 10
(130 – 120 is also 10)
V
between 110 and 120 there are 5 intervals
(between 120 and 130 there are also 5 intervals).
110
Divide this number by how many intervals there are:
10 ÷ 5 = 2
This tells you how much each interval is worth. So, starting from the bottom, the lines
go up by 2 each time, like this: 110, 112, 114, 116 and so on.
So B = 112 and A = 126
Example 1b
Here is a horizontal scale. We follow the same procedure.
Number of intervals between two numbered lines: 10
Difference between the numbers: 18 – 17 = 1
Difference ÷ Intervals = 1 ÷ 10 = 0.1
So the lines go 17 (or, better, 17.0), 17.1, 17.2, 17.3, and so on.
So A = 17.2 and B = 17.9
4
OUTCOME 1: NUMERACY/INT 2
17
V
V18
A
B
READING SCALES
?1
For each scale write down the numbers represented by the letters A and B:
(a)
(b)
(c)
43
38
300
(d)
42
200
(e)
300
V
V
A
B
(g)
(f)
300
350
(h)
V
V
A
B
400
320
600
60
B
V
V
B
V
A B
27.4
48
V
(i)
V
B
V
A
36
V
A
V
A
V
V
B
37
V
B
B
47
A
V
46
V
A
V
A
27.3
50
OUTCOME 1: NUMERACY/INT 2
5
BAR CHARTS (SIMPLE)
SECTION 2
A bar chart is a series of bars (horizontal or vertical) where the length or height of each bar
represents the quantity or frequency of the category or item to which it refers. There is
only one scale on a bar chart. The scale must start at 0 (although you will see this rule being
frequently broken in real life !).
Example 2a
This table of cashmere sales:
Cashmere Sales at Blenkinsop’s
Month
January
February
March
April
Unit Sales
60
40
30
15
will result in the following bar chart:
Cashmere Sales
60
Unit Sales
50
40
30
20
10
0
January
February
March
April
In your assessment you could be asked the following sorts of questions:
1.
How many cashmere items were sold in March ?
Solution:
Read off the chart. The answer is 30 items.
2.
How many cashmere items were sold in April ?
Solution:
For this you have to interpolate, i.e. read between the lines. The height of the April bar
appears to be exactly half way between 10 and 20, so we can safely offer 15 as the answer.
6
OUTCOME 1: NUMERACY/INT 2
BAR CHARTS (SIMPLE)
3.
What percentage of the items were sold in February ?
Solution:
We first have to add up all the items, 60 + 40 + 30 + 15 = 145.
February
× 100%
Totalsold
40
=
× 100%
145
= 27.6%
So percentage sold in February =
[Calculator sequence: 40 ÷ 145 × 100 = (display shows 27.586 etc., to be rounded off)]
NB If you are unsure about the calculation of percentages, ask your tutor for extra
assistance.
4.
Estimate how many cashmere items will be sold in May, giving a reason for your
answer.
Solution:
The sales are steadily decreasing, i.e. the trend is downwards, so you could estimate the
May sales as being certainly less than 10, perhaps even less than 5. This is presumably
because of the warmer weather.
OUTCOME 1: NUMERACY/INT 2
7
BAR CHARTS (SIMPLE)
Example 2b
This horizontal bar chart shows a company’s total sales for years 1991 to 1996.
Company Sales 1991-1996
1996
Ye a r
1995
1994
1993
1992
1991
0
200
400
600
800
1,000 1,200
1,400 1,600
1,800
Sales (£000)
1.
What were the total sales for 1994 ?
Solution:
Not £1,400 as you might first think. Look carefully at the label. The scale is in thousands of
pounds. So the sales are actually 1,400 thousand, i.e. £1,400,000.
Always look carefully at the axis labels to see what units you are operating in.
2.
What were the sales in 1993?
Solution:
Interpolation is harder here. The bar looks like it is 3/4 of the way up from 1,000 towards
1,200, so 1,150 or, correctly, £1,150,000 is as near as you’ll get to the answer.
A graph of any sort drawn on computer will not allow fine interpolation like this. You
would need more subdivisions of the scale. Alternatively, you could have a graph drawn by
hand on squared paper which would allow you to pin down the answer more accurately.
But then, you wouldn’t be expecting an accurate reading off a graph anyway. A graph only
gives a quick, visual snapshot of what is happening.
Ideally, any answer you give based on a graph such as the one above should be qualified
with a word such as ‘about’ or ‘approximately’.
8
OUTCOME 1: NUMERACY/INT 2
BAR CHARTS (SIMPLE)
?2
1.
This bar chart shows the diameters (to the nearest mm) of a sample of tomatoes:
Diameters of a Sample of Tomatoes
Number of Tomatoes
16
14
12
10
8
6
4
2
0
37
38
39
40
41
42
Diameter (mm)
(a)
(b)
(c)
This bar chart shows the sales of ice cream (£00) per month by a shop in 1998.
Ice Cream Sales 1998
Month
2.
How many tomatoes measure 38 mm (to the nearest mm)?
How many tomatoes are in the sample altogether?
What percentage of tomatoes measure 40 mm or more ?
Dec
Nov
Oct
Sept
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
0
1
2
3
4
5
6
7
8
9
10
11 12
13
Value of Sales (£00)
(a)
(b)
(c)
(d)
(e)
During which month were sales (i) highest (ii) lowest?
What is the value of sales in (i) October (ii) February?
What is the average monthly value of sales during the year?
What percentage of the year’s sales were during the months of June - August?
Explain the pattern of sales over the year.
OUTCOME 1: NUMERACY/INT 2
9
PERCENTAGE BAR CHARTS
SECTION 3
A percentage bar chart differs from an ordinary bar chart only in that the scale always
refers to a percentage, so the highest it will ever reach is 100, and usually it doesn’t get
anywhere near that.
Example 3
This shows a percentage bar chart which illustrates the principal activities of participants
at an outdoor centre one summer. The table shows the actual percentages calculated to 2
decimal places. How accurately can you read the graph, which is drawn by computer?
Activity
Walking Climbing Canoeing
Yachting Pot-holing
Hang-gliding Total
Actual
Numbers
Percentage
231
104
76
39.83
17.93
13.10
82
24
63
580
14.14
4.14
10.86
100
Percentage Breakdown by Activity
45%
40%
35%
30%
25%
20%
15%
10%
5%
0%
Walking
Climbing
Canoeing
Yachting
Activity
10
OUTCOME 1: NUMERACY/INT 2
Pot-holing Hang-gliding
PERCENTAGE BAR CHARTS
?3
This percentage chart shows the breaking strain, in tonnes to the nearest tonne, of ropes
manufactured in a factory:
%
V
45
40
35
30
25
20
15
10
5
2-3
3-4
4-5
5-6
6-7
V
0
Breaking Strain (tonnes)
(a)
What percentage of ropes have a breaking strain of between 3 and 4 tonnes?
(b)
What percentage have a breaking strain of between 4 and 6 tonnes?
(c)
If there were 650 ropes manufactured altogether,
• how many are represented by the first column?
• how many are represented by the middle column?
OUTCOME 1: NUMERACY/INT 2
11
COMPOUND BAR CHARTS
SECTION 4
A compound (or multiple) bar chart is a method of drawing several bar charts within one,
used for comparing, say, items within a year as well as items between years.
Example 4a
The output of PQR plc over a period of time, in thousands of units, is as follows:
1996
1997
1998
Product A
180
130
50
Product B
90
110
170
Product C
180
180
125
Total
450
420
345
There are two possible compound (or multiple) bar charts you could draw.
Here is the first chart:
Output of PQR plc
200
150
Product A
100
Product B
Product C
50
0
1996
1997
1998
Ye a r
This is perhaps the more normal way of displaying the data: we can see how the output of
each product fared compared with the others within the same year.
Thus the bars for the products are together with no space between them, but the years
themselves are separated by spaces.
Think how confusing the chart would be if there were no spaces at all!
On the next page we have our second option, showing exactly the same information but
with a different emphasis:
12
OUTCOME 1: NUMERACY/INT 2
COMPOUND BAR CHARTS
This second chart shows that we are now comparing how the output changed over the
years for each product, looking at each product separately.
Output of PQR plc
200
150
1996
100
1997
1998
50
0
Product A
Product B
Product C
Product
And now the question you’re just dying to ask: Which of the two charts is more ‘correct’?
The answer, which you probably won’t be too happy with, is that they’re both equally
right; it depends entirely on who is using them and what comparisons are to be made.
That said, you’ll find the first more common than the second, i.e. comparing items within
years, rather than years within items.
They are both read in the same way as a simple bar chart. Notice, however, that a key (or
legend) at the side is essential with this type of chart, otherwise you don’t know what each
shading refers to.
OUTCOME 1: NUMERACY/INT 2
13
COMPOUND BAR CHARTS
Example 4b
The following chart shows the changes in the membership of a tennis club over a period of
three years:
V
60
55
50
45
40
35
30
25
20
15
10
5
0
1.
Seniors
Men
Women
Children
1996
1997
Year
1998
How many more men were members in 1998 than in 1996?
Solution:
55 men in 1998, 35 men in 1996, so 55 – 35 = 20 more men.
2.
Which category of membership remained steady over a period of time?
Solution:
The number of women stayed steady at 35 for 1996 and 1997.
3.
Assuming all the women except for 2 from 1997 rejoined in 1998, how many new
women members were there in 1998?
Solution:
35 in 1997 less 2 is 33 rejoining, then 40 – 33 = 7 new members
4.
What is the general trend in membership?
Solution:
In general, the trend is upwards, with all categories being the same or better over the years
except for Seniors who fell in number during 1998.
5.
If the annual membership fees are: £15 for children, £35 for men and women, and
£20 for seniors, what is this club’s income for 1998?
Solution:
Children: 20 × £15 = £300
Seniors: 15 × £20 = £300
14
Men+Women: 95 × £35 = £3,325
Total income: £3,925
OUTCOME 1: NUMERACY/INT 2
COMPOUND BAR CHARTS
?4
1.
The chart below compares the average prices of a selection of consumer goods in
1990 with the prices of similar goods in 1985.
Costs of Selected Goods
Hi-fi System
Home Computer
1985
1990
Television
Vacuum Cleaner
Washing Machine
0
50
100
150
200
250
300
350
400
450
500
Costs in Pounds (£)
(a)
(b)
(c)
Which item showed a decrease, and by how much did it decrease?
Which item showed the largest increase, and by what percentage?
What is the average annual increase in the price of a hi-fi system over this
period of time?
OUTCOME 1: NUMERACY/INT 2
15
COMPOUND BAR CHARTS
2.
The following compound bar chart shows how the percentage of the population of a
third world country is changing from being rural to being urban.
%
80
70
V
Urban population
Rural population
60
50
40
30
20
10
0
(a)
(b)
(c)
(d)
16
1950
1970
1990
2000
What is the general trend of population movement?
In which years was the urban percentage below 65%?
Estimate the relative percentages for the year 2010 [this is called
extrapolation].
If the population of the country in 1990 was 56,800,000 (to the nearest
hundred thousand), how many of them could be described as being ‘rural’?
OUTCOME 1: NUMERACY/INT 2
COMPONENT BAR CHARTS
SECTION 5
Component bar charts (sometimes known as sectional or stacked bar charts) have the
bars divided into sections or components.
Example 5
This component bar chart illustrates exactly the same information as the compound bar
chart on page 14.
Number of Members
Tennis Club Membership
130
120
110
100
90
80
70
60
50
40
30
20
10
0
Seniors
Men
Women
Children
1996
1997
Ye a r
1998
The height of each column represents the total number of members for each year.
The shaded parts represent the individual components (members in different categories).
NOTE: In this chart, the shadings in the legend are in the same order (top to bottom) as the
shadings in the bars, but sometimes you get them the other way round, so beware !
The changes in the numbers of children (being at the bottom of each column) and seniors
(being at the top) are easy to see, but it’s not quite so easy to see the changes in the number
of men or women.
