1.1.pdf
- 3. 1-3
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Copyright Notice
© Copyright. All worldwide rights reserved. No part of this publication may be reproduced,
stored in a retrieval system or transmitted in any form by any other means whatsoever: i.e.
photocopy, electronic, mechanical recording or otherwise without the prior written permission of
ST Aerospace Ltd.
Knowledge Levels — Category A, B1, B2, B3 and C Aircraft
Maintenance Licence
Basic knowledge for categories A, B1, B2 and B3 are indicated by the allocation of knowledge levels indicators (1,
2 or 3) against each applicable subject. Category C applicants must meet either the category B1 or the category B2
basic knowledge levels.
The knowledge level indicators are defined as follows:
LEVEL 1
A familiarisation with the principal elements of the subject.
Objectives: The applicant should be familiar with the basic elements of the subject.
The applicant should be able to give a simple description of the whole subject, using common words and
examples.
The applicant should be able to use typical terms.
LEVEL 2
A general knowledge of the theoretical and practical aspects of the subject.
An ability to apply that knowledge.
Objectives: The applicant should be able to understand the theoretical fundamentals of the subject.
The applicant should be able to give a general description of the subject using, as appropriate, typical
examples.
The applicant should be able to use mathematical formulae in conjunction with physical laws describing the
subject.
The applicant should be able to read and understand sketches, drawings and schematics describing the
subject.
The applicant should be able to apply his knowledge in a practical manner using detailed procedures.
LEVEL 3
A detailed knowledge of the theoretical and practical aspects of the subject.
A capacity to combine and apply the separate elements of knowledge in a logical and comprehensive
manner.
Objectives: The applicant should know the theory of the subject and interrelationships with other subjects.
The applicant should be able to give a detailed description of the subject using theoretical fundamentals
and specific examples.
The applicant should understand and be able to use mathematical formulae related to the subject.
The applicant should be able to read, understand and prepare sketches, simple drawings and schematics
describing the subject.
The applicant should be able to apply his knowledge in a practical manner using manufacturer's
instructions.
The applicant should be able to interpret results from various sources and measurements and apply
corrective action where appropriate.
- 5. 1-5
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Table of Contents
Module 1.1 Arithmetic________________________________________________________9
Fractions ________________________________________________________________9
Types of Fractions ________________________________________________________9
Working with Fractions_____________________________________________________9
Decimals________________________________________________________________19
Working with Decimals____________________________________________________19
Conversion Between Fractions and Decimals _________________________________25
Convert a Decimal to a Fraction ____________________________________________25
Convert a Fraction to a Decimal ____________________________________________25
Percentages_____________________________________________________________31
Definition ______________________________________________________________31
Changing a Fraction to a Percentage ________________________________________31
Changing a Percentage to a Fraction ________________________________________31
Changing a Percentage to a Decimal ________________________________________31
Changing a Decimal to a Percentage ________________________________________31
Values of a Percentage of a Quantity ________________________________________32
Expressing one Quantity as a Percentage of Another ____________________________32
Rounding, Significant Figures, and Decimal Places ____________________________37
Rounding ______________________________________________________________37
Significant Figures _______________________________________________________38
Decimal Places _________________________________________________________39
Mean, Median, Mode and Range ____________________________________________41
Definitions _____________________________________________________________41
Calculating Mean ________________________________________________________41
Calculating Median ______________________________________________________42
Calculating Mode ________________________________________________________42
Calculating Range _______________________________________________________43
Angles _________________________________________________________________49
Definitions and Conversions _______________________________________________49
Degrees and Radians: Measuring Angles _____________________________________50
Acute Angles ___________________________________________________________51
Obtuse Angles __________________________________________________________51
Reflex angles ___________________________________________________________51
Right Angles ___________________________________________________________52
Complementary Angles ___________________________________________________52
Supplementary Angles____________________________________________________53
Perpendicular Lines ______________________________________________________53
Triangles _______________________________________________________________59
Properties of shapes._____________________________________________________59
Definitions _____________________________________________________________59
- 6. 1-6
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Areas and Volume of Common Shapes_______________________________________63
Rectangle______________________________________________________________63
Square ________________________________________________________________63
Triangle _______________________________________________________________64
Parallelogram___________________________________________________________64
Rhombus ______________________________________________________________65
Trapezium _____________________________________________________________66
Kite___________________________________________________________________67
Circle _________________________________________________________________67
Other Regular Polygons___________________________________________________68
Summary of Quadrilaterals ________________________________________________69
Surface Area and Volume of Common Solids _________________________________71
Introduction ____________________________________________________________71
Common Solids _________________________________________________________71
Common Conversions ____________________________________________________81
Length ________________________________________________________________81
Area __________________________________________________________________81
Volume________________________________________________________________81
Mass _________________________________________________________________81
- 7. 1-7
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Module 1.1 Enabling Objectives and Certification Statement
Certification Statement
These Study Notes comply with the syllabus of EASA Regulation (EC) No.2042/2003 Annex III
(Part-66) Appendix I, as amended by Regulation (EC) No.1149/2011, and the associated
Knowledge Levels as specified below:
Objective
Part-66
Reference
Licence Category
B1 B2
Arithmetic 1.1
Arithmetical terms and signs, methods of
multiplication and division, fractions and
decimals, factors and multiples, weights,
measures and conversion factors, ratio and
proportion, averages and percentages, areas
and volumes, squares, cubes, square and cube
roots.
2 2
- 9. 1-9
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Module 1.1 Arithmetic
Fractions
Types of Fractions
1. Proper Fractions. Proper fractions may be defined as fractions less than 1.
For example:
1
2
,
1
3
,
2
5
,
11
17
etc
2. Improper Fractions. These are fractions which are greater than 1.
For example:
7
3
,
5
3
,
17
11
,
8
5
etc
3. Mixed Numbers. These include whole numbers and vulgar fractions. For example:
1
1
2
, 2
3
5
, 6
4
11
, 27
6
7
etc
4. For all fractions, the number above the bar is called the numerator and the number below
the bar is called the denominator.
