PIA B_Basic_Module 1.1 (MATHS) Final.pdf

PIA Training Centre (PTC) Module 1 – MATHEMATICS
Category – A/B1/B2
ISO 9001:2008 Certified For Training Purpose Only
PTC/CM/B Basic/M01/01 Rev. 00
1.1 ‐ i Mar, 2014
MODULE 1
Sub Module 1.1 ARITHMETIC
Sub Module 1.2 ALGEBRA
Sub Module 1.3 LOGARITHMS
Sub Module 1.4 GEOMETRY
Sub Module 1.5 INTRODUCTION TO STATISTICS

1.1 ‐ ii Mar, 2014
List of Amendments

Amendment No.
Sub-Module &
Pages:
Issue Date: Date Inserted: Inserted By: Date Removed: Removed By:
Issue 01, Rev-00 All 31 March 2014

1.1 ‐ iii Mar, 2014

Category – A/B1/B2 Sub Module 1.1 – Arithmetics
1.1 ‐ 1 Mar, 2014
MODULE 1
Sub Module 1.1
ARITHMETIC

1.1 ‐ 2 Mar, 2014
Contents

ARITHMETICAL TERMS AND SIGNS .................................................. 5
Rational and Irrational Numbers .................................................. 5
Absolute Value of a Number ......................................................... 6
PRACTICE QUESTIONS ..................................................................... 6
ARITHMETIC OPERATIONS .............................................................. 7
The Laws of Signs .......................................................................... 7
The Use of Symbols ....................................................................... 8
The Commutative, Associative and Distributive Laws .................. 9
Long Multiplication ....................................................................... 9
Sequence of Arithmetical Operations ......................................... 11
PRACTICE QUESTIONS ................................................................... 11
FACTORS AND MULTIPLES ............................................................. 12
Factors & Multiples ..................................................................... 12
Lowest Common Multiple (L.C.M.) ............................................. 12
Highest Common Factor (H.C.F.)
................................................. 12
FRACTIONS ................................................................................... 13
Vulgar Fractions .......................................................................... 13
Types of Fractions ....................................................................... 14

Lowest Common Denominator .................................................. 15
Addition of Fractions .................................................................. 15
Subtraction of Fractions ............................................................. 16
Multiplication of Fractions ......................................................... 17
Cancellation of Fractions ............................................................ 17
Division of Fractions ................................................................... 17
Operations with Fractions .......................................................... 18
DECIMALS ..................................................................................... 20
The Decimal System ................................................................... 20
Fraction to Decimal Conversion ................................................. 20
Conversion of Decimals to Fractions .......................................... 21
Operations of Decimal Numbers ................................................ 21
Powers of Ten ............................................................................. 21
Estimation Techniques ............................................................... 23
WEIGHTS, MEASURES AND CONVERSION FACTORS ....................... 25
The International System of Units .............................................. 25
Factors of Multiples & Sub‐multiples: ........................................ 25
Space & Time: ............................................................................. 25

1.1 ‐ 3 Mar, 2014
Mechanics: .................................................................................. 25
Heat: ............................................................................................ 26
Expressing SI Units ...................................................................... 26

Conversion Factors
...................................................................... 26
RATIO AND PROPORTION ............................................................. 27
Ratio ............................................................................................ 27
Proportional Parts ....................................................................... 28
Direct Proportion ........................................................................ 28
Inverse Proportion ...................................................................... 29
AVERAGES .................................................................................... 31
PERCENTAGES ............................................................................... 32
Percentage of a Quantity ............................................................ 33
AREAS AND VOLUMES .................................................................. 35
Areas ........................................................................................... 35
Volumes ...................................................................................... 38
SQUARES, CUBES AND SQUARE & CUBE ROOTS ............................ 41
Squares ....................................................................................... 41
Cubes .......................................................................................... 41
Square Roots............................................................................... 41
Cube Roots.................................................................................. 41