1.
How many members are there in each category in 1997?
Solution:
The numbers here on the 1997 column are relatively simple to pick out, but they can be
much more difficult, so I will show you the solution on the assumption that the numbers are
harder than they really are.
Starting at the bottom section, the number of children = 15
The next line up is at 50, and is the total of women + children, so take off the 15 and we
have 35 women.
OUTCOME 1: NUMERACY/INT 2
17
COMPONENT BAR CHARTS
The next line up is at 90 and is the total of men + women + children, but the last line was at
50, so 50 from 90 gives us 40 men.
Finally the top line gives the total for the whole year = 115. Subtract the value of the
previous line from this (90) and we are left with 25 seniors.
So the number of members in each category is: 15 children, 35 women, 40 men, 25 seniors.
?5
1.
This component bar chart shows the average thinking, braking (and total stopping)
distances for a certain car travelling at various speeds:
Stopping Distances at Various Speeds
Speed in km per hour
Thinking
100
85
65
50
30
0
18
Braking
5
10 15 20 25 30
35 40 45 50 55 60
Distance in metres
65
70
75 80
85
(a)
What is the total stopping distance when travelling at 30 km per hour?
(b)
At 100 km per hour, how much of the stopping distance is ‘thinking distance’?
(c)
At 65 km per hour, how much of the stopping distance is ‘braking distance’?
(d)
If 5 miles = 8 kilometres, roughly what is the stopping distance when the car
is travelling at 30 miles per hour?
OUTCOME 1: NUMERACY/INT 2
COMPONENT BAR CHARTS
A company is divided into 5 sectors, boringly named A, B, C, D and E.
The chart below shows how the employees of the company are divided up among
the sectors:
Annual Employment
No. of Employees
2.
110
100
90
80
70
60
50
40
30
20
10
0
Sector A
Sector B
Sector C
Sector D
Sector E
1998
Ye a r
1999
(a)
Has the total number of employees changed over the two years ?
(b)
Just by looking at the various shadings (try to ignore the numbers if you can),
say which sectors employ more people, which employ less, and which
employ the same numbers in 1999 as they did in 1998.
(c)
Now use the chart to write down the actual numbers of employees per sector,
in each year. (The numbers are all multiples of 5 to make it easier!)
(d)
For each year, calculate the percentage of employees who work for Sector D.
OUTCOME 1: NUMERACY/INT 2
19
HISTOGRAMS
SECTION 6
A histogram is a diagram which is often used to illustrate grouped data. It is a specialised
diagram, and is perhaps used more by statisticians than by the general public.
It initially appears to be a bar chart but there are several major differences. The most
important feature is that the area of each bar is proportional to the frequency of that group
(whereas with an ordinary bar chart, it is the height of the bar that is proportional). Other
differences you will see in Example 6 below are that
(i)
there are no spaces between the bars.
(The spaces between the bars in a bar chart are purely cosmetic – you can have the
bars against each other if you wish – but with a histogram there must be no spaces.
The reasons for this are fairly complex and go way beyond what you need to know
for this Unit.)
(ii)
the vertical scale does not (or rather should not, because you will see many in print
that are wrong) have a label such as ‘frequency’. Instead, there should be an area key
somewhere (see Example below). But the scale must start at 0.
(iii)
the horizontal axis usually starts a half or one whole bar-width from its left hand end,
and is a continuous scale which shows the boundaries of each class or interval.
Example 6
The table below shows the amount of money spent per week by a number of families on
their normal shopping:
Amount spent (£)
No. of families
30-40
40-50
50-60
60-70
70-80
80+
15
27
32
16
9
3
Here is the resulting histogram:
Weekly Shopping Spending
35
30
= 5 families
25
20
15
10
5
0
30
40
50
60
70
Amount Spent (£)
20
OUTCOME 1: NUMERACY/INT 2
80
90
HISTOGRAMS
Points to note:
(i)
Since the first group or class or interval is 30-40 and the second is 40-50, by
convention the first stops just short of £40, i.e. is up to and including £39.99, and the
second starts at £40.00. Similarly for the others.
These intervals can also be written 30 < 40 and 40 < 50 (the symbol < means
‘less than’).
Another way you might see them written is ‘30 less 40’ and ‘40 less 50’.
(ii)
The last interval in the table is written ‘80 +’. This is called an open-ended interval
and means ‘£80.00 and over’. Strictly speaking, it could go up to £100 or beyond.
But in order to graph the information we must make it stop somewhere. The
convention is to make it the same width (i.e. £10 wide) as the one next to it. This is
why the histogram shows it stopping at £90.
(iii)
The diagram should include somewhere a key (or legend) showing what a particular
area represents, even if it is really quite obvious from the picture.
(iv)
Notice how the horizontal axis has a continuous scale. The edges of each bar line up
with the boundaries of the group it represents.
The types of questions we can ask are very similar to the ones asked of a bar chart, e.g.
‘What percentage of the families spent between £40 and £60 per week ?’
(27+32)
× 100%
(15+27+32+16+9+3)
59
=
× 100%
102
= 57.8%
Percentage =
The questions in the next exercise are all based on groups similar to the ones in the
example, e.g. 30-40, 40-50 and so on.
If the groups are 30-39, 40-49, etc., where each group starts on a different number from the
number on which the previous group ended, things get a lot more difficult.
If the groups are of different widths, e.g. 30-40, 40-50, 50-70, 70-100, then matters get
more complicated still, but I think you really have to be specialising in the study of statistics
to cope with that, so we’ll body-swerve them.
OUTCOME 1: NUMERACY/INT 2
21
HISTOGRAMS
?6
1.
This histogram shows the age structure of a group of people flying on a large
aeroplane:
Ages of Passengers
150
140
= 10 passengers
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
Years
22
(a)
How many of the passengers are under the age of 20?
(b)
How many are aged between 30 and 50?
(c)
If all those under 10 years travel free, those between 10 and 60 pay £280
each, and those over 60 pay an extra £20 each to cover the extra cost of
insurance, what is the total income for the airline from this flight?
(d)
What, then, is the average price paid per passenger (to the nearest penny)?
OUTCOME 1: NUMERACY/INT 2
HISTOGRAMS
2.
One Saturday there were 200 goals scored in all the games in the English football
league.
All the games were split into 10-minute intervals, and the number of goals scored
during each interval was noted.
The histogram below shows the results.
No. of Goals Scored in 10-minute Intervals
56
52
48
44
40
36
32
28
24
20
16
12
8
4
0
represents 4 goals
0
10
20
30
40
50
60
70
80
90
Minutes
(a)
Remembering that each half of a game lasts 45 minutes, do you think there
were more goals scored during the first half or during the second half ? Can
you tell for sure?
(b)
How many goals were scored in the first 20 minutes?
(c)
What fraction of all the goals were scored during the last 20 minutes?
(d)
What percentage were scored within 5 minutes of half-time?
OUTCOME 1: NUMERACY/INT 2
23
PIE CHARTS
SECTION 7
A pie chart is in the shape of a circle, with the angle of each sector representing the quantity
or frequency. This quantity or frequency is the same proportion of the total as the angle is
of 360° (degrees). It is the circular equivalent of the component bar chart, and so it follows
that you have to know the total which is to be represented by the circle.
Reading a pie chart to any accuracy at all requires the use of a protractor. If you don’t know
how to use one, ask your tutor.
Example 7a
This example involves the use of a pie chart for rough comparisons only.
This chart shows the breakdown of the production costs of an item in a factory.
Describe some of the information conveyed.
Production Cost Breakdown
Direct Materials
Direct Labour
Production Overheads
Office Costs
••
••
Solution:
Just by eye you can see various things, for instance:
• Direct Materials and Office Costs together account for half the production cost.
• Likewise, Direct Labour and Overheads together account for half the cost.
• The four components, in decreasing order of size, are Direct Labour (largest, accounting
for about one third of the total), followed by Direct Materials (also close to one third),
Office Costs, and Overheads.
• Office Costs and Overheads together form about one third of the total, but it isn’t easy
(by eye) to see how this relates to the other two components.
24
OUTCOME 1: NUMERACY/INT 2
PIE CHARTS
Example 7b
We use the same pie chart as in Example 7a, but this time we are told that the total
production cost is £135,400 and we want a more accurate breakdown.
Solution:
The first step is to measure, as accurately as we can, the sizes of the angles at the centre of
the circle where the two radii meet (ask if you need help with this):
Direct Labour
Office Costs
135°
60°
Direct Materials
Production Overheads
120°
45°
Production Cost Breakdown
Direct Materials
60°
120°
45°
135°
Direct Labour
Production Overheads
•
••• Office Costs
To express these angles in terms of money, we take each as a fraction of 360° and then
multiply this by the total cost, i.e. by £135,400.
So for, say, Direct Materials, the cost is
No. of degrees in sector
120
× total cost =
× £135, 400
360
360
= £45,133.33
The calculation is slightly easier if you notice that 120 is exactly one third of 360, so
dividing 135,400 by 3 is a quicker method.
However, you should note that an answer as precise as this is not a sensible one for several
reasons.
• The total figure of £135,400 is being quoted only to the nearest hundred pounds, so
giving an answer to the nearest penny is a bit silly. A more sensible answer would be,
say, £45,100 which is also to the nearest £100.
OUTCOME 1: NUMERACY/INT 2
25
PIE CHARTS
• It is highly doubtful if the actual figure for Direct Materials is so convenient that it would
measure at exactly 120°. It is probably anything between 119° and 121° (which, if you
care to work them out, gives answers between £44,757.22 and £45,509.44).
Having worked out the figure for Direct Materials, it is helpful if you notice that the angle
for Office Costs is 60°.
This is half of the Direct Materials angle of 120°.
This makes the Office Costs amount equal to £45,133 ÷ 2 = £22,566.50 which rounds off
sensibly as £22,600. You should always look out for such shortcuts!
The other figures are:
Direct Labour:
Overheads:
135
× £135, 400 = £50,775 (round to £50,800)
360
1
45° is of 135°, so overhead costs are £50,775 ÷ 3 = £16,925 (round to £16,900)
3
If you are having to work them all out, it is useful to check that the monetary figures for all
four components do actually add up to £135,400.
If percentages are also required, they can be calculated in the usual way, but use the nonrounded figures if possible.
In this particular example, because we didn’t round off very much at all in relation to the
total amount, the answers for Direct Materials are the same to 1 decimal place:
45,133.33
45,100
×100% and
×100% are both 33.3% correct to 1 decimal place
135,400
135,400
If you are lucky, the pie chart may have the percentages already calculated for you:
Office Costs
16.7%
Direct Materials
33.3%
Production Overheads
12.5%
Direct Labour
37.5%
26
OUTCOME 1: NUMERACY/INT 2
PIE CHARTS
?7
1.
The pie chart shows the number of viewers who watched the four TV channels
provided in a certain country.
Measure the angles at the centre and hence calculate the percentage of viewers who
watched each channel.
Viewing Figures
Channel 1
Channel 2
Channel 3
Channel 4
OUTCOME 1: NUMERACY/INT 2
27
PIE CHARTS
2.
This pie chart shows how the same viewers are distributed shortly after the
introduction of a fifth channel.
Viewing Figures
Channel 1
Channel 2
Channel 3
12 Channel 4
12 Channel 5
12
(a)
Which channel still has the same proportion of viewers?
(b)
Which channel appears to have suffered most from the competition provided
by Channel 5?
(c)
By what percentage have the viewing figures for Channel 4 dropped?
(d)
If there are 5,666,400 viewers altogether, how many are watching
(i)
(ii)
(e)
Channel 1
the new Channel 5?
Channel 5 is funded by advertising. XYZ plc will only advertise on a channel if
the channel reaches more than 20% of the viewing public.
Will this company advertise on Channel 5?
28
OUTCOME 1: NUMERACY/INT 2
LINE GRAPHS
SECTION 8
Origin
Horizontal Axis
x
V
So we always plot the independent variable on
the horizontal (or x) axis and the dependent
variable on the vertical (or y) axis.