5. Simplest Form. The simplest form of
30
60
is
1
2
. Fractions can be expressed in simplest
form by dividing numerator and denominator by equal numbers until they will not divide
further. For example:
8
12
=
2
3
in simplest form (after dividing numerator and
denominator by 4).
Working with Fractions
6. Cancelling. The process of dividing numerator and denominator by equal values is
called cancelling. For example:
27
81
=
9
27
=
3
9
=
1
3
- 10. 1-10
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
7. Converting. To convert mixed numbers to improper fractions, multiply the whole
number by the denominator and add to the numerator. For example:
2
3
5
=
13
5
To convert improper fractions to mixed numbers, divide the numerator by the
denominator to give a whole number - the remainder gives a new numerator. For
example:
25
4
= 6
1
4
8. Cancelling. Cancelling improper fractions involves exactly the same process as
cancelling vulgar fractions. For example:
28
4
=
7
1
= 7 and
45
6
=
15
2
= 7
1
2
9. Multiplication
(a) Express all mixed numbers as improper fractions
(b) Cancel vertically if possible
(c) Cancel across the multiplication sign if possible
(d) Multiply numerators together, multiply denominators together
(e) If the result is an improper fraction, convert to a mixed number
(f) Check that your answer is in the simplest form
Examples:
(1)
2
9
x 4 =
2
9
x
4
1
=
8
9
(2) 1
4
5
x 2
1
3
x
5
14
=
9
5
x
7
3
x
5
14
=
3
2
= 1
1
2
10. Division
(a) Convert all mixed numbers to improper fractions
(b) Invert the fraction you are dividing by
(c) Proceed as for multiplication.
1 1 2
1
1
3
- 11. 1-11
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Examples:
(1)
3
4
1
5
7
=
3
4
12
7
=
3
4
x
7
12
=
7
16
(2)
3
4
7 =
3
4
x
1
7
=
3
28
(3)
3
8
5
16
=
3
8
x
16
5
=
6
5
= 1
5
1
11. Mixed Multiplication and Division
(a) Invert all the fractions preceded by a division sign
(b) Treat the calculations as multiplication only.
Example:
1
3
4
4
1
2
x 1
5
7
=
7
4
2
9
x
12
7
=
7
4
x
2
9
x
12
7
=
7
4
x
2
9
x
12
7
=
2
3
12. Addition
(a) Express all fractions as mixed numbers in lowest terms
(b) Add the whole numbers together
(c) To add the vulgar fractions, you must convert each fraction so that their
denominators are all the same. This is done by finding the lowest common
multiple (LCM) of the denominators.
1
4
2
1
You only turn upside down the fraction
you are dividing by, i.e. the fraction after
the division sign
1
3
1
1
3
1
- 12. 1-12
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Examples:
(1)
1
5
+
1
6
+
3
10
=
6 5 9
30
=
20
30
=
2
3
(2)
9
4
+
5
12
+ 1
3
8
= 2
1
4
+
5
12
+ 1
3
8
= 3 +
1
4
+
5
12
+
3
8
= 3 +
6 10 9
24
= 3 +
25
24
= 3 + 1
1
24
= 4
1
24
Note: If your addition of fractions results in an improper fraction, you must convert this to
a mixed number as shown in example (2).
13. Subtraction
The same basic procedure should be used for subtraction as for addition.
Examples:
(1)
8
9
-
2
3
=
8 6
9
=
2
9
(2)
8
3
- 1
4
7
= 2
2
3
- 1
4
7
= 1 +
2
3
-
4
7
= 1 +
14 12
21
= 1
2
21
(3) 4
1
3
- 1
3
4
= 3 +
1
3
-
3
4
= 3 +
4 9
12
= 2 +
12 4 9
12
= 2 +
16 9
12
As numerator (4 - 9) give a negative
value, one whole unit has to be
converted to
12
12
before the
subtraction of fractions is carried out.
- 13. 1-13
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
= 2
7
12
14. Mixed addition and subtraction can be carried out exactly as above.
Examples:
(1) 4
1
2
- 5
7
12
+ 3
3
4
= 2 +
6 7 9
12
= 2 +
8
12
= 2
2
3
(2) 2
1
8
- 1
3
4
+ 4
1
3
= 5 +
3 18 8
24
= 4 +
24 3 18 8
24
= 4
17
24
15. Remember that your final step in any calculation must be to simplify (cancel fractions).
Example:
3
3
5
+ 1
9
10
- 2
3
4
= 2 +
12 18 15
20
= 2
15
20
= 2
3
4
- 15. 1-15
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Worksheet
1. Convert the following mixed numbers to improper fractions:
(a) 2
6
7
(b) 3
4
9
(c) 21
3
5
(d) 5
21
25
(e) 2
1
7
2. Convert the following improper fractions to mixed numbers:
(a)
11
3
(b)
21
5
(c)
53
7
(d)
210
4
(e)
99
8
3. Multiply and simplify the following:
(a)
6
7
x
14
15
(b)
2
3
x 2
1
7
x 1
2
5
(c) 1
2
5
x
3
8
x
10
21
4. Divide and simplify the following:
(a)
3
10
9
25
(b) 3
2
3
3
1
7
(c) 9
1
3
3
1
9
5. Evaluate the following:
(a)
7
10
x
5
6
x
3
14
(b) 7
1
5
x
15
24
3
8
(c)
17
29
2
1
8
x 7
1
4
6. Add the following fractions and mixed numbers:
(a)
2
3
+
3
7
(b)
3
4
+
1
2
+
2
3
(c)
2
3
+
3
4
+
4
5
(d)
1
2
+
1
3
+
1
4
+
1
5
+
1
6
(e) 3
1
8
+
7
16
7. Subtract the following:
(a) 1
1
2
-
3
5
(b) 1
3
4
- 2
2
5
(c) 1
2
5
- 2
1
7
(d)
4
11
-
2
7
(e) 7
4
9
- 8
2
3
- 16. 1-16
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
8. Evaluate the following:
(a) 2
2
3
3
5
8
1
4
9
(b) 1
1
2
2
1
3
5
42
(c)
1
1
3
2
7
5
6
3
4
3
7
x
(d)
1
1
5
2
1
3
4
5
2
1
2
5
6
x
(e)
1
4
1
3
5
1
7
8
1
2
(f) 3
1
3
1
1
2
2
1
2
(g) 2
2
7
7
24
3
1
2
4
2
3
x
(h)
5
16
3
4
1
3
2
3
1
4
1
6
(i)
3
1
2
2
3
1
3
5
1
1
3
3
7
1
3
- 17. 1-17
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Answers
1. a)
20
7
b)
31
9
c)
108
5
d)
146
25
e)
15
7
2. a) 3
2
3
b) 4
1
5
c) 7
4
7
d) 52
1
2
e) 12
3
8
3. a)
4
5
b) 2 c)
1
4
4. a)
5
6
b) 1
1
6
c) 3
5. a)
1
8
b) 12 c) 2
6. a) 1
2
21
b) 1
11
12
c) 2
13
60
d) 1
9
20
e) 3
9
16
7. a)
9
10
b) -
13
20
c) -
26
35
d)
6
77
e) -1
2
9
8. a) 4
61
72
b) -
5
7
c) 2
4
9
d)
3
5
e) 1
19
55
f)
5
6
g) 1
5
12
h) -
5
36
i) 1
11
24
- 19. 1-19
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Decimals
Working with Decimals
1. Decimals are a very important and particular set of fractions. They are fractions whose
denominators are powers of 10, i.e. 10, 100, 1000, 10000 etc (do not be concerned
about the meaning of 'powers of 10', you will deal with this later in the course). Decimals
are not written in the usual fraction form, but in shorthand using a decimal point.
Examples:
a)
1
10
= 0.1 b)
1
100
= 0.01 c)
1
1000
= 0.001
d) 5
7
10
= 5.7 e) 63
7
100
= 63.07
2. If you have difficulty in relating decimals to fractions, the following table may help.
THOUSANDS HUNDREDS TENS UNITS TENTHS HUNDREDTHS THOUSANDTHS
1000 100 10 1
1
10
1
100
1
1000
5 3 4 6 7 9 2
The number in the table is 5346.792; it consists of 5 thousands, 3 hundreds, 4 tens, 6
units, 7 tenths, 9 hundredths and 2 thousandths.
3. The number of digits after the decimal point is called decimal places.
Examples:
a) 27.6 has one decimal place
b) 27.16 has two decimal places
c) 27.026 has three decimal places
d) 101.2032 has four decimal places
4. In addition of decimals, particular care must be taken to ensure that decimal points are
in line.
Example: Evaluate 27.3 + 0.021 + 68.3 27.3
0.021
+ 68.3
95.621
- 20. 1-20
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
5. Similarly, in subtraction, ensure that decimal points are in line.
Example: Evaluate 27.3 - 4.36 27.3
- 4.36
22.94
6. When multiplying decimals, ignore the decimal point until the final answer is obtained,
then count the number of decimal places in both the numbers being multiplied. This is
the number of decimal places in the answer.
Example: Evaluate 27.3 x 9.31 Note: Total of 3 decimal places.
273
931
245 700
8 190
273
254 163 Answer = 254.163 (3 dpl.)
Note: It does not matter which you multiply first (i.e. the 9, the 3 or the 1) providing „00‟ is
placed before the answer when multiplying out the 100‟s (in this case the 9) and „0‟ is
placed before the answer when multiplying out the 10‟s (in this case the 3) So the above
calculation could have looked like this:
273
931
273
8 190
245 700
254 163 The answer is the same
7. In division, it is easier to divide by a whole number than by a decimal. To make the
divisor (the number you are dividing by) into a whole number, move the decimal point a
specific number of places to the right. You must then also move the decimal point in the
dividend (the number you are dividing into) to the right by the same number of decimal
places.
Example: Evaluate 24.024 4.62 We have moved the decimal
point 2 places in both the
2402.4 462 divisor and the dividend, but the
answer is unaltered
5.2
= 462)2402.4
2310
924
924
An approximate answer could be calculated as follows: 24.024 24
4.62 5
27.3 can also be
written as 27.30
5
- 21. 1-21
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Worksheet
1. Calculate the sum of the following:
a) 0.251 + 10.298 b) 18.098 + 210.099 c) 0.025 + 10.995
2. Evaluate:
a) 21.76 - 18.51 b) 32.76 - 20.086 c) 10.75 - 19.999
3. Find the product of:
a) 5.05 x 13.8 b) 1.27 x 0.871 c) -1.01 x 0.89
4. Calculate:
a) 42.39 0.09 b) 3.375 1.5 c) 0.002 0.8
- 23. 1-23
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Answers
1. a) 10.549 b) 228.197 c) 11.02
2. a) 3.25 b) 12.674 c) -9.249
3. a) 69.69 b) 1.10617 c) -0.8989
4. a) 471 b) 2.25 c) 0.0025
- 25. 1-25
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Conversion Between Fractions and Decimals
Convert a Decimal to a Fraction
Write the number over the appropriate power of 10 and, if possible, cancel to lowest terms.
Examples:
a) 0.8 =
8
10
=
4
5
b) 6.25 = 6 +
25
100
= 6
1
4
c) 0.037 =
37
1000
Convert a Fraction to a Decimal
Divide the numerator by the denominator.