1.1 ‐ 4 Mar, 2014
Page Intentionally Left Blank

1.1 ‐ 5 Mar, 2014
ARITHMETICAL TERMS AND SIGNS
Rational and Irrational Numbers
It is generally believed that our present number system began
with the use of the natural numbers, such as 1, 2, 3, 4, . . .
These whole numbers, known as the positive integers, were
used primarily for counting. However, as time went on, it
became apparent that whole numbers could not be used for
defining certain mathematical quantities. For example, a period
in time might be between 3 and 4 days or the area of a field
might be between 2 and 3 acres (or whatever unit of measure
was used at the time). So the positive fractions were introduced,
e.g.
2
1
,
4
1
and
4
3
. These two groups of numbers, the positive
integers and the positive fractions, constitute what we call the
positive rational numbers. Thus, 317 is an integer or whole
number, is a positive fraction and
4
1
3 is a rational number.
In fact a rational number is any number that can be expressed
as the quotient of two integers, i.e. any number that can be
written in the form
b
a
where a and b represent any integers.
Thus
5
4
,
9
7
and 1 are all rational numbers.
The natural numbers are positive integers, but suppose we wish
to subtract a larger natural number from a smaller natural
number, e.g. 10 subtracted from 7, we obviously obtain a
number which is less than zero, i.e. 3
10
7 

 . So our idea of
numbers must be enlarged to include numbers less than zero
called negative numbers. The number zero (0) is unique, it is
not a natural number because all natural numbers represent
positive integer values, i.e. numbers above zero and quite
clearly from what has been said, it is not a negative number
either. It sits uniquely on its own and must be added to our
number collection.
So to the natural numbers (positive integers) we have added
negative integers, the concept of zero, positive rational numbers
and negative natural numbers. What about numbers like 2 ?
This is not a rational number because it cannot be represented
by the quotient of two integers. So yet another class of number
needs to be included, the irrational or non-rational numbers.
Together all, the above kinds of numbers constitute the broad
class of numbers known as real numbers.

1.1 ‐ 6 Mar, 2014
Absolute Value of a Number
Although we have mentioned negative numbers, we have not
considered their arithmetic manipulation. All positive and
negative numbers are referred to as signed numbers and they
obey the arithmetic laws of sign. Before we consider these laws,
let us first consider what we mean by signed numbers.
Conventional representation of signed numbers is shown below,
with zero at the midpoint. Positive numbers are conventionally
shown to the right of zero and negative numbers to the left:
· · ·, −4, −3, −2, −1, 0, +1, +2, +3, +4, · · ·
The number of units a point is from zero, regardless of its
direction, is called the absolute value of the number
corresponding to the point on the above number system when
points are drawn to scale. Thus the absolute value of a positive
number, or of zero, is the number itself. While the absolute
value of a negative number is the number with its sign changed.
For example, the absolute value of +10 is 10 and the absolute
value of −10 is also 10. Now the absolute value of any number n
is represented by the symbol |n|. Thus |+24| means the absolute
value of +24. Which is larger, |+3| or |−14|? The answer is |−14|
because its absolute value is 14, while that of |+3| is 3 and of
course 14 is larger than 3.
PRACTICE QUESTIONS
1. 6, 7, 9, 15 are ___________ numbers.
2.
5
8
,
4
1
and
64
7
are ___________ numbers.
3. Rewrite the numbers 5, 13, 16 in the form
b
a
, where
6

b .
4. Express the negative integers −4, −7, −12 in the form
b
a
, where 4

b .
5. 16
 can be expressed as a positive ___________. It
is ___________.
6. 10
 cannot be expressed as a/an ___________
number; however, it is a/an ___________.

1.1 ‐ 7 Mar, 2014
ARITHMETIC OPERATIONS
The Laws of Signs
First law: To add two numbers with like signs, add their absolute
values and prefix their common sign to the result.
This law works for ordinary arithmetic numbers and simply
defines what we have always done in arithmetic addition. For
example: 7
4)
(
3)
( 



 ; -12
(-5)
(-7) 
 and so on.
Second law: To add two signed numbers with unlike signs,
subtract the smaller absolute value
from the larger and prefix the sign of the number with the larger
absolute value to the results.
So following this rule, we get for example: 3
(-2)
5)
( 