Vertical Axis
Of the two variables, one is usually dependent
on the other.
Dependent Variable
A line graph shows the relationship between two variables (the quantities or sets of values
you want to plot) by means of lines joining the plotted data points. The lines joining these
points are normally drawn straight, although
curves may be used in certain circumstances.
V y
Independent Variable
Thus, say we are graphing profit against the number of units of production; the profit will
clearly depend on how many units are produced, so profit is plotted vertically and
production horizontally.
Time is always plotted on the horizontal axis (unless we are drawing a learning curve).
Line graphs are often drawn when bar charts are actually more appropriate, but the line
graph can have more visual impact.
In the examples that follow, you will notice that the scales on the axes
don’t have to start at 0; they can start at any convenient number.
Purists, however, will often put a 0 at the origin (the point where the
axes meet) and then show a break in the axis before the first number of
the scale.
110
105
100
0
Something like this:
OUTCOME 1: NUMERACY/INT 2
29
LINE GRAPHS
Example 8a
This line graph shows company sales in millions of pounds over a period of nine years.
You’ll notice that I have chosen to start the vertical scale at 18. Some textbooks might well
disagree with this and insist on the axis starting at 0 with a break in it as illustrated on the
previous page. It’s only a matter of convention and choice. Don’t lose any sleep over it.
(BUT remember that bar charts must start at 0!)
If we had started our line graph at 0 and kept the scale consistent, as we should, the graph
might look like this.
Year
The differences in values are not very noticeable at all.
This is why we can start the axes at a convenient number and not necessarily at 0.
30
OUTCOME 1: NUMERACY/INT 2
LINE GRAPHS
What kind of questions are you likely to be asked about line graphs?
Here are a few:
1.
What were the sales in 1993?
Solution:
£18.6 million
To get this you go
2.
right along the Year axis till you get to 1993
then up till you reach the graph
then left till you reach the Sales axis and read off the answer.
In what year were the sales £19 million?
Solution:
1995
To get this you go
3.
up the Sales axis till you reach 19 (million)
then across till you reach the graph
then down till you reach the Year axis and read off the answer.
Estimate the sales for 1999.
Solution:
Assuming all conditions remain constant, probably somewhere between £19.8 m and £20 m.
This is an example of extrapolation, or using your common sense, looking at the general
trend, then making a good guess about the future.
4.
When were the sales £19.4 m?
Solution:
The answer looks like half-way between 1996 and 1997 but this is nonsense. In fact, the
question doesn’t make any sense, and you wouldn’t be asked it (unless someone was trying
to really make you think!).
The sales are year totals so you cannot interpolate like this and split the horizontal axis into
months or any other sub-unit.
The main problem with line graphs is that it looks as if you can always interpolate, but in
fact you have to be very careful indeed!
This information could just as easily have been shown as a bar chart, but line graphs are
often used for visual impact, especially in situations where time is one of the variables.
OUTCOME 1: NUMERACY/INT 2
31
LINE GRAPHS
Example 7b
The graph below shows the temperature of a patient taken at half-hourly intervals starting
at 2 a.m.
Temperature (°C)
Because the horizontal scale is continuous we can interpolate and estimate the temperature
at intermediate points between those actually marked. It is also reasonable to assume that
the temperature will move in a steady fashion and not fluctuate wildly. For example it won’t
suddenly zoom up to 45° just after 5 o’clock and then zoom back down again (although
there are illnesses where this sort of thing can happen!).
1.
What is the temperature at 6.30 a.m.?
Solution:
About 40.8°C
2.
What is the temperature at 4.15 a.m.?
Solution:
About 39.5°C
3.
For how long is the temperature above 40°C?
Solution:
It’s very difficult to do this accurately here, but it first hits 40°C at about 4.30 a.m. and
drops back down to 40°C a little after 8.30 a.m., so the answer is ‘just over 4 hours’. If the
graph had been drawn by hand, the answer would have been more accurately extracted
from the graph, but it would not necessarily have been more correct ! Look at the following
picture. The graph assumes that the line joining the two dots is straight, but it could have
curved above or below the assumed straight line, giving totally different readings!
And I apologise very sincerely if you are a student of medicine in any way and I have got the
technicalities of patient temperature wrong. The graph’s the thing!
32
OUTCOME 1: NUMERACY/INT 2
LINE GRAPHS
?8
1.
The graph below shows the daily sales in a shop for the period of a week.
(a)
What were the sales on Monday?
(b)
By how much did Thursday’s sales exceed Wednesday’s sales?
(c)
What, roughly, were the total sales for the week? Why ‘roughly’?
(d)
What were the average daily sales for the week?
(e)
Explain the pattern of sales. What would you expect next week’s graph to
look like?
OUTCOME 1: NUMERACY/INT 2
33
LINE GRAPHS
The graph shows the height of a plant in millimetres for the first month of its growth.
Height
2.
Days after Sowing
(a)
For how many days did it not grow at all?
(b)
Estimate its height on day 17.
(c)
Estimate on which day it was 40 mm tall.
(d)
During which 5-day period did the plant experience
(i)
(e)
34
the most growth
(ii)
Estimate its height 40 days after sowing.
OUTCOME 1: NUMERACY/INT 2
the least growth?
DISTANCE–TIME GRAPHS
SECTION 9
A distance–time graph is a special line graph where the distance travelled is plotted on the
vertical axis and the time taken is plotted on the horizontal axis. Various points on the
graph are labelled A, B, etc. for reference.
Example 9
The graph below charts the progress of a group of walkers from Broughty Ferry to
Arbroath. Discuss some of the features of the graph.
Solution:
Here is a discussion of some of the features of the graph.
The vertical axis label tells you that the distance is measured from Broughty Ferry.
The point A shows that the walkers are starting at 7 a.m. It also shows that they are 0 km
from Broughty Ferry, i.e. they are at Broughty Ferry.
The point B shows they have reached a point 5 km from the start by 8 a.m.
OUTCOME 1: NUMERACY/INT 2
35
DISTANCE–TIME GRAPHS
The line BC is horizontal. This means that the distance from Broughty Ferry is not
changing, so they aren’t going anywhere; perhaps they are having a rest. At any rate, they’re
standing still.
This rest lasts for 1 interval on the horizontal axis. Since 2 intervals equal half an hour, one
interval represents 15 minutes. Then, at 8.15 a.m., they set off again.
Their next rest (points D to E) lasts from 9.30 to 10 a.m., and they are 10 km from the start
at that time.
Clearly, point F represents Arbroath. Once there, they wait 15 minutes, then they arrive
back at point H, distance 0 (i.e. Broughty Ferry) by 11.37 or so, so presumably they
returned by bus (?).
We can calculate their average speed during any part of their journey using the formula
Speed =
Distance
Time
so their return trip is at a speed of 14÷ 22 = 0.6363 km per minute (because the 22 is in
minutes). If we multiply this by 60 we get 38.2 km per hour.
Of course, this represents their average speed. They won’t be travelling at 38 km/h or so
for every minute of their journey back. At some points the bus will travel faster, at other
points it will be stationary (picking up passengers, waiting at traffic lights, etc.) so an
average speed is the best that we can do.
We can compare average speeds roughly by looking at the slope (or gradient) of the line.
GH has the steepest slope so represents the fastest speed, CD has the least steep line so
represents the slowest speed.
In fact, if you look carefully at the graph, CD and EF both have the same slope, but AB is
marginally steeper, showing a slightly higher average speed between those two points.
36
OUTCOME 1: NUMERACY/INT 2
DISTANCE–TIME GRAPHS
?9
The following graph shows the progress of a cyclist and a motorist from Ayton to Seaton
which is 60 kilometres away. Both travelled along exactly the same roads.
The graph does not say which is the cyclist and which is the motorist, but it should be
obvious to you which is which.
60
55
•
50
Distance (km) from Ayton
45
40
35
•
•
30
25
20
15
10
•
5
0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
1p.m.
2p.m.
3p.m.
4p.m.
5p.m.
At what time does the cyclist leave Ayton?
Does the cyclist actually arrive at Seaton in the time scale shown on the graph?
At what time is the cyclist 26 km from Ayton?
How far from Seaton is the cyclist at 4.15 pm?
Calculate the cyclist’s average speed before he has a rest.
At what two times did the motorist meet the cyclist on the road, and how far were
both from Ayton at each meeting?
Calculate the motorist’s average speed on
(i)
the outward journey only
(ii)
the homeward journey only
(iii) the total journey including the stop.
OUTCOME 1: NUMERACY/INT 2
37
OGIVES OR CUMULATIVE FREQUENCY CURVES
SECTION 10
The ogive is a fairly specialist curve, used very extensively in statistics, to analyse data for a
large sample or population. The word ogive comes from architecture and describes the
kind of S-shaped curve you often get in cornices. The ogive represents cumulative
frequency, which is simply a running total.
The graph shows you how many items there are below a certain value.
You will notice that the vertical axis (which is the cumulative frequency axis) always starts
at 0. The horizontal axis starts at whatever is the number below which there are zero
items.
Example 10a
The graph below shows the heights of primary school children at the start of Primary 1.
The questions and solutions below should help to explain how this graph is used.
1.
How many children are less than 60 cm tall?
Solution:
600 children
Remember that the graph tells you the frequency below a certain value.
38
OUTCOME 1: NUMERACY/INT 2
OGIVES OR CUMULATIVE FREQUENCY CURVES
2.
How many children are less than 70 cm tall?
Solution:
1,150 children
(This is the best estimate you can make here. In fact, some would say that it is too good an
estimate!)
3.
How many children are between 60 cm and 70 cm tall?
Solution:
Subtract the two previous answers: 1,150 – 600 = 550
4.
What percentage of children are over 70 cm tall ?
Solution:
The total number of children involved is 2,900 (that’s how high the graph goes).
The number below 70 cm is 1,150, so subtracting from 2,900 gives us 1,750 children who
are over 70 cm tall.
Then the calculation of the percentage:
1,750
× 100% = 60.3%
2,900
Note
Since the graph tells you how many children are below a certain height, strictly speaking
question 4 should be asking for the number who are ‘70 cm or over’, which is not the same
as ‘over 70 cm’. But with such a large number of children and such poor graphical
precision it is not worth splitting hairs over this. With smaller numbers and/or more
precise graphs, however, such hairs do have to be split, so be warned!
OUTCOME 1: NUMERACY/INT 2
39
OGIVES OR CUMULATIVE FREQUENCY CURVES
Example 10b
This ogive illustrates the weekly wages as paid by a company:
1.
How many employees are there in the company?
Solution:
The highest value the graph reaches is 210, so that’s it.
2.
How many employees earned between £100 and £200 a week?
Solution:
40 employees earned less than £100 and 190 earned less than £200, so subtracting the two
gives us 190 – 40 = 150 employees.
3.
Copy this table and use the graph to complete it:
Wage (£)
0-50
50-100 100-150 150-200 200-250 250-300
Wage (£)
0-50
50-100 100-150 150-200 200-250 250-300
No. of employees
20 - 0
= 20
40 - 20 150 - 40 190 - 150 200 - 190 210 - 200
= 20
= 110
= 40
= 10
= 10
No. of employees
Solution:
40
OUTCOME 1: NUMERACY/INT 2
OGIVES OR CUMULATIVE FREQUENCY CURVES
4.
If all the employees were lined up in order of wage, roughly how much does the
middle employee earn?
Solution:
There are 210 employees altogether so the middle one is the 105th employee in line.
Go up the vertical scale till you reach 105, then go horizontally till you reach the graph and
go back down again. You hit the £ axis at approximately £130 (slightly under that actually).
This means that half of the people earn less than £130 and half of them earn more than
£130.
(We cannot use this graph to find how many people earned exactly £130!)
NOTE: we call the wage of £130 the median wage.