Examples:
a)
4
5
=
b)
3
8
=
c)
5
6
=
Here, we cut off the result to the number of decimal places required.
Thus
5
6
= 0.83 correct to 2 decimal places
or
5
6
= 0.8333 correct to 4 decimal places
80
.
0
00
.
4
5
375
0
000
3
8
.
.
8333
0
0000
5
6
.
.
.......the 3 will re-occur for ever
- 27. 1-27
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Worksheet
1. Convert the following decimals to proper fractions in their simplest form:
a) 0.73 b) 0.02 c) 0.004
2. Convert the following proper fractions to decimals to 2 decimal places:
a)
5
8
b)
13
15
c)
3
200
3. Place in ascending order of magnitude:
a)
1
6
, 0.167 and
3
20
b)
2
5
, 0.44 and
7
16
- 29. 1-29
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Answers
1. a)
73
100
b)
1
50
c)
1
250
2. a) 0.63 0.87 0.02
3. a)
3
20
,
1
6
and 0.167 b)
2
5
,
7
16
and 0.44
- 31. 1-31
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Percentages
Definition.
A percentage is a fraction whose denominator is 100.
Example: 3% means
3
100
Changing a Fraction to a Percentage
To change a fraction to a percentage, multiply by 100%.
Example:
3
5
as a percentage =
3
5
x 100% = 60%
Changing a Percentage to a Fraction
To change a percentage to a fraction, divide by 100%.
Examples: a) 8% as a fraction =
8%
100%
8
100
2
25
b) 12 ½ % as a fraction =
12
100%
25
2
1
100
25
200
1
8
1
2
%
x
Changing a Percentage to a Decimal
To convert a percentage to a decimal, firstly, convert the percentage to a fraction, then convert
the fraction to a decimal.
Examples: a) 65% as a fraction =
65
100
, as a decimal = 0.65
b) 32 ½ % as a fraction =
32
100
1
2
, as a decimal = 0.325
Changing a Decimal to a Percentage
To convert a decimal to a percentage, firstly, convert the decimal to a fraction, then convert the
fraction to a percentage.
Examples:
a) 0.021 as a fraction =
21
1000
, =
21
100
.
, as a percentage = 2.1%
- 32. 1-32
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
b) 0.037 as a fraction =
37
1000
, =
37
100
.
, as a percentage = 3.7%
Values of a Percentage of a Quantity
To find the value of a percentage of a quantity, express the percentage as a fraction and
multiply by the quantity.
Examples:
a) 4% of 60 =
4
100
x 60 =
12
5
= 2
2
5
b) 3 ½ % of 1500 =
3
100
1
2
x 1500 =
7
200
x 1500 =
105
2
= 52 ½
Expressing one Quantity as a Percentage of Another
To express one quantity as a percentage of another, make a fraction of the 2 quantities and
multiply by 100.
Examples:
a) 12 as a percentage of 50 =
12
50
x 100 = 24%
b) 4 as a percentage of 60 =
4
60
x 100 = 6.67%
2
5
- 33. 1-33
Module 1.1 Arithmetic
For Training Purposes Only
ST Aerospace Ltd
© Copyright 2013
Worksheet
1. Calculate:
a) 4% of 30 b) 0.8% of 360 c) 1.5% of 60
d) 120% of 75 e) 80% of 90
2. Express:
a) 30 as a percentage of 50
b) 24 as a percentage of 16
c) 0.5 as a percentage of 12.5
d) 3.2 as a percentage of 2.4
e) 0.08 as a percentage of 0.72
3. Express as a proper fraction:
a) 0.6 b) 0.35 c) 0.48 d) 0.05
e) 0.325 f) 25% g) 13% h) 4.5%
i) 16 ⅓ %
4. Express as a percentage:
a) 0.43 b) 0.025 c) 1.25 d)
2
3
e)
3
7
f)
1
12
g)
3
8
- 35. 1-35
Module 1.1 Arithmetic
For Training Purposes Only
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© Copyright 2013
Answers
1. a) 1
1
5
or 1.2 b) 2.88 c) 0.9 d) 90 e) 72
2. a) 60% b) 150% c) 4% d) 133
1
3
% e) 11
1
9
%
3. a)
3
5
b)
7
20
c)
12
25
d)
20
1
e)
13
40
f)
1
4
g)
13
100
h)
9
200
i)
49
300
4. a) 43% b) 2.5% c) 125% d) 66
2
3
% e) 42
6
7
%
f) 8
1
3
% g) 37
1
2
%
- 37. 1-37
Module 1.1 Arithmetic
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ST Aerospace Ltd
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Rounding, Significant Figures, and Decimal Places
Rounding
Rounding is the process of reducing the number of significant digits in a number. The result of
rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude. The result
is less precise but easier to use.
For example: 73 rounded to the nearest ten is 70, because 73 is closer to 70 than to 80.
Methods of Rounding
Common Method: This method is commonly used in mathematical applications, for example in
accounting. It is the one generally taught in elementary mathematics classes. This method is
also known as Symmetric Arithmetic Rounding or Round-Half-Up (Symmetric Implementation)
Decide which is the last digit to keep.
Increase it by 1 if the next digit is 5 or more (this is called rounding up)
Leave it the same if the next digit is 4 or less (this is called rounding down)
Examples:
3.044 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
3.045 rounded to hundredths is 3.05 (because the next digit, 5, is 5 or more).
3.0447 rounded to hundredths is 3.04 (because the next digit, 4, is less than 5).
For negative numbers the absolute value is rounded.
Examples:
−2.1349 rounded to hundredths is −2.13
−2.1350 rounded to hundredths is −2.14
Round to Even Method: This method is also known as unbiased rounding, convergent
rounding, statistician's rounding or bankers' rounding. It is identical to the common method of
rounding except when the digit(s) following the rounding digit start with a five and have no non-
zero digits after it. The new algorithm is:
Decide which is the last digit to keep.