 ;
-4
6)
(
(-10) 

 and so on.
Third law: To subtract one signed number from another, change
the sign of the number to be subtracted and follow the rules for
addition.
For example, if we subtract 5from 3
- , we get
-8
5)
(
(-3) 





 )
5
(
)
3
( .
Now what about the multiplication and division of negative and
positive numbers, so as not to labour the point the rules for
these operations are combined in our fourth and final law.
Fourth law: To multiply (or divide) one signed number by
another, multiply (or divide) their absolute values; then, if the
numbers have like signs, prefix the plus sign to the result; if they
have unlike signs, prefix the minus sign to the result.
Therefore, applying this rule to the multiplication of two positive
numbers, e.g. 12
4
3 
 ; 63
9
7 
 and so on, which of
course, is simple arithmetic! Now applying the rule to the
multiplication of mixed sign numbers we get e.g.
-24
-3)
8)
( 

 ( ; -35
7
5
- 
 and so on.

1.1 ‐ 8 Mar, 2014
The Use of Symbols
We have introduced earlier the concept of symbols to represent
numbers when we defined rational numbers where the letters a
and b were used to represent any integer. Look at the symbols
below, do they represent the same number?
IX; 9; nine; 81

The answer is ‘yes’, since each expression is a perfectly valid
way of representing the positive integer 9. In algebra we use
letters to represent Arabic numerals such numbers are called
general numbers or literal numbers, as distinguished from
explicit numbers like 1, 2, 3, etc. Thus a literal number is simply
a number represented by a letter, instead of a numeral. Literal
numbers are used to state algebraic rules, laws and formulae;
these statements being made in mathematical sentences called
equations.
If a is a positive integer and b is 1, what is
b
a ? Ofcourse
a
b
a  . Any number divided by 1 is always itself. Thus a
a 
1
, c
c 
1
, 7
1
7  and so on.
Suppose a is again any positive integer, but b is 0. What is the
value of a/b? What we are asking is, what is the value of any
positive integer divided by zero? Well the answer is that we
really do not know! The value of the quotient a/b, if b=0, is not
defined in mathematics. This is because there is no such
quotient that meets the conditions required of quotients.
For example, you know that to check the accuracy of a division
problem, you can multiply the quotient by the divisor to get the
dividend. For example, if 21/7 = 3, then 7 is the divisor, 21 is the
dividend and 3 is the quotient and so 3 × 7 = 21, as expected.
So, if 17/0 were equal to 17, then 17 × 0 should again equal 17
but it does not! Or, if 17/0 were equal to zero, then 0 × 0 should
equal 17 but again it does not. Any number multiplied by zero is
always zero. Therefore, division of any number by zero (as well
as zero divided by zero) is excluded from mathematics. If b=0,
or if both a and b are zero, then a/b is meaningless.
When multiplying literal numbers together we try to avoid the
multiplication sign ( ), this is because it can be easily mistaken
for the letter x .
Thus, instead of writing b
a  for the product of two general
numbers, we write b
a. (the dot notation for multiplication) or
more usually just ab to indicate the product of two general
numbers a and b . We can also write

1.1 ‐ 9 Mar, 2014
The Commutative, Associative and Distributive Laws
We all know that 30
5
6 
 and 30
6
5 
 , so is it true that
when multiplying any two numbers together, the result is the
same no matter what the order? The answer is yes. The above
relationship may be stated as: The product of two real numbers
is the same no matter in what order they are multiplied. That is,
ba
ab  ; this is known as the commutative law of multiplication.
If three or more real numbers are multiplied together, the order
in which they are multiplied still makes no difference to the
product. For example, 24
4
3
2 

 and 24
3
2
4 

 . This
relationship may be stated formally as: The product of three or
more numbers is the same no matter in what manner they are
grouped. That is, c
ab
bc
a )
(
)
(  ; this is known as the associative
law of multiplication.
These laws may seem ridiculously simple, yet they form the
basis of many algebraic techniques, which we will be using
later! We also have commutative and associative laws for
addition of numbers, which by now will be quite obvious to us,
here they are:
The sum of two numbers is the same no matter in what order
they are added. That is, a
b
b
a 

 . This is known as the
commutative law of addition.
The sum of three or more numbers is the same no matter in
what manner they are grouped. That is, )
(
)
( c
b
a
c
b
a 