The median is the value of the variable (here, wages)which splits the wages (when taken in
order of size) in half. 50% of the data set has a value below the median, and 50% has a
value above the median.
The median is a type of average used where a ‘true’ average (such as the sum of the data set
divided by the number of values, technically called the mean) might give a false impression.
This often happens in calculating average wages, where one or two extra large (or extra
small) numbers can drag the average off-beam and present a false picture.
5.
Look at the lowest 25% of earners. What is the highest wage that someone in this
group can earn?
Solution:
There are 210 employees altogether, and 25% of this is 52.5.
So go up the vertical scale till you reach approximately 52.5 (i.e. the 52.5th employee, if
there is such a thing !), then go across till you reach the graph and back down to the £ axis.
You hit it at roughly £110. This means that the ‘poorest’ quarter of the workforce earn
below £110.
We can call this wage of £110 the ‘25th percentile wage’, i.e. the wage earned by the
people who are 25% of the way up from the bottom of the heap.
Similarly, the ‘90th percentile’ wage is that earned by those 90% from the bottom of the
heap, in other words 10% from the top.
Percentiles are quantities which are used quite often, especially in the fields of psychology
and sociology.
OUTCOME 1: NUMERACY/INT 2
41
OGIVES OR CUMULATIVE FREQUENCY CURVES
? 10
1.
This ogive shows the ages of school pupils on a holiday trip:
(a)
How many pupils went altogether?
(b)
How old was the eldest pupil?
(c)
How many were aged
(i)
(ii)
(iii)
(iv)
(v)
42
under 11
11 or over
under 16
16 or over
between 11 and 16?
OUTCOME 1: NUMERACY/INT 2
OGIVES OR CUMULATIVE FREQUENCY CURVES
This ogive shows the cumulative daily takings in a shop over the period April–June:
cf
V
100
90
•
•
•
•
80
Number of days
•
70
•
60
50
•
40
30
•
10
8,000
7,500
7,000
6,500
6,000
5,500
5,000
4,500
V
0 •
4,000
2.
Takings (£)
(a)
(b)
(c)
(d)
(e)
(f)
How many days’ takings are represented by this graph?
For how many days were the takings below £5,000?
For how many days were the takings above £6,500?
The takings on the worst 10 days were below how much?
The takings on the best 10 days were above how much?
The owner of the store demands that the turnover for the top 20% of days
must be more than £6,000. Is the owner satisfied?
OUTCOME 1: NUMERACY/INT 2
43
SCATTER DIAGRAMS
SECTION 11
A scatter diagram (or scatter graph) is a method of plotting the values of two variables,
using axes and plotting points just as you did in line graphs, but without trying to join any of
the points together (this is because usually there are too many dots for any obvious line to
be drawn). The resulting distribution of dots or crosses can then give us an idea whether
the relationship between the two variables is weak or strong. The technical word for this
relationship is correlation, and the calculations and study of correlation form a significant
part of many maths and statistics courses.
Here is a very simple scatter diagram which plots
four models of aeroplane. One axis shows the
number of passengers the plane can carry, the other
axis shows the hourly cost of operating the plane.
Carrying Capacity
Example 11a
•B
•D
•C
A shows a plane which costs little to operate and
•A
carries few passengers.
O
Operating Costs
Operating
Costs
B shows a plane which costs little to operate (but a
bit more than A) and carries many passengers.
C shows a plane which costs a lot to operate and
carries few passengers (but a few more than A).
D shows a plane which costs a lot to operate (but slightly less than C) and carries many
passengers (but not quite as many as B).
The diagram does not show any relationship whatever between the two variables (i.e.
between operating costs and carrying capacity).
Here is another scatter diagram. Each dot
represents an aeroplane in an airline’s fleet. Clearly
they show that there is a fairly high correlation
(i.e. a strong relationship) between the number of
passengers a plane can carry and the profit which
that plane generates for the airline.
The correlation is positive, i.e. the higher the
value of one variable, the higher the value of the
other.
44
OUTCOME 1: NUMERACY/INT 2
Profit
Example 11b
•
•
•
0
•
•
•
•
•
•
•
•
•
•
•A
Carrying
Capacity
Carrying
Capacity
SCATTER DIAGRAMS
The plane marked A is clearly a rogue well away from the rest of the group: a large plane –
the largest in the fleet actually – which is also bringing in very poor profits. The reasons for
this could be many – it could fly a route which is costly in terms of fuel but has few
passengers, or it has been suffering peculiar maintenance problems. Whatever the reason, it
stands out like a sore thumb and so the company can investigate the reason.
Example 11c
In fact it is higher here than in 11b. How do we
know? Because the dots lie closer to a straight line
here. If the dots actually all lie on a straight line
then we have perfect correlation; you can’t get
better than that.
•
Efficiency Index
Here, a factory manager has plotted the age of
each machine in the factory against the efficiency of
that machine. Again, as with the aeroplanes in
Example 11b, the correlation is high.
• •
• •
•A
•
• • •
•
•
•
Age
ofofMachine
Age
Machine
0
But in this case our correlation is negative. The higher the age of the machine, the lower
is the efficiency index. (In other words, the older the machine is, the worse it works.)
The machine marked A is not very old (in fact it’s the third youngest machine in the factory)
but it has lousy efficiency. Clearly, an investigation is in order.
Here, someone has plotted the relationship
between the wages paid to the employees of a
company and the physical height of those
employees.
Clearly, we would expect there to be no
relationship at all, and the scatter diagram shows
this with a ‘plum pudding’ effect.
Height of Employees
Example 11d
0
•
•
•
•
•
•
•
•
•
•
• •
•
•
•
Wage
paid
by by
Firm
Wage
paid
Firm
There is no correlation at all, either positive or
negative. We say that the correlation is zero.
If the dots had happened to lie reasonably close to a straight line, we would hope that the
correlation was spurious, i.e. completely coincidental, and that the firm did not have any
policy of paying by height!
OUTCOME 1: NUMERACY/INT 2
45
SCATTER DIAGRAMS
? 11
For each diagram say whether it displays positive correlation, negative correlation, or zero
correlation. Write a sentence or two about each one. Try to put the strengths of the six
correlations in order of size.
(2)
Volume of Production
Company Sales (£)
(1)
0
(3)
Absenteeism by Employees
(4)
Examination Marks
Crop Yield (tonnes)
0
Rainfall (mm)
(5)
0
Age of Student
(6)
Traffic Speed (mph)
Production Costs (£)
0
46
0
Company Profits (£)
Production Level (units)
OUTCOME 1: NUMERACY/INT 2
0
Volume of Traffic (000s)
DOT PLOTS AND STEM–LEAF CHARTS
SECTION 12
Dot plots and stem–leaf charts are two possible ways of noting down data quickly and
illustrating it equally quickly in order to see what the data is ‘trying to tell you’.
Example 12a
Suppose you are a teacher collecting the marks of 30 of your pupils alphabetically. These
are the results in your mark book:
Anderson
Black
Brown ***
Chalmers
Chan
Cooke
De Marco
Dundas
Dunn
Durie
35
75
20
63
42
65
50
25
35
73
Eaton
Edwards
Elliot
Emerson ***
Falconer
Garland
Haddow
Jackson
Kerr ***
McArthur
65
35
31
80
65
62
68
69
58
37
MacCormack
McDonald
MacKay
Patterson
Robertson
Scott
Singh
Thomson
Wilson
Woods
32
62
71
68
65
59
60
62
63
74
I have put asterisks at three of the names and will show their relative positions in the
diagrams coming up.
To do a dot plot you simply draw a horizontal axis encompassing all the marks awarded,
then go through the list, one at a time, starting with Anderson and ending with Woods and
draw a dot to represent each pupil, stacking them up if more than one gets a certain mark:
Brown
•
20
•
Kerr
•
• • •• •
30
•
40
•
50
Emerson
•
• • • •• ••
60
•
•
•• ••• • •••
•
70
80
You can very quickly see that there is a large clustering round the middle 60s and another,
smaller cluster, in the 30–40 band. If the dots are drawn carefully enough, the dot plot
almost looks like a bar chart as well.
If I want to do a very quick visual analysis for the marks gained in an exam by my own class,
I’ll very often do a dot plot.
OUTCOME 1: NUMERACY/INT 2
47
DOT PLOTS AND STEM–LEAF CHARTS
For the stem–leaf chart, the 10s become the stem and the units become the leaves.
Details of how to do this are in the section for Outcome 2. The completed chart looks like
this:
V
Brown
5
2 5 5 5 7
Kerr
V
0
1
2
0
0
1
0
8 9
2 2 2 3 3 5 5 5 5 8 8 9
2 3 4
V
2
3
4
5
6
7
8
Emerson
where
3 2 means a mark of 32
5 0 means a mark of 50
6 3 3
means two pupils scored 63 each
and so on.
A stem–leaf chart must have some sort of key or legend to tell us that, in this particular
example, 3|2 means 32, because it could just as easily mean 3.2, or 320, or 3,200.
Notice how the numbers in each leaf are in increasing order of size as you move away from
the stem.
48
OUTCOME 1: NUMERACY/INT 2
DOT PLOTS AND STEM–LEAF CHARTS
Example 12b
A researcher is looking at the heights (in centimetres) reached by crops which have been
treated with two different fertilisers. She can compare these by drawing two dot plots one
lined up under the other, or by drawing a back-to-back stem–leaf chart.
Maxigrow Dot Plot:
Plant X is 144.9 cm tall
143
••
144
145
V
•
•
• •
•• •
•• • • • •
146
147
148
Nutribloom Dot Plot:
Plant Y is 145.7 cm tall
143
144
•
•
•
• • • •
V
•
145
•
•
•
•
146
•
•
147
•
148
Back-to-back Stem–Leaf Chart
Maxigrow
Nutribloom
9 4 143
9 6 6 6 5 5 5 5 3 3 2 0 144
0 145
146
Plant X
147
0 144 4 means that one plant has
148
reached 144.4 cm with Nutribloom,
4
1
0
2
0
7
3 3 3 5 7
0 3 7
6
Plant Y
and another plant has reached
144.0 cm with Maxigrow
Both charts clearly show that the plants grow taller having been treated with Nutribloom.
However, they also show (and this would not be so obvious just from the list of numbers)
that the Maxigrow plants are clustered quite closely together, whereas the range of heights
of the Nutribloom plants is much greater, i.e. they are spread out a lot more. This may
merit further investigation. A farmer with a mechanised harvester may well prefer the
slightly smaller plant if it is more consistent in height.
OUTCOME 1: NUMERACY/INT 2
49
DOT PLOTS AND STEM–LEAF CHARTS
? 12
1.
The dot plot below shows the marks of 20 students in a maths exam which was
marked out of a total of 70 marks.
•
•
•
•
20
(a)
(b)
(c)
(d)
2.
30
••
50
60
•
•
•
70
The stem–leaf chart below shows the age distribution of members of a table tennis
club:
(a)
(b)
(c)
8
0
0
0
2
1
0
8
0
1
1
2
4
1
9
1
1
1
2
4
3
1
2
3
3
5
1
2
4
5
7
2
2
4
8
7
2
2
5
8
346667889
3334555567779
5799
9
Note:
5
2 means a person aged
52 years.
How many members does this club have?
Give the ages of
(i) the youngest member of the club (ii) the oldest (iii) the middle one.
Teenagers pay nothing at all for annual membership.
Members aged 20-39 pay £17.50 a year each, those aged 40-65 pay £35.50 a
year each, and the over 65s pay a flat £10 a year each.
How much is raised annually from membership fees?
The back-to-back stem–leaf chart shows the heights of male and female students in a
class of 37. It shows that the smallest woman is 1.50 metres tall and that the tallest
man is 1.97 metres tall.
Men
8
9
9 7 5 5
9 8 6
7 4
50
40
•
Did anyone score 100% in the exam ? If ‘yes’, how many did so ? If ‘no’, how
do you know?
What was the lowest mark scored?