Increase it by 1 if the next digit is 6 or more, or a 5 followed by one or more non-zero
digits.
Leave it the same if the next digit is 4 or less
Otherwise, all that follows the last digit is a 5 and possibly trailing zeroes; then change
the last digit to the nearest even digit. That is, increase the rounded digit if it is currently
odd; leave it if it is already even.
With all rounding schemes there are two possible outcomes: increasing the rounding digit by
one or leaving it alone. With traditional rounding, if the number has a value less than the half-
way mark between the possible outcomes, it is rounded down; if the number has a value exactly
half-way or greater than half-way between the possible outcomes, it is rounded up. The round-
- 38. 1-38
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to-even method is the same except that numbers exactly half-way between the possible
outcomes are sometimes rounded up - sometimes down.
Although it is customary to round the number 4.5 up to 5, in fact 4.5 is no nearer to 5 than it is to
4 (it is 0.5 away from both).
When dealing with large sets of scientific or statistical data, where trends are important,
traditional rounding on average biases the data upwards slightly. Over a large set of data, or
when many subsequent rounding operations are performed as in digital signal processing, the
round-to-even rule tends to reduce the total rounding error, with (on average) an equal portion
of numbers rounding up as rounding down. This generally reduces the upwards skewing of the
result.
Round-to-even is used rather than round-to-odd as the latter rule would prevent rounding to a
result of zero.
Examples:
3.016 rounded to hundredths is 3.02 (because the next digit (6) is 6 or more)
3.013 rounded to hundredths is 3.01 (because the next digit (3) is 4 or less)
3.015 rounded to hundredths is 3.02 (because the next digit is 5, and the hundredths
digit (1) is odd)
3.045 rounded to hundredths is 3.04 (because the next digit is 5, and the hundredths
digit (4) is even)
3.04501 rounded to hundredths is 3.05 (because the next digit is 5, but it is followed by
non-zero digits)
Significant Figures
Rounding to n significant figures is a form of rounding. Significant figures (also called significant
digits) can also refer to a crude form of error representation based around significant figure
rounding.
Rounding to n significant figures is a more general-purpose technique than rounding to n
decimal places, since it handles numbers of different scales in a uniform way.
Rules of Significant Figures
All non-zero digits are significant. Example: '123.45' has five significant figures: 1,2,3,4
and 5.
Zeros appearing in between two non-zero digits are significant. Example: '101.12' has
five significant figures: 1,0,1,1,2.
All zeros appearing to the right of an understood decimal point or non-zeros appearing to
the right of a decimal after the decimal point are significant. Example: '12.2300' has six
significant figures: 1,2,2,3,0 and 0. The number '0.00122300' still only has six significant
figures (the zeros before the '1' are not significant). In addition, '12.00' has four significant
figures.
All zeros appearing in a number without a decimal point and to the right of the last non-
zero digit are not significant unless indicated by a bar. Example: '1300' has two
significant figures: 1 and 3. The zeros are not considered significant because they don't
have a bar. However, 1300.0 has five significant figures.
- 39. 1-39
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However, this last convention is not universally used; it is often necessary to determine from
context whether trailing zeros in a number without a decimal point are intended to be significant.
Digits may be important without being 'significant' in this usage. For instance, the zeros in '1300'
or '0.005' are not considered significant digits, but are still important as placeholders that
establish the number's magnitude. A number with all zero digits (e.g. '0.000') has no significant
digits, because the uncertainty is larger than the actual measurement.
Examples:
Rounding to 2 significant figures:
12,300 becomes 12,000
13 stays as 13
0.00123 becomes 0.0012
0.1 becomes 0.10 (the trailing zero indicates that we are rounding to 2 significant
figures).
0.02084 becomes 0.021
0.0125 becomes 0.012 in unbiased rounding, while it is 0.013 in biased.
One issue with rounding to n significant figures is that the value of n is not always clear. This
occurs when the last significant figure is a zero to the left of the decimal point. For example, in
the final example above, when 19 800 is rounded to 20 000, it is not clear from the rounded
value what n was used - n could be anything from 1 to 5. The level of rounding can be specified
explicitly. The abbreviation s.f. is sometimes used, for example "20,000 to 2 s.f."
Scientific notation could be used to reduce the ambiguity, as in (2.0 × 104
). As always, the best
approach is to state the uncertainty separately and explicitly, as in 20,000 ± 1%, so that
significant-figures rules do not apply.
A less common method of presenting ambiguous significant figures is underlining the last
significant figure of a number, for example "20000"
Decimal Places
The precision of a value describes the number of digits that are used to express that value. In
a scientific setting this would be the total number of digits (sometimes called the significant
digits) or, less commonly, the number of fractional digits or places (the number of digits
following the point). This second definition is useful in financial and engineering applications
where the number of digits in the fractional part has particular importance.
In both cases, the term precision can be used to describe the position at which an inexact result
will be rounded. For example, in floating-point arithmetic, a result is rounded to a given or fixed
precision, which is the length of the resulting significand. In financial calculations, a number is
often rounded to a given number of places (for example, to two places after the point for many
world currencies).
As an illustration, the decimal quantity 12.345 can be expressed with various numbers of
significant digits or decimal places. If insufficient precision is available then the number is
- 40. 1-40
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rounded in some manner to fit the available precision. The following table shows the results for
various total precisions and decimal places, with the results rounded to nearest where ties
round up or to an even digit (the most common rounding modes).
Note that it is often not appropriate to display a figure with more digits than that which can be
measured. For instance, if a device measures to the nearest gram and gives a reading of
12.345 kg, it would create false precision if you were to express this measurement as
12.34500 kg.
n
Rounded to ‘n’
significant digits
Rounded to ‘n’
decimal places
Five 12.345 12.34500
Four 12.35 12.3450
Three 12.3 12.345
Two 12 12.35
One 1 × 101
12.4
Zero n/a 12
- 41. 1-41
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Mean, Median, Mode and Range
Definitions
The Mean, Median and Mode of a set of numbers are three types of “average” of the set.