 .
This is known as the associative law of addition.
The above laws are valid no matter whether or not the number
is positive or negative. So, for example, 3
)
5
16
(
8 


 and
3
5
)
16
8
( 


 .
In order to complete our laws we need to consider the following
problem: 
 )
6
5
(
4 ? We may solve this problem in one of two
ways, firstly by adding the numbers inside the brackets and then
multiplying the result by 4, this gives: 44
)
11
(
4  . Alternatively,
we may multiply out the bracket as follows:
44
24
20
)
6
4
(
)
5
4
( 




 . Thus, whichever method we
choose, the arithmetic result is the same. This result is true in all
cases, no matter how many numbers are contained within the
brackets! So in general, using literal numbers we have:
ac
ab
c
b
a 

 )
( . This is the distributive law.
Remember that the distributive law is valid no matter how many
numbers are contained in the brackets, and no matter whether
the sign connecting them is a plus or minus. As we will see
later, this law is one of the most useful and convenient rules for
manipulating formulae and solving algebraic expressions and
equations.
Long Multiplication

1.1 ‐ 10 Mar, 2014
Suppose we wish to multiply 35 by 24, i.e. 35
24 . The
numbers are first set out, one under the other, like this:
24
35
where the right-hand integers 5 and 4 are the units and the
left-hand integers are the tens, i.e. 10
3 and 10
2 . We
multiply the tens on the bottom row by the tens and units on the
top row. So to start this process, we place a nought in the units
column underneath the bottom row, then multiply 2 by 5 to get
10
1 , carry the 1 into the tens column and add it to the product
3
2 ; i.e.:
24
35
0
then multiply the 10
5
2 
 , put in the nought of the ten and
carry the one
24
35
00
1
now multiply 6
3
2 
 (the tens) and add the carried ten to it, to
give 7, then
24
35
700
We now multiply the 4 units by 35 . That is 20
5
4 
 put down
the nought carry 2 into the ten column, then multiply the 4
units by the 3 tens or, 12
3
4 
 and add to it the 2 we carried
to give 140, i.e.:
24
35
700
140
All that remains for us to do now is add 700 to 140 to get the
result by long multiplication, i.e.:
24
35
700
140
840

1.1 ‐ 11 Mar, 2014
Sequence of Arithmetical Operations
Numbers are often combined in a series of arithmetical
operations. When this happens a definite sequence must be
observed.
1. Brackets are used if there is any danger of ambiguity.
The contents of the bracket must be evaluated before
performing any other operation. Thus:
2  (7 + 4) = 2  11 = 22
15 - (8 - 3) = 15 - 5 = 10
2. Multiplication and division must be done before addition
and subtraction. Thus:
5 8 + 7 = 40 + 7 = 47 (not 5  15)
8 ÷ 4 + 9 = 2 + 9 = 11 (not 8 ÷ 13)
5  4 - 12 ÷ 3 + 7 = 20 - 4 + 7 = 27 - 4 =
23
So far we have used the standard operations of add, subtract,
multiply and divide.
Practice Questions
1. Find the value of:
a. )
( d
c
b
a 
 , where 3

a , 4


b , 6

c and
1


d .
b. 3
)
7
6
21
( 

c. 3
5
4
6 


d. 2
2
2 

2. Which of the following has the largest absolute value:
−7, 3, 15, −25, −31?
3. 





 28
)
3
(
)
4
(
16 ?
4. Find the absolute value of 




 )
82
(
)
38
14
(
4 ?
5. What is: (a)
3
15

; (b)
2
12


; (c)
2
14
1


 .
6. What is: (a) )
5
)(
2
)(
3
( 

 ; (b) )
15
(
2
3 

 .
7. Evaluate )
3
2
(
2 d
c
b
a 
 , when 4

a , 8

b , 2


c
and 2

d .
8. Use long multiplication to find the products of the
following:
a. 82
234
b. 236
1824