If the teacher said that the pass mark was 40 out of 70, what percentage of
the class passed?
The most common or frequently occurring mark is called the mode or the
modal mark. What is the mode or modal mark of this distribution?
1
2
3
4
5
6
7
3.
••
•
•
•
•
• •• •
6
5
3
6
1
5
2
3
2
0
Women
15
16
17
18
19
0
1
2
0
1 2 4 4 5 6
3 3 5 7
6
1
OUTCOME 1: NUMERACY/INT 2
(a) How tall is the tallest woman?
(b) How tall is the smallest man?
(c) What does the chart tell you about
the relative sizes of this group of
men and women?
(d) Calculate the average female height.
TABLES (GENERAL)
SECTION 13
Tables are an extremely convenient way of organising numerical and other data which needs
to be known reasonably accurately before (perhaps) being illustrated by a graph or chart
(which, as you have seen, can lose a lot of the precision of the original data).
Example 13a
This table shows the results of a survey of spectacle use of a sample of 267 men and women.
Need spectacles
Wear
Don’t wear
Men
Women
TOTAL
42
34
76
Don’t need
spectacles
5
27
32
76
83
159
TOTAL
123
144
267
Here are the kinds of questions you could get asked in your assessment:
1.
How many men need spectacles?
Solution:
42 + 5 = 47
2.
What fraction of the women who need spectacles don’t actually wear them?
Solution:
34 + 27 = 61 women need spectacles but 27 don’t wear them, so the required fraction is
27/61 which can be decimalised to give 0.443 or 44.3%.
3.
What percentage of all the women do not need spectacles at all?
Solution:
83/144 = 57.6%
(Note the fraction: women who don’t need spectacles/total all women)
4.
What percentage of all the people surveyed are men who wear spectacles?
Solution:
42/267 = 15.7%.
(Note fraction: men who wear spectacles/total all people
The most important thing is to watch what number goes on the bottom of the fraction (i.e.
the denominator of the fraction).
OUTCOME 1: NUMERACY/INT 2
51
TABLES (GENERAL)
Example 13b
This table shows how many attempts a group of men and women made to pass their driving
test. The table is not complete, so the first job is to complete it.
Passed on
1st attempt
Men
Women
18 and under
Over 18
18 and under
Over18
Total
10
32
83
Took 2-3
attempts
15
12
24
59
Took 4 or more
attempts
0
5
1
13
Total
24
63
155
Solution:
I have labelled the blank spaces A to F for reference.
Men
Women
Total
18 and under
Over 18
18 and under
Over18
Passed on
1st attempt
A
B
10
32
83
Took 2-3
attempts
C
15
12
24
59
Took 4 or more
attempts
0
5
1
D
13
Total
24
E
F
63
155
To get F, add 10 and 12 and 1 to get 23.
To get E, add 24 and 23 (F) and 63, then subtract the answer from 155 to get 45.
To get D, add 0 and 5 and 1, then subtract the answer from 13 to get 7.
To get C, add 15 and 12 and 24, then subtract the answer from 59 to get 8.
To get A, add 8(C) and 0 and subtract from 24 to get 16.
To get B you can either add 16(A) and 10 and 32 and subtract the answer from 83 or you
can add 15 and 5 and subtract the answer from 45(E). Either way, B is 25.
You could have calculated some of these values in a different order, but you can see that you
cannot try for either A or B or E first.
You can then be asked for all sorts of fractions and percentages just as in the previous
example.
52
OUTCOME 1: NUMERACY/INT 2
TABLES (GENERAL)
? 13
1.
This table shows an analysis of the promotion prospects of staff in a company. They
have been grouped by age with 30 being the boundary age between the two groups.
Unpromoted
Promoted grades 1-3
Promoted grades 4-6
Total
(a)
(b)
(c)
(d)
(e)
2.
Male
Under 30 30& over
16
18
10
23
9
19
35
60
Female
Under 30 30& over
21
29
15
17
12
14
48
60
Total
84
65
54
203
How many staff are involved in the survey?
How many males, aged 30 or over, have been promoted to grades 1-3?
What percentage of the entire work force here is male?
What percentage of all the females is unpromoted?
Perform another calculation and make a comment on the relative promotion
prospects of men and women in this company.
Here, an office manager has analysed the number of errors in documents of various
sizes. ‘< 5’ means ‘less than 5’ and ‘5 ≤ 10’ means ‘between 5 and 10 inclusive’ and
so on.
Number of
errors
in the
document
(a)
(b)
(c)
(d)
(e)
(f)
(g)
<5
5 ≤ 10
10 < 20
20 & over
Total
Category of document
Small (under Medium (100- Large (over
100 pages)
200 pages)
200 pages)
15
34
42
8
54
1
10
19
35
Total
67
80
79
30
256
Copy and complete the table to fill in the six missing numbers.
How many documents have been dealt with?
What fraction of all the documents are categorised as ‘medium’?
What fraction of all the documents have between 10 and 20 errors in them?
What fraction of the large documents have less than 5 errors in them?
What fraction of the documents with 20 errors or over are in the ‘small’
category?
What fraction of all the documents are medium-sized with under 5 errors?
OUTCOME 1: NUMERACY/INT 2
53
FLOW CHARTS
SECTION 14
A flow chart is a diagram which shows the order in which instructions have to be carried
out in order to complete a task.
Specialist flow charts (e.g. fault diagnostic charts) can have many types of boxes, but simple
flow charts have three types of box.
• Terminator boxes are rectangles with rounded corners, with the words START or STOP
in them.
START
STOP
• Instruction or operation boxes are ordinary rectangles.
In here you are told to DO
something, perhaps
CALCULATE a formula or
WORK SOMETHING OUT.
• Decision boxes, which ask a question, are diamond shaped. Each decision box can only
ask one question, to which there are only two answers, YES or NO.
The answer
to the question in here
MUST BE EITHER YES OR NO,
you have no other
choices!
This means that you need two decision boxes to cater for an ‘in between’ situation – see
Example 14b later on.
54
OUTCOME 1: NUMERACY/INT 2
FLOW CHARTS
Example 14a
This is a simple flow chart for crossing the road.
You start at the box labelled ‘start’ and follow the
arrows, doing whatever the operation boxes tell
you to do.
When you reach a decision box, ask yourself the
question and follow the appropriate arrow.
As you see, the chart incorporates a loop – if the
road is not clear you wait and rejoin the routine
above the ‘look left’ instruction box.
You then go round and round the loop until the
answer to the question is ‘Yes’. The road is then
clear, and you can cross the road.
START
LookLeft
left
Look
LookRight
right
Look
Look Left
left again
Is the
road
clear ?
No
Wait
Yes
Cross
Crossthe
theRoad
road
STOP
OUTCOME 1: NUMERACY/INT 2
55
FLOW CHARTS
Example 14b
This chart calculates the amount of
commission payable for sales of
various amounts.
START
Three different rates are payable:
Read the sales figure
5% on sales below £10,000
7% on sales between £10,000 and
£20,000
10% on sales over £20,000
and we need two decision boxes to
cope with that.
There is no decision box for a question
such as ‘are sales between £10,000
and £20,000?’ We only have questions
involving ‘less than’ or ‘more than’ and
use these to cope with situations
involving ‘between’ as you see below.
1.
Are the
sales more than
£10,000 ?
No
Commission =
5% of sales
Y es
Are the
sales more than
£20,000 ?
No
Commission =
7% of sales
Y es
Commission =
10% of sales
STOP
Calculate the commission for sales of £12,000.
Solution:
Start at the top and follow the arrow.
The answer to the first question is ‘yes’, so go down to the second question and the answer
to that is ‘no’ so go right. The commission is 7% of sales, i.e. 7% of £12,000 which is
£840.
2.
Calculate the commission on sales of £30,000.
Solution:
The answer to both questions is ‘yes’ so we calculate 10% of £30,000 and get £3,000.
Note: If the sales were exactly £10,000 or exactly £20,000 what would be the answers to
the questions?
Think about it! Is ‘more than £10,000’ the same as ‘£10,000 or more’?
56
OUTCOME 1: NUMERACY/INT 2
FLOW CHARTS
? 14
START
1.
The chart on the right calculates the
discount, in a sale, calculated on the
original price of the goods.
Original
cost of
of Item
item
Original cost
Does item
cost less than
£40 ?
The discount depends on the original
price of the goods.
Y es
Discount = 5%
No
Calculate the discount on
(a)
(b)
(c)
(d)
Does item
cost less than
£80 ?
a blouse normally costing £36
a suit normally costing £140
a jacket normally costing £96
a sweater normally costing £64
Y es
Discount = 7.5%
No
Does item
cost less than
£120 ?
Y es
Discount = 10%
No
Discount = 12%
STOP
START
Read length of hire
Is hire f or
more than 3
d a yss?
day
No
Cost = £33 x
ays
number of dday
No
Cost = £29 x
ays
number of dday
Y es
Is hire f or
more than 5
d a yss?
day
Y es
Is hire f or
more than 10
d a yss?
day
No
Cost = £20 x
a yss
number of dday
2.
The chart on the left shows the cost
of hiring a car. The daily hire charge
depends on the length of the hire.
Find the cost of hiring a car for
(a)
(b)
(c)
(d)
6 days
2 days
10 days
5 days
Y es
Cost = £15 x
ays
number of dday
STOP
OUTCOME 1: NUMERACY/INT 2
57
FLOW CHARTS
3.
The flow chart on the right calculates
a salesperson’s monthly salary, which
is a basic wage of £500 plus
commission which depends on the
value of the sales.
Notice the instruction box for
calculating B.
START
Are sales less
than £20,000 ?
No
Calculate A =
4% of £20,000
Y es
This is how flow charts get you to
find, for example, a percentage only of
sales which are over £20,000. A
finds 4% of sales £20,000, whereas B
finds 6% but only of that proportion
of sales which exceeds £20,000.
Commission = 4% of Sales
Calculate B = 6% of
(Sales – £20,000)
Salary = Commission + £500
Salary = A + B + £500
STOP
Find the monthly salary of a person who sells goods to the value of
(a)
(b)
(c)
(d)
4.
£16,200
£28,400
£37,200
£5,000
The flow chart below is used to calculate the mileage allowance for employees of a
company. The allowance depends on the number of miles travelled.
Reminder: the symbol > means ‘greater than’.
START
N = number of miles
No
Is N > 1,200 ?
No
Is N > 2,000 ?
Y es
Y es
Allowance =
£N x 0.36
Allowance = £432 +
£(N – 1200) x 0.25
STOP
Calculate the mileage allowance due for these distances:
(a)
(b)
(c)
(d)
58
2,100 miles
475 miles
1,580 miles
1,850 miles
OUTCOME 1: NUMERACY/INT 2
Allowance = £312 + £N
£N x 0.16
FLOW CHARTS
5.
The flow chart on the right
shows calculation of mileage
allowance which depends on
the engine size as well as the
number of miles travelled.
ws
ch
s
(b)
(c)
(d)
6.
This chart shows the cost
of joining a squash club.
No
Rate is 25.7 p
per mile
Rate is 30.4 p
per mile
350 miles in a 1,500
cc car
173 miles in a 980 cc
car
79 miles in a 2 litre car
475 miles in a 1.1 litre
car
An easier one to finish
with!
Is engine
capacity > 1199
cc ?
Y es
Find the allowances due for
the following:
(a)
START
Is no of miles >
200 ?
Y es
No
Allowance = Miles x
Rate per mile
Allowance = 200 x Rate
plus (miles–200) x 11.3p
STOP
START
h
g
Was the person a
member during the
prev ious y ear ?
Y es
No
y
Find the cost of joining for
a family of 2 adults, an 18
year old, and a 15 year
old, all of whom have
been members before, and
a 12 year old who is
joining for the first time.
Joining Fee is
£26.50
Y es
Is the person 18
or under ?