However, the “Mean” is the term most commonly taken as the average.
Mean: The sum of a set of data divided by the
number of data
Median: The middle value or the mean of the
middle two values, when the data is
arranged in numerical order.
Mode: The value (number) that appears the most.
It is possible to have more than one mode,
and it is possible to have no mode.
Calculating Mean
To find the mean, you need to add up all the data, and then divide this total by the number of
values in the data.
Example 1: Find the Mean of 2, 2, 3, 5, 5, 7, 8
Adding the numbers up gives: 2 + 2 + 3 + 5 + 5 + 7 + 8 = 32
There are 7 values, so you divide the total by 7: 32 ÷ 7 = 4.57...
So the mean is 4.57 (2 d.p.)
Example 2: Find the Mean of 2, 3, 3, 4, 6, 7
Adding the numbers up gives: 2 + 3 + 3 + 4 + 6 + 7 = 25
There are 6 values, so you divide the total by 6: 25 ÷ 6 = 4.33...
So the mean is 4.33 (2 d.p.)
- 42. 1-42
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Calculating Median
To find the median, you need to put the values in order, then find the middle value. If there are
two values in the middle then you find the mean of these two values.
Example 1: Find the median of 2, 2, 3, 5, 5, 7, 8
The numbers in order: 2 , 2 , 3 , (5) , 5 , 7 , 8
The middle value is marked in brackets, and it is 5.
So the median is 5
Example 2: Find the median of 2, 3, 3, 4, 6, 7
The numbers in order: 2 , 3 , (3 , 4) , 6 , 7
This time there are two values in the middle. They have been put in brackets. The median is
found by calculating the mean of these two values: (3 + 4) ÷ 2 = 3.5
So the median is 3.5
Calculating Mode
The mode is the value which appears the most often in the data. It is possible to have more
than one mode if there is more than one value which appears the most.
Example 1: Find the mode of 2, 2, 3, 5, 5, 7, 8
The data values: 2 , 2 , 3 , 5 , 5 , 7 , 8
The values which appear most often are 2 and 5. They both appear more time than any of the
other data values.
So the modes are 2 and 5
Example 2: Find the mode of 2, 3, 3, 4, 6, 7
The data values: 2 , 3 , 3 , 4 , 6 , 7
This time there is only one value which appears most often - the number 3. It appears more
times than any of the other data values.
So the mode is 3
- 43. 1-43
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Calculating Range
To find the range, you first need to find the lowest and highest values in the data. The range is
found by subtracting the lowest value from the highest value.
Example 1: Find the range of 2, 2, 3, 5, 5, 7, 8
The data values: 2 , 2 , 3 , 5 , 5 , 7 , 8
The lowest value is 2 and the highest value is 8. Subtracting the lowest from the highest gives:
8 - 2 = 6
So the range is 6
Example 2: Find the range of 2, 3, 3, 4, 6, 7
The data values: 2 , 3 , 3 , 4 , 6 , 7
The lowest value is 2 and the highest value is 7. Subtracting the lowest from the highest
gives:…7 - 2 = 5
So the range is 5
- 45. 1-45
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Worksheet
1. A data set contains these 12 values: 3, 5, 9, 4, 5, 11, 10, 5, 7, 7, 8, 10
(a) What is the mean?
(b) What is the median?
(c) What is the mode?
(d) What is the range?
2. Calculate the mean, median, mode and range for each set of data below:
(a) 3, 6, 3, 7, 4, 3, 9
(b) 11, 10, 12, 12, 9, 10, 14, 12, 9
(c) 2, 9, 7, 3, 5, 5, 6, 5, 4, 9
- 47. 1-47
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Answers
1 (a) 7 (b) 7 (c) 5 (d) 8
2 (a) Mean = 5
Median = 4
Mode = 3
Range = 6
(b) Mean = 11
Median = 11
Mode = 12
Range = 5
(c)
Mean = 5.5
Median = 5
Mode = 5
Range = 7
- 49. 1-49
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Angles
Definitions and Conversions
We can specify an angle by using a point on each ray and the vertex. The angle below may be
specified as angle ABC or as angle CBA; you may also see this written as
ABC or as CBA. Note how the vertex point is always given in the middle.
Example:
Many different names exist for the same angle. For the angle below, PBC, PBW, CBP,
and WBA are all names for the same angle.
- 50. 1-50
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Degrees and Radians: Measuring Angles
We measure the size of an angle using degrees. We can also use radians to measure angles.
There are 2 radians in 360o
The radius of a circle fits around the circumference 6.28 (or 2) times. 1 radian = 57.3 degrees.
To convert from degrees to radians, use
360
no
x 2 where no
is the number of degrees
Note: Degrees can be further subdivided into minutes and seconds.
60 seconds = 1 minute
60 minutes = 1 degree
- 51. 1-51
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Acute Angles
An acute angle is an angle measuring between 0 and 90 degrees.
Example:
The following angles are all acute angles.
Obtuse Angles
An obtuse angle is an angle measuring between 90 and 180 degrees.
Example:
The following angles are all obtuse.
Reflex angles
A reflex angle is an angle measuring between 180 and 360 degrees
- 52. 1-52
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Right Angles
A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right
angle are said to be perpendicular. Note that any two right angles are supplementary angles (a
right angle is its own angle supplement).
Complementary Angles
Two angles are called complementary angles if the sum of their degree measurements equals
90 degrees. One of the complementary angles is said to be the complement of the other.
Example:
These two angles are complementary.
Note that these two angles can be "pasted" together to form a right angle!
- 53. 1-53
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Supplementary Angles
Two angles are called supplementary angles if the sum of their degree measurements equals
180 degrees. One of the supplementary angles is said to be the supplement of the other.