No
Annual Fee is £38
Annual Fee is £23
Cost of Membership =
Joining Fee + Annual
Fee
STOP
OUTCOME 1: NUMERACY/INT 2
59
MISCELLANEOUS CHARTS
SECTION 15
A wide variety of charts are used in all walks of life, and there is no hope here of covering
more than a tiny fraction of them. Here are a few.
Example 15a – Distance charts
This distance chart shows the shortest distance by road between certain towns in Scotland
– similar charts are found in the backs of road
atlases.
To find the distance between Edinburgh and
Perth:
Ab
d
er
ee
119
Ed
142
look DOWN the column from Edinburgh
look ALONG the row from Perth
and where the two meet is the answer, 44
miles.
n
i
u
nb
43
rg
h
G
s
la
go
w
v
n
er
es
s
104
158
169
In
81
44
61
115
Pe
226
125
84
251
145
rt
h
St
n
ra
ra
er
? 15a
(a)
How far is it from Aberdeen to Perth?
(b)
How far is it from Stranraer to Edinburgh?
(c)
A salesman drives from Aberdeen to Glasgow to Inverness and back to Aberdeen.
His car averages 30 miles to the gallon. There are 4.5 litres to the gallon and petrol
costs him 79.5p per litre. Calculate the cost of the round trip.
• Inverness
Here is an alternative way to illustrate
the same information, although half of
it has been omitted for the sake of
neatness.
104
81
251
Perth
60
Glasgow •
84
Stranraer
OUTCOME 1: NUMERACY/INT 2
•
119
•
43
61
Questions (a), (b) and (c) above
would be answered in exactly the
same way.
• Aberdeen
115
44
125
• Edinburgh
MISCELLANEOUS CHARTS
Example 15b – Weight/Height Chart
A weight/height chart is frequently found in health magazines and doctors’ waiting rooms.
It shows, for instance, that the average weight of a female of small build who is
5 feet 2 inches tall is either 51 kilograms or 8 stones.
You will of course note that its accuracy is severely limited, partly because it is trying to fit a
lot of information onto one chart.
Male weight
Small build
Large build
kg
st
lb
kg
st
lb
53
54
55
57
59
60
62
64
66
68
69
71
73
75
77
8
8
8
8
9
9
9
10
10
10
10
11
11
11
12
4
7
10
13
3
7
11
1
5
9
13
3
8
12
2
61
62
63
65
67
69
71
73
75
77
79
81
83
86
88
9
9
10
10
10
10
11
11
11
12
12
12
13
13
13
8
11
0
4
7
12
3
7
11
1
6
11
2
7
11
Height
metres ft
1.475
4
1.500
4
1.525
5
1.550
5
1.575
5
1.600
5
1.625
5
1.650
5
1.675
5
1.700
5
1.725
5
1.750
5
1.775
5
1.800
5
1.825
6
1.850
6
1.875
6
1.900
6
in
10
11
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
Female weight
Small build
Large build
kg
st
lb kg
st lb
45
7
2 53
8
5
47
7
5 54
8
8
48
7
8 56
8 11
49
7 11 57
9
0
51
8
0 59
9
4
52
8
3 61
9
8
54
8
7 63
9 12
56
8 11 64
10
2
58
9
1 66
10
6
59
9
5 68
10 10
61
9
9 70
11
0
63
9 13 72
11
5
65
10
3 74
11 10
? 15b
Use the chart to estimate the weights of:
(a)
a male, large build, 1.7 metres tall
(b)
a female, large build, 5 feet 6 inches tall
(c)
a male, small build, 5 feet 11 inches tall
(d)
a female, medium build, 1.6 metres tall
OUTCOME 1: NUMERACY/INT 2
61
MISCELLANEOUS CHARTS
Example 15c – Wallpaper Chart
Height of Wall
Distance round the room in metres (doors and windows included)
from Skirting
9
2.15-2.30 m
2.30-2.45 m
2.45-2.60 m
2.60-2.75 m
2.75-2.90 m
2.90-3.05 m
3.05-3.20 m
4
5
5
5
6
6
6
10 12 13 14 15 16 17 18 19 21 22 23 24 26 27 28 30
5
5
5
5
6
6
7
5
6
6
6
7
7
8
6
6
7
7
7
8
8
6 7 7 8
7 7 8 8
7 8 9 9
7 8 9 9
8 9 9 10
8 9 10 10
9 10 10 11
8
9
10
10
10
11
12
9
9
10
10
11
12
13
9
10
11
11
12
12
13
10
10
12
12
12
13
14
10
11
12
12
13
14
15
11
11
13
13
14
14
16
12
12
14
14
14
15
16
12
13
14
14
15
16
17
13
13
15
15
15
16
18
13
14
15
15
16
17
19
A wallpaper chart is frequently seen in DIY stores. It allows you to calculate how many
rolls of wallpaper you need for the walls of a room. (Ceilings are worked out separately.)
For instance, how many rolls of wallpaper would you need if your room is a rectangle
measuring 4.3 metres by 3.8 metres and the wall height is 2.83 metres?
Solution:
You start by getting the distance round the room.
2 lengths + 2 breadths = 4.3 + 4.3 + 3.8 + 3.8 = 16.2 m
You’ll have to round this UP to the next number on the chart, which is 17 m. So go along
the top row till you reach the column headed 17 m, then down until you reach the interval
in which 2.83 finds itself. This is the 2.75 - 2.90 m interval.
The number you see there is 10, so you need 10 rolls of wallpaper.
? 15c
Calculate the number of rolls you need for each of these rooms:
(a)
4.5 m long by 3.7 m broad, 2.58 m high
(b)
3.7 m long by 2.9 m broad, 2.61 m high
(c)
5.9 m long by 5.7 m broad, 3.01 m high
(d)
8.8 m long by 6.3 m broad, 3.10 m high
62
OUTCOME 1: NUMERACY/INT 2
MISCELLANEOUS CHARTS
Example 15d – Loan Repayment Table
A loan repayment table is frequently seen in banks and building societies.
It shows the monthly repayment for a loan of £100.
Term
(months)
6
12
18
24
30
36
42
48
54
60
APR
(%)
22.2
23.6
23.9
23.8
23.6
23.3
23.1
22.8
22.5
22.3
With Insurance
Without Insurance
Total amount Includes
Monthly
Total amount Monthly
payable (£)
premium (£) repayment (£) payable (£) repayment (£)
109.32
3.17
18.22
105.96
17.66
117.60
5.06
9.80
111.96
9.33
125.46
6.40
6.97
117.90
6.55
133.20
7.46
5.55
123.84
5.16
149.70
8.44
4.99
129.90
4.33
148.68
9.52
4.13
135.72
3.77
156.66
10.50
3.73
141.96
3.38
164.64
11.52
3.43
147.84
3.08
173.34
12.65
3.21
153.90
2.85
181.80
13.82
3.03
159.60
2.66
Suppose you wish to borrow £500 and pay it back over 3 years without any insurance.
What is the monthly repayment and the total paid back?
Solution:
Three years is 36 months, and to pay back £100 over 36 months is £3.77 per month, giving
a total repayment of £135.72 .
So, for £500, multiply these figures by 5.
5 × £3.77 = £18.85 per month
5 × £135.72 = £678.60 repaid in total (the extra £178.60 you paid back
over and above the £500 borrowed is the interest)
? 15d
Calculate the monthly repayment and the total paid back for the following loans:
(a)
£800 borrowed, repaid over 5 years, with insurance
(b)
£2,300 borrowed, repaid over 3½ years, without insurance
(c)
£3,250 borrowed, repaid over 4½ years, with insurance
OUTCOME 1: NUMERACY/INT 2
63
MISCELLANEOUS CHARTS
Example 15e – Compound Interest Table
A compound interest table is used by accountants. It shows how much £1 will amount to
when invested for a certain number of years at a particular rate of interest.
Year
1%
2%
3%
4%
5%
6%
7%
8%
9%
1
2
3
4
5
1.010
1.020
1.030
1.041
1.051
1.020
1.040
1.061
1.082
1.104
1.030
1.061
1.093
1.126
1.159
1.040
1.082
1.125
1.170
1.217
1.050
1.103
1.158
1.216
1.276
1.060
1.124
1.191
1.262
1.338
1.070
1.145
1.225
1.311
1.403
1.080
1.166
1.260
1.360
1.469
1.090
1.188
1.295
1.412
1.539
6
7
8
9
10
1.062
1.072
1.083
1.094
1.105
1.126
1.149
1.172
1.195
1.219
1.194
1.230
1.267
1.305
1.344
1.265
1.316
1.369
1.423
1.480
1.340
1.407
1.477
1.551
1.629
1.419
1.504
1.594
1.689
1.791
1.501
1.606
1.718
1.838
1.967
1.587
1.714
1.851
1.999
2.159
1.677
1.828
1.993
2.172
2.367
11
12
13
14
15
1.116
1.127
1.138
1.149
1.161
1.243
1.268
1.294
1.319
1.346
1.384
1.426
1.469
1.513
1.558
1.539
1.601
1.665
1.732
1.801
1.710
1.796
1.886
1.980
2.079
1.898
2.012
2.133
2.261
2.397
2.105
2.252
2.410
2.579
2.759
2.332
2.518
2.720
2.937
3.172
2.580
2.813
3.066
3.342
3.642
It shows that, for example,
£1 invested for 5 years at 8% interest per annum (compound) amounts to £1.469
So £10 invested for 5 years at 8% will amount to 10 × £1.469 = £14.69
and if we subtract the original £10 from this we find that £4.69 is interest, i.e. money gained
by the investment over that period of time.
? 15e
Calculate what the following amounts of money will be worth after investment:
(a)
£200 after 11 years at 9%
(b)
£2,000 after 6 years at 4%
(c)
£6,540 after 3 years at 6%
64
OUTCOME 1: NUMERACY/INT 2
MISCELLANEOUS CHARTS
Example 15f – Conversion Tables
A conversion table is very popular in diaries and in travel brochures:
Millimetres – Inches
This line
shows that:
5 inches = 127.0 mm
and also that
5 mm = 0.20 inches
mm or
mm inches
25.4
1
50.8
2
76.2
3
101.6
4
V
127.0
5
152.4
6
177.8
7
203.2
8
228.6
9
254.0
10
inches
0.04
0.08
0.12
0.16
0.20
0.24
0.28
0.32
0.35
0.39
mm or
mm inches inches
508
20
0.79
762
30
1.18
1016
40
1.58
1270
50
1.97
1524
60
2.36
1778
70
2.76
2032
80
3.15
2286
90
3.54
2540 100
3.94
5080 200
7.88
? 15f
1.
Use the table above to express
(a)
83 millimetres in inches (find the quantities for 80 and for 3, then add them)
(b)
142 inches in millimetres
2.
Here is another table, which converts litres to gallons and vice versa:
(a)
A British motorist in
France fills up his petrol
tank and buys 35 litres how many gallons is this?
Litres – Gallons
litres or
litres gallons gallons
4.55
1
0.22
9.09
2
0.44
13.64
3
0.66
18.18
4
0.88
22.73
5
1.10
27.28
6
1.32
31.82
7
1.54
36.37
8
1.76
40.91
9
1.98
45.46
10
2.20
(b)
A motorist in Britain fills
up her tank with 40 litres
of petrol and pays £30.32
for it. This is equivalent
to how much per gallon?
(c)
The government is
worried about public
reaction when the price
for one gallon reaches £4.
What is the equivalent price per litre?
litres or
litres gallons gallons
90.9
20
4.40
136.4
30
6.60
181.8
40
8.80
227.3
50 11.00
272.8
60 13.20
318.2
70 15.40
363.7
80 17.60
409.1
90 19.80
454.6
100 22.00
909.2
200 44.00
OUTCOME 1: NUMERACY/INT 2
65
MISCELLANEOUS CHARTS
Holiday charts can be a bane in the life of anyone going on holiday!