Example:
These two angles are supplementary.
Note that these two angles can be "pasted" together to form a straight line!
Perpendicular Lines
Two lines that meet at a right angle are perpendicular. They are also said to be “normal” to each
other.
- 55. 1-55
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Worksheet
1. Convert the following angles to radians
a) b) c)
d) e) f)
- 57. 1-57
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Answers
1. a) ¼ radians b) ¼ radians c) ½ radians
d)
180
23
radians e) 2/5 radians f) 2/3 radians
- 59. 1-59
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Triangles
Properties of shapes.
The 3 properties of shapes that we are going to look at are:
1. The number of sides
2. The interior angles (the angles inside).
3. The length of the sides.
These properties help use to remember which shapes are which and why they are so called (in
some cases).
Let's start with a shape that has 3 sides: TRIANGLES (tri- means 3).
Triangles ALWAYS have 3 sides.
The interior angles of a triangle add up to 180 degrees.
Definitions
Here are the triangles you are expected to know about:
1. Equilateral Triangle
2. Isosceles Triangle
3. Right- Angled Triangle
4. Scalene Triangle
2. Congruent and Similar Triangles
Congruent and Similar are two words usually applied to triangles but can equally be applied to
other shapes.
Congruent triangles are two triangles which have equal angles and are the same size (i.e.
identical in every way) but may be oriented differently.
Similar triangles are two triangles which have the same angles but are of different size.
- 60. 1-60
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3. Equilateral Triangle
An equilateral triangle has got 3 sides of equal length and 3 angles that are equal.
Since ALL the angles in a triangle add up to 180º then 180 divided by 3 must be 60º.
The clue is in the name EQUILateral.
4. Isosceles Triangle
An Isosceles triangle has got two sides of equal length and 2 angles equal.
What is the value of the angle at the top of this Isosceles triangle?
The answer is 80˚. All angles in a triangle add up to 180˚ so 180 - (50 + 50) = 80˚
So an isosceles triangle has only got two sides of equal length and two angles the same.
- 61. 1-61
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5. Right- Angled Triangle
The right angled triangle contains a right angle (an angle of 90˚)
In a right angled triangle what must the other two angles add up to 90˚ because all the angles in
a triangle add up to 180˚ and a Right Angled Triangle has got one angle of 90˚.
6. Scalene Triangle
A scalene triangle is the easiest of them all. The scalene triangle has got NO sides of equal
length and NO angles the same.
- 63. 1-63
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Areas and Volume of Common Shapes
Rectangle
The area A of any rectangle is equal to the product of the length l and the width w.
Formula: A = lw
Square
The area A of any square is equal to the square of the length s of a side.
Formula: A = s2
- 64. 1-64
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Triangle
The area A of any triangle is equal to one-half the product of any base b and corresponding
height h.
Formula: A = ½ bh
Parallelogram
A parallelogram is a 4 sided shape with the 2 opposing sides parallel to each other.
The area A of any parallelogram is equal to the product of any base b and the corresponding
height h.
Formula: A = bh
- 65. 1-65
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Rhombus
A rhombus is a parallelogram with all 4 sides equal length. The diagonals bisect the interior
angles equally and the diagonals intersect each other at right angles.
The area A of any rhombus is equal to one-half the product of the lengths d1 and d2 of its
diagonals.
Formula: A = ½ d1d2
or
Formula: A = bh as in the parallelogram
- 66. 1-66
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Trapezium
A trapezium has only 2 sides parallel. (UK definition)
The area A of any trapezium is equal to one-half the product of the height h and the sum of the
bases, b1 and b2.
Formula: A = ½ h (b1 + b2)
- 67. 1-67
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Kite
A Kite shape has no sides parallel. The area can be found by
Formula: A = ½ d1d2
Circle
The area A of any circle is equal to the product of and the square of the radius r.
Formula: A = r2
Sector Area Theorem
The area A of any sector with an arc that has degree measure n and with radius r is equal to the
product of the arc's measure divided by 360 multiplied by times the square of the radius.
Formula: A = (n/360)(r2
)
- 68. 1-68
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Other Regular Polygons
Regular polygons are any polygons that are equilateral and equiangular.
The area A of any regular polygon with perimeter P and apothem of measure a is equal to one-
half the product of the perimeter and the apothem.
This formula can be derived if you make 5 triangles inside the shape. The area of each triangle
is ½ Sa (1/2 base x height).
The total area is therefore 5 x ½ Sa (in the case of the pentagon shown – the 5 only represent
the number of sides). But 5 x S is the total perimeter of the shape, so:
Formula: A = ½ aP
The angle shown is one equal portion of 360 degrees. In the case of the pentagon, it is 360/5
= 72o
. Hence the internal angle of any polygon can be found by calculating the supplement of
the external angle.
- 69. 1-69
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Summary of Quadrilaterals
SQUARE
4 Sides equal
4 right angles
Diagonals bisect each other at right angles
Diagonals are equal
RECTANGLE
2 pairs of opposite sides equal and parallel
4 right angles
Diagonals are equal and bisect each other
RHOMBUS
4 sides equal , opposite sides parallel
Diagonals bisect each other but are not of equal length
PARALLELOGRAM
2 pairs opposite sides equal and parallel
Diagonals bisect each other but are not of equal length
TRAPEZIUM / TRAPEZOID
1 pair opposite sides parallel
KITE
2 pairs of adjacent sides equal
Longer diagonal bisects shorter at right angles
or
- 71. 1-71
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Surface Area and Volume of Common Solids
Introduction
There are special formulas that deal with solids, but they only deal with right prisms. Right
prisms are prisms that have two special characteristics - all lateral edges are perpendicular to
the bases, and lateral faces are rectangular. The figure below depicts a right prism.