Example 15g – Holiday Charts
HOTEL
GRECIAN HOTEL
SPARTAN ARMS
MARATHO
ON BEACH PHAROS HOTEL
Board Basis
HalfBoard Bed & Br
HalfBoard Bed & Br
HalfBoard Bed & Br
Adults sharing
2
2
No. of Nights
7 14
2
2
2
2
2
2
2
2
3
HalfBoard Bed & Br
3
2
2
2
2
7 14
7 14
7 14
7 14
7 14
7 14
06 Apr - 12 Apr 289 385
n/a n/a
285 380
250 339
295 409
239 299
279 379
n/a n/a
13 Apr - 16 Apr 309 399
n/a n/a
304 395
270 341
319 429
265 315
290 395
n/a n/a
17 Apr - 19 Apr 299 375
n/a n/a
293 369
273 348
315 419
265 295
290 395
n/a n/a
20 Apr - 03 May 275 369
265 325
270 359
251 349
300 419
239 279
269 369
n/a n/a
04 May - 12 May 299 389
272 339
285 379
225 351
320 435
245 299
290 399
239 285
13 May - 24 May 339 439
301 418
330 420
319 389
340 465
275 339
334 440
269 325
25 May - 31 May 359 475
336 442
351 464
321 438
345 485
289 349
355 479
279 339
01 Jun - 14 Jun 369 482
338 451
362 471
341 447
349 489
279 339
355 485
275 325
15 Jun - 28 Jun 375 491
350 460
368 479
344 451
359 500
285 349
369 505
276 329
29 Jun - 05 July 385 500
355 465
379 489
353 458
369 520
299 379
385 525
279 345
06 July - 12 July 400 525
385 500
392 515
368 499
379 535
305 385
390 535
285 349
13 July - 19 July 415 549
399 510
400 525
374 513
395 559
315 399
415 559
299 359
Departures on or between
7 14
ULT
ALL PRICES ARE PER ADU
hild 50%, 3rd
d child 75%, 4th child free
e
Deductions for children (sharing with adultss): 1st child 25%, 2nd ch
Flight Supplements per person including ch
hildren: Hea
athrow nil; M anchester £8; Glasgow /Edinburgh £19
Insurance (per person, including children): 7 days £17..50 ; 14 day
ys £35
Find the cost of a 7-day holiday for 2 adults and 1 child at the Marathon Beach hotel, flying
from Glasgow on 10 June and staying on a bed and breakfast basis.
Solution:
The table shows the basic price per person to be £279. So 2 adults cost £558. The child
gets a 25% reduction, so pays 75% of the price. 75% of £279 is £209.25.
Furthermore, we need 3 flight supplements from Glasgow of £19 each = £57 and then 3
lots of insurance for 7 days, 3 × £17.50 = £52.50.
So total price is £558 + £209.25 + £57 + £52.50 = £876.75.
? 15g
(a)
Find the cost for 2 adults flying for 7 days from Heathrow to the Spartan Arms Hotel
(half board basis) on 1 May.
(b)
Find the cost for 2 adults and 3 children flying from Manchester for a 14-day holiday
to the Pharos Hotel on 3 June (half board).
66
OUTCOME 1: NUMERACY/INT 2
MISCELLANEOUS CHARTS
That brings us to the end of the first Outcome.
You should now try the Tutor Assignment which follows. Your tutor will contact you with
the results, and will also make arrangements with you for your assessment.
I hope you enjoyed working through this pack.
OUTCOME 1: NUMERACY/INT 2
67
TUTOR ASSIGNMENT
Note: Don't submit only the answers to your tutor. If you make a mistake your tutor
won't know where the mistake lies and will be unable to help you. Show all your
working. This means writing down the numbers you use in any calculations and also
showing what the calculations are, even though you use a calculator to carry them out.
T1
A company has three divisions A, B and C. The graph below gives details of the
number of people employed in each division during a 3-year period. Numbers have
been rounded to the nearest hundred.
Employees by Division
No. of Employees (hundred)
35
30
25
Division A
20
Division B
15
Division C
10
5
0
1997
1998
1999
Calendar Year
(a)
(b)
(c)
Comment on the general trend of employment in the three divisions.
What is the total number of people employed in each year?
For each year express the division B number as a percentage of the total for that
year.
T2
Three farmers specialise in arable farming. The component graph shows how they use
their land area for each of the crops they grow.
Armour has 500 hectares, Blazer has 750 and Cardigan has 900 hectares.
68
OUTCOME 1: NUMERACY/INT 2
TUTOR ASSIGNMENT
How many hectares does each farmer devote to barley?
Note - you will have to estimate the percentages so there will be a variety of ‘correct’
answers!
Land Use
100%
90%
80%
Percentage
70%
60%
Oats
50%
Barley
40%
Wheat
30%
20%
10%
0%
Armour
Blazer
Cardigan
Farmer
T3
The histogram below shows information about the average time in minutes taken for a
group of personnel to carry out a task.
Task Times – Histogram
25
represents 5 people
20
15
10
5
0
0
5
10
15
20
25
30
35
40
45
50
Time (minutes)
(a)
(b)
(c)
How many people took between 20 and 30 minutes?
How many people took between 30 and 50 minutes?
How many people took less than 20 minutes?
OUTCOME 1: NUMERACY/INT 2
69
TUTOR ASSIGNMENT
T4
A large group of students were asked how much money they had on their person at
that moment, and the pie chart below shows the result.
(a)
(b)
70
Use a protractor to measure the angle at the centre for each sector
If there was a total of 132 students interviewed, calculate the number of students
represented by each sector.
OUTCOME 1: NUMERACY/INT 2
TUTOR ASSIGNMENT
T5
The line graph below shows a salesman's attempts to get away from home one day.
During the journey he is, at varying times, stuck in a traffic jam, on a dual carriageway,
on a motorway, and on an ordinary road in heavy traffic.
Progress from Home
60
G
Distance from Home
50
F
40
E
30
20
C
10
D
B
A
0
0
15
30
45
60
75
90
Minutes from Starting
(a)
(b)
(c)
Use your skill and judgement to decide which section of graph refers to which
part of the journey.
Use the formula ‘speed equals distance divided by time’ to calculate his speed
during the fastest part of the journey, where the distances are given in miles.
Find the average speed over the whole journey.
T6
The ogive shows how many people earned less than a certain wage in a place of work.
(a)
(b)
(c)
(d)
(e)
(f)
Roughly how many staff are there altogether?
How many people earn less than £250?
What is the lowest wage and what is the highest wage earned in this
establishment?
How many staff earn above £300?
How many earn between £225 and £275?
What is the top wage earned by the most poorly paid 30 people employed here?
OUTCOME 1: NUMERACY/INT 2
71
TUTOR ASSIGNMENT
No. of staff earning less than a certain wage
100
90
80
No. of Staff
70
60
50
40
30
20
10
0
125
150
175
200
225
250
275
300
325
350
Wage (£)
T7
The scatter diagram below shows the relationship between the number of secondary
school pupils and teachers in the various Scottish regions in 1990/91. (Figures are
taken from the Scottish Abstract of Statistics 1991.)
Pupil/Teacher numbers in Scottish Secondary
Schools by Region 1990/91
No. of Teachers (thousand)
3.5
3
2.5
2
1.5
1
0.5
0
0
10
20
30
40
No. of Pupils (thousand)
(a)
(b)
(c)
72
Would you say the correlation was strong or not?
Give a point estimate for the number of teachers you would expect in a region
with 27,000 pupils.
Using the point on the extreme right of the scatter diagram, calculate (roughly)
the pupil : teacher ratio which appeared to be the norm in Scottish regions in
1990/91.
OUTCOME 1: NUMERACY/INT 2
TUTOR ASSIGNMENT
T8
The stem–leaf chart gives the values of sales contracts of a firm over a period of time.
2
3
4
5
6
7
8
9
(a)
(b)
(c)
0
2
0
0
3
0
2
0
1
2
0
0
3
1
2
0
3
4
1
1
3
1
3
2
5
5
2
1
4
1
3
2
5
7
2
1
4
2
5
3
6
8
2
5
5
2
6
5
8
8
3
6
6
3
9
8
5
7
6
5
where
9
5 7 7 9
7
6 7 8 8 9
5 7 7 7 9 9
5
2
means a contract worth
£52,000 to the nearest £
5 6 7
How many contracts were there altogether?
What is the value of the poorest contract?
What was the total value of the most lucrative 10% of the contracts?
T9
The number of faulty items made per day by a machine is catalogued as follows:
No of faulty items
No of days (frequency)
0
5
1
2
2
3
3
7
4
4
7
3
9
2
10
1
11
3
15
1
The first entry shows that on 5 days no faulty items were reported.
(a)
(b)
(c)
Over how many days was the record kept?
What was the total number of faulty items reported in that time?
If the average number of faulty items per day exceeds 4.2 in any month
management gets upset. What will be their reaction to this month's figures?
OUTCOME 1: NUMERACY/INT 2
73
TUTOR ASSIGNMENT
T10
This combined chart compares total numbers of rail and air passengers in Scotland
1985-1990. The left-hand vertical axis refers to train journeys (bar chart) and the
right-hand axis refers to air journeys (line graph). (I know you haven't seen a graph
like this in the Unit, but try the question anyway!)
Rail/Air Passenger Numbers - Scotland
58
12
57
55
8
54
6
53
52
4
51
Air Journeys (million)
Train Journeys (million)
10
56
2
50
49
0
1985
1986
1987
1988
1989
1990
Year
Rail
(a)
(b)
(c)
Air
Discuss briefly the general trend of each mode of transport.
The total receipts from rail passengers in 1985 was £94.2 million. What was
the average price per rail journey in 1985? Why does it appear low?
In 1987 there were 453,000 aircraft movements. What appears to be the
average number of passengers per movement? Why does it appear low?
T11
Look at the compound interest table on page 64. How much will £25,000 amount to
after 10 years’ investment at 7% per annum, compounded annually?
T12
Look at the holiday chart on page 66. Calculate the total cost of a holiday for 2 adults
and 2 children going for 14 days half board at the Grecian Hotel, departing from
Edinburgh on 10 July.
74
OUTCOME 1: NUMERACY/INT 2
TUTOR ASSIGNMENT
T13
This flowchart shows how commission is calculated by a company for its sales
persons.
START
YES
TOTAL
TOTAL
SALES <<
SALES
£60,000?
£60,000?
NO
TOTAL SALES <
£100,000?
NO
YES
SALARY = £1,500 PLUS
0.6% OF SALES
SALARY = £1,860 PLUS
0.8% OF (SALES - £60,000)
SALARY = £2,660 PLUS
1% OF (SALES - £100,000)
STOP
Calculate the salary payable for sales of
(a) £45,000
(b) £82,000
(c) £119,000
OUTCOME 1: NUMERACY/INT 2
75
ANSWERS TO SAQs
ANSWERS
? 1: Answers
(a)
(c)
(e)
(g)
(i)
A
A
A
A
A
= 36.4
= 230
= 310
= 46.4
= 53
B
B
B
B
B
=
=
=
=
=
37.2
280
314
47.8
58
(b)
(d)
(f)
(h)
A
A
A
A
=
=
=
=
42.1
310
440
27.31
B
B
B
B
=
=
=
=
42.6
335
480
27.37
? 2: Answers
1.
2.
(a)
(b)
7 tomatoes
5 + 7 + 10 + 15 + 10 + 3 = 50 tomatoes
(c)
28 tomatoes are 40mm or more: 28 × 100 = 56%
50
(a)
(i) August (ii) March
(b)
(i) £900 (ii) £350
The numbers you see on the scale do not necessarily mean exactly what
they say! Always look for an indication of units. Here, each number
actually represents £100.
(c)
Average monthly sales
= Total sales for year ÷ 12
= £7,850 ÷ 12 = £654 to the nearest £
It would not make sense to include pennies in the answer because of the
approximate nature of the original data.
(d)
3000 ÷ 7850 = 38.2%
(e)
Low in March (cold weather) gradually increasing to a high in August
(summer).