Common Solids
1. Right Prism Area
The lateral area L (area of the vertical sides only) of any right prism is equal to the perimeter of
the base times the height of the prism (L = Ph).
The total area T of any right prism is equal to two times the area of the base plus the lateral
area.
Formula: T = 2B + Ph
- 72. 1-72
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2. Right Prism Volume
The volume V of any right prism is the product of B, the area of the base, and the height h of the
prism.
Formula: V = Bh
- 73. 1-73
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3. Pyramid Volume
A pyramid is a polyhedron with a single base and lateral faces that are all triangular. All lateral
edges of a pyramid meet at a single point, or vertex.
The volume V of any pyramid with height h and a base with area B is equal to one-third the
product of the height and the area of the base. This applies even if the prism is not a „right-
prism‟ i.e. the axis is not perpendicular to the base. The height however, is still measured
perpendicular to the base as shown below.
Formula: V = 3
1 Bh
A regular pyramid is a pyramid that has a base that is a regular polygon and with lateral faces
that are all congruent isosceles triangles.
- 74. 1-74
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4. Cylinder Volume
The volume V of any cylinder with radius r and height h is equal to the product of the area of a
base and the height.
Formula: V = r2
h
5. Cylinder Surface Area
For any right circular cylinder with radius r and height h, the total surface area T is two times the
area of the base (2 r2
) plus the curved surface area (2 rh).
Formula: T = 2 r h + 2 r2
- 75. 1-75
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6. Cone Volume
The volume V of any cone with radius r and height h is equal to one-third the product of the
height and the area of the base.
Formula: V = 3
1 r2
h
7. Cone Surface Area
The total surface area T of a cone with radius r and slant height l is equal to the area of the
base ( r2
) plus times the product of the radius and the slant height.
Formula: T = r l + r2
- 76. 1-76
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8. Sphere Volume and Surface Area
The volume V for any sphere with radius r is equal to four-thirds times the product of and the
cube of the radius. The area A of any sphere with radius r is equal to 4 times the square of
the radius.
Volume Formula: V = 3
4 r3
Surface Area Formula: A = 4r2
- 77. 1-77
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Worksheet
1. A grave is dug 2m x 1m x 1m deep. The earth removed is piled into a pyramid of circular
base 2 m diameter. Find the height of the pyramid (in terms of ).
Give the answer in m, cm and mm.
2. A right prism has ends 10 cm x 10 cm and is 50 cm long. It is drilled lengthwise with an 8
cm drill through its full length. Find
a) the remaining volume of the prism material. Give the answer in terms of and in
mm3
b) the surface area of the inside of the hole. Give the answer in terms of and in
mm2
3. Find the surface area of a right cone with base radius 3 inches and perpendicular height
of 4 inches. Leave the answer in terms of and include the base area.
4. Find the ratio of the „surface area to volume‟ of spheres of the following diameters:
a) 2m
b) 4m
c) 8m
- 79. 1-79
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Answers
1. 6/ m, 600/ cm, 6000/ mm
2. a) (5 x 106
) - (8 x 105
) mm3
b) 400 cm2
, 40,000 mm2
3. 24 in2
4. a) 3:1
b) 1½ :1
c) ¾ : 1
- 81. 1-81
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Common Conversions
Length
metric > imperial
1 millimetre [mm] 0.0394 in
1 centimetre [cm] 10 mm 0.3937 in
1 metre [m] 100 cm 1.0936 yd
1 kilometre [km] 1000 m 0.6214 mile
Area
Volume
metric > imperial
1 cu cm [cm3
] 0.0610 in3
1 cu decimetre [dm3
] 1,000 cm3
0.0353 ft3
1 cu metre [m3
] 1,000 dm3
1.3080 yd3
1 litre [l] 1 dm3
1.76 pt
1 hectolitre [hl] 100 l 21.997 gal
Mass
imperial > metric
1 inch [in] 2.54 cm
1 foot [ft] 12 in 0.3048 m
1 yard [yd] 3 ft 0.9144 m
1 mile 1760 yd 1.6093 km
1 nautical mile 1.15 mile 1.852 km
imperial > metric
1 sq inch [in2
] 6.4516 cm2
1 sq foot [ft2
] 144 in2
0.0929 m2
1 sq yd [yd2
] 9 ft2
0.8361 m2
1 acre 4840 yd2
4046.9 m2
1 sq mile [mile2
] 640 acres 2.59 km2
metric > imperial
1 sq cm [cm2
] 100 mm2
0.1550 in2
1 sq m [m2
] 10,000 cm2
1.1960 yd2
1 hectare [ha] 10,000 m2
2.4711 acres
1 sq km [km2
] 100 ha 0.3861 mile2
imperial > metric
1 cu inch [in3
] 16.387 cm3
1 cu foot [ft3
] 1,728 in3
0.0283 m3
1 fluid ounce [fl oz] 28.413 ml
1 pint [pt] 20 fl oz 0.5683 l
1 gallon [gal] 8 pt 4.5461 l
USA measure > metric
1 fluid ounce 1.0408 UK fl oz 29.574 ml
1 pint (16 fl oz) 0.8327 UK pt 0.4731 l
1 gallon 0.8327 UK gal 3.7854 l
imperial > metric
1 ounce [oz] 437.5 grain 28.35 g
1 pound [lb] 16 oz 0.4536 kg
1 stone 14 lb 6.3503 kg
1 hundredweight [cwt] 112 lb 50.802 kg
1 slug 14.6 kg
1 long ton (UK) 20 cwt 1.016 t
1 short ton (US) 2,000 lb 0.907 t
metric > imperial
1 milligram [mg] 0.0154 grain
1 gram [g] 1,000 mg 0.0353 oz
1 kilogram [kg] 1,000 g 2.2046 lb
1 kilogram [kg] 1,000 g 0.068 slug
1 tonne [t] 1,000 kg
0.9842 long ton
(UK)