? 3: Answers
(a)
(b)
(c)
76
approx 9%
approx 32% + 45% = 77%
(i)
approx 2% of 650 = 13 ropes
(ii)
approx 32% of 650 = 208 ropes
OUTCOME 1: NUMERACY/INT 2
ANSWERS TO SAQs
? 4: Answers
1.
(a)
(b)
Home computer dropped by £130 or £125
Washing machine increased by £150
This represents
2.
150
300
× 100 = 50%
(c)
Increased by £50 over 5 years, an annual average of £10
(a)
In general, urban population is steadily increasing whereas rural
population is decreasing.
Urban percentage below 60% in 1950 and reached 60% by 1970
(probably reached 65% round about 1980 but that is an educated guess).
Rural percentage down to probably 15% or even 10%, with urban
percentage correspondingly 85% or 90%.
About 22%, i.e. about 12.5 million
(b)
(c)
(d)
Did you notice that the horizontal scale in question 2 is not consistent? The bars
for 2000 should really be 3cm or so to the left, showing a steeper decline in the
rural population than it does at present. Watch out for this kind of misleading
scale in real life!
? 5: Answers
1.
(a)
(b)
(c)
(d)
2.
(a)
(b)
About 18 metres
Also about 18 metres
About 48 metres (total) minus 12 metres (thinking) = 36 metres braking
distance.
5mph = 8km/h so 30mph = 48km/h
The nearest bar to 48km/h is the 50km/h one
Stopping distance is about 31 metres.
No, total of each column is about 105
More:
sectors A and C
Less:
sectors D and E
Sector B looks roughly the same
Did you notice that the order of sectors in the legend is the opposite way round
from their order in the columns, unlike the example on page 17?
(c)
Sector
1998
1999
A
15
20
B
25
25
C
10
35
D
35
20
E
20
10
Total
105
105
OUTCOME 1: NUMERACY/INT 2
77
ANSWERS TO SAQs
(d)
1998: sector D: 35 × 100 = 33.3% (exactly 1 )
3
105
1999: sector D: 20 × 100 = 19.0% (to one decimal place)
105
? 6: Answers
1.
(a)
(b)
(c)
(d)
2.
25 + 5 = 30
110 + 145 = 255
Children
5 × £0
Adults
440 × £280
Over 60s
35 × £300
Total
£133,700
Divide answer (c) by total no. of passengers, i.e. by 480, answer £278.54
to nearest penny.
(a)
Between 0 and 40 mins, 90 goals were scored.
Between 50 and 90 mins, 54 goals were scored.
56 goals were scored within 5 mins either way of half time. For all we
know, all 56 might have been scored after half time, though this is unlikely.
So, yes, there were probably more goals scored in the first half than in the
second, but we can’t be 100% sure of this based purely on the graph.
(b)
(c)
(d)
22 goals
18/200 = 9/100 or 9%
56/200 = 28%
? 7: Answers
1.
Channel 1
Channel 2
Channel 3
Channel 4
Total
2.
(a)
(b)
90°
60°
67°
143°
25%
16.7%
18.6%
39.7%
100%
Channel 2 still has a 60° angle, i.e. 16.7%
New angles are:
Ch 1 70°
Ch 3 48°
Ch 4 123°
Ch 5 59°
Channel 1 appears to have suffered the most, though Channel 3 is a close
second.
78
OUTCOME 1: NUMERACY/INT 2
ANSWERS TO SAQs
(c)
(d)
Channel 4 has dropped from 144° to 124°; this is a drop of
20
× 100 = nearly 14%
144
(i)
Ch 1 = 70 of 5,666,400 = 1,101,800 viewers
360
(ii)
(e)
Ch 5 = 59°, i.e.
59
of 5,666,400 = 928,660
360
59
as a percentage is 16.4%. This does not reach the 20% level
360
required by XYZ so they will not advertise (yet).
? 8: Answers
1.
2.
(a)
£2,000
(b)
£3,000 – £500 = £2,500
(c)
£2,000 + £1,500 + £500 + £3,000 + £4,000 + £7,500 + 0 = £18,500
We can only read the graph with any certainty to within £200 or so, so
every day ’s takings could be plus or minus £200, say. Thus the final
answer could be quite far out. Graphs like this were never meant to give
accurate results, only impressions.
(d)
Divide £18,500 by the number of days. Aha! But is it by 7 ( = all the days
of the week) or is it by 6 (looks like it was shut on Sunday) so that
‘average daily sales’ means ‘average daily sales for those days the shop
was actually open’. I think it makes more sense to divide by 6, giving
approximately £3,083 per day.
(e)
Higher sales towards weekend, more people shopping, longer opening
hours. Closed Sunday, few people shopping at start of week. Next
week’s graph will possibly look very similar.
(a)
(b)
(c)
First two days
35mm roughly
Day 18
(d)
(i)
(ii)
(e)
Very tricky. It may well have reached its maximum height by day 30,
there is no evidence that it will continue increasing in height.
Days 10-15 (graph steepest)
Days 25-30 (graph least steep)
OUTCOME 1: NUMERACY/INT 2
79
ANSWERS TO SAQs
? 9: Answers
Clearly the dashed line represents the motorist, the solid line represents the cyclist.
(a)
(b)
(c)
(d)
(e)
(f)
1.15p.m.
No, at 5p.m. he still has 10km to go
3.05p.m.
38km from Ayton, so 22km from Seaton
35km in 2¼ hours = 15.2km/h
The graphs cross
(i)
at 2.20p.m., 12km from Ayton
(ii)
at 4.20p.m., 40km from Ayton
(g)
(i)
(ii)
(iii)
60km ÷ ¾ hr = 80km/h
60km ÷ 1hr = 60km/h
120km ÷ 2¾hrs = 43.6km/h
? 10: Answers
1.
(a)
(b)
(c)
145 pupils (highest point of graph)
19 years old
(i)
65 aged under 11
(ii)
145 – 65 = 80 aged 11 or over
(iii) 130 aged under 16
(iv) 145 – 130 = 15 aged 16 or over
(v)
Subtract answer (i) from answer (iii)
130 – 65 = 65 aged between 11 and 16
2.
(a)
(b)
(c)
(d)
(e)
(f)
Looks like 91 days (April, May, June)
45 days
Below £6,500 for 80 days, so answer 11 days
About £4,250
About £6,600
20% of 91 is 18 days
18 from 91 is 73
73 days → £6,200
Owner is satisfied.
? 11: Answers
Strongest to weakest: 5, 1, 6, 2, 3, 4.
1 and 5 show positive correlation – as one variable goes up so does the other.
2, 3 and 6 show negative correlation – as one variable goes up the other goes down.
4 appears to show no correlation at all.
80
OUTCOME 1: NUMERACY/INT 2
ANSWERS TO SAQs
? 12: Answers
1.
(a)
(b)
(c)
(d)
Yes, one person got 70/70 = 100%
20 out of 70
5 students got below 40, 15 out of 20 passed, i.e. 75%
Looks as if 51 out of 70 is the modal mark
2.
(a)
Age group
No. of members
10s
3
20s
16
30s
20
40s
11
50s
8
60s
6
70s
3
i.e. 67 members
3.
(b)
(i)
(ii)
(iii)
(c)
36 × £17.50 plus 23 × £35.50 plus 5 × £10 = £1,496.50
(a)
(b)
(c)
Tallest woman 1.81m
Smallest man 1.55m
Women are shorter on average than men, with most women in the
smallest size group.
Total height of all women ÷ number of women
= (1.50 + 1.51 + … + 1.80 + 1.81) ÷ 16
= 26m ÷ 16 = 1.625m
(d)
The youngest is 18.
The oldest is 73.
The middle one is the 34th person in the line, i.e. 35 years old.
? 13: Answers
1.
(a)
(b)
(c)
(d)
(e)
2.
(a)
203
23
95/203 = 46.8%
50/108 = 46.3%
Calculate the equivalent to (d) for males, i.e. what percentage of all the
males is unpromoted: 34/95 = 35.8%. Looks as if men are more likely to
be promoted than women.
<5
5≤10
10<20
20 & over
Total
Small
15
11
8
1
35
Medium
34
42
17
10
103
Large
18
27
54
19
118
Total
67
80
79
30
256
OUTCOME 1: NUMERACY/INT 2
81
ANSWERS TO SAQs
(b)
(c)
(d)
(e)
(f)
(g)
256
103/256
79/256
18/118
1/30
34/256
Page 56
If the sales are exactly £10,000, the answer to the question “Are sales more than
£10,000?” is ‘No’.
? 14: Answers
1.
(a)
(b)
(c)
(d)
5% of £36 = £1.80 discount
12% of £140 = £16.80 discount
10% of £96 = £9.60 discount
7½% of £64 = £4.80 discount
2.
(a)
(b)
(c)
(d)
6 × £20 = £120
2 × £33 = £66
10 × £20 = £200
5 × £29 = £145
3.
(a)
Commission = 4% of £16,200
= £648
Salary = £648 + £500 = £1,148
(b)
A = 4% of £20,000 = £800
B = 6% of £8,400 = £504 (8,400 comes from 28,400 – 20,000)
Salary = £800 + £504 + £500 = £1,804
(c)
(d)
£2,332
£700
4.
(a)
(b)
(c)
(d)
£312 + 2,100 × £0.16 = £648
475 × £0.36 = £171
£432 + 380 × £0.25 = £527
£432 + 650 × £0.25 = £594.50
5.
(a)
(b)
(c)
(d)
200 × 30.4p + 150 × 11.3p = £77.75
173 × 25.7p = £44.46
79 × 30.4p = £24.02
200 × 25.7p + 275 × 11.3p = £82.48
82
OUTCOME 1: NUMERACY/INT 2
ANSWERS TO SAQs
6.
2 adults
18 year old
15 year old
12 year old (new)
Total
=
=
=
=
=
2 × £38
£23
£23
£26.50 + £23
£171.50
? 15a: Answers
(a)
(b)
(c)
81 miles
125 miles
(142 + 169 + 104) miles = 415 miles
415 ÷ 30
×
4.5
×
↑
↑
turns miles
turns gallons
into gallons
into litres
79.6 = £49.55
↑
times cost
per litre
Note that the ‘real’ answer is close to £49.55 but you cannot think of this an
exact answer. All the figures in the question, apart from the petrol cost, have
been rounded off.
? 15b: Answers
(a)
(b)
(c)
(d)
71kg or 11st 3lb
66kg or 10st 6lb
69kg or 10st 13lb
About 56kg or 8st 12lb
(roughly splitting the difference between small and large build)
? 15c: Answers
(a)
(b)
(c)
(d)
9 rolls
7 rolls
14 rolls
19 or 20 rolls
OUTCOME 1: NUMERACY/INT 2
83
ANSWERS TO SAQs
? 15d: Answers
(a)
Payment
Total
8 × £3.03 = £24.24
8 × £181.80 = £1,454.40
(b)
Payment
Total
23 × £3.38 = £77.74
23 × £141.96 = £3,265.08
(c)
Payment
Total
32.5 × £3.21 = £104.33
32.5 × £173.34 = £5,633.55
? 15e: Answers
(a)
(b)
(c)
£200 × 2.580 = £516
£2,000 × 1.265 = £2,530
£6,540 × 1.191 = £7,789.14
? 15f: Answers
1.
(a)
(b)
3.15 + 0.12 = 3.27 inches
2,540 + 1,016 + 50.8 = 3,606.8mm
2.
(a)
7.7 gallons
(b)
30.32
= £3.45 per gallon
8.8
(c)
£4
= 87.91p per litre
4.55
? 15g: Answers
(a)
(b)
£270 × 2 plus £17.50 × 2 = £575
£485 × 2 plus 75% of £485 plus 50% of £485 plus 25% of £485 plus 5 × £8
plus 5 ×£35
Total £1,912.50
NB First child gets a 25% discount so pays 75% of the price.
84
OUTCOME 1: NUMERACY/INT 2