1.1 ‐ 12 Mar, 2014
FACTORS AND MULTIPLES
Factors & Multiples
If one number divides exactly into a second number the first
number is said to be a factor of the second. Thus:
35 = 5 7 ………….5 is a factor of 35 and so is 7.
240 = 3  8  10 ………….3, 8 and 10 are all factors of
240.
63 = 3 21 = 7  9……..…… 63 is said to be a multiple
of any of the numbers 3, 7, 9 and 21 because each of them
divides 63 exactly.
Finally, it is to remember that any number n multiplied by 1 is
itself, or n × 1 = n . So every number has itself and 1 as factors;
1 and n are considered trivial factors and when asked to find
the factors of an explicit or literal number, we will exclude the
number itself and 1. If a number has no other factors apart from
these, it is said to be prime number. Thus 2, 3, 7, 11, 13, 17, 19
and so on are all prime numbers.
Lowest Common Multiple (L.C.M.)
The L.C.M. of a set of numbers is the smallest number into
which each of the given numbers will divide. Thus the L.C.M. of
3, 4 and 8 is 24 because 24 is the smallest number into which
the numbers 3, 4 and 8 will divide exactly.
The L.C.M. of a set of numbers can usually be found by
inspection.
Highest Common Factor (H.C.F.)
The H.C.F. of a set of numbers is the greatest number which is
a factor of each of the numbers. Thus 12 is the H.C.F. of 24, 36
and 60. Also 20 is the H.C.F. of 40, 60 and 80.
Example: Find the LCM and HCF of 12 and 18.
The multiples of 12 are 12, 24, 36, 48, 60, 72, and so on;
whereas the multiples of 18 are 18, 36, 54, 72, 90, and so on.
Therefore the LCM of 12 and 18 is 36.
The factors of 12 are 2, 3, 4 and 6; whereas the factors of 18
are 2, 3, 6, and 9. Therefore the HCF of 12 and 18 is 6.
Practice Questions
1. What numbers are factors of:
(a) 24 (b) 56 (c) 42
2. Which of the following numbers are factors of 12: 2, 3, 4,
5, 6, 12, 18 and 24?
3. Write down all the multiples of 3 between 10 and 40.
4. Find the L.C.M. of the following set of numbers:
(a) 8 and 12 (b) 3, 4 and 5 (c) 2, 6 and 12
5. Find the H.C.F. of each of the following sets of numbers:
a) 8 and 12 (b) 24 and 36 (c) 10, 15
and 30

1.1 ‐ 13 Mar, 2014
FRACTIONS
A fraction is a division of one number by another. Thus, the
fraction 2/3 means two divided by three. The fraction x/y means
the literal number x divided by y. The number above the line is
called the numerator; the number below the line is the
denominator, as you learnt before. Thus, fractions are
represented as:
r
denominato
numerator
One question arises, why do we need to use fractions at all?
Why not use only decimal fractions? Well, one very valid reason
is that fractions provide exact relationships between numbers.
For example, the fraction 1/3 is exact, but the decimal fraction
equivalent has to be an approximation, to a given number of
decimals 0.3333, is corrected to four decimal places. Thus, 1/3
+ 1/3 + 1/3 = 1 but 0.3333 + 0.3333 + 0.3333 = 0.9999, not
quite 1.
Vulgar Fractions
The circle in the diagram below has been divided into eight
equal parts. Each part is called one-eighth of the circle and
written as
1
8 . If five of the eight equal parts are taken then we
have taken
5
8 of the circle.
From what has been said above we see that a fraction is always
a part of something. The number below the line (the
denominator) gives the fraction its name and tells us the
number of equal parts into which the whole has been divided.
The top number (the numerator) tells us the number of these
equal parts that are to be taken. For example the fraction
3
4
means that the whole has been divided into four equal parts and
that three of these parts are to be taken.
The value of a fraction is unchanged if we multiply or divide both
its numerator and denominator by the same amount.
3
5 =
12
20 (by multiplying the numerator and denominator
by 4)

1.1 ‐ 14 Mar, 2014
2
7 =
10
35 (by multiplying the numerator and denominator
by 5)
12
32 =
3
8 (by dividing the numerator and denominator by 4)
16
64 =
1
4 (by dividing the numerator and denominator by
16)
Example: Write down the fraction
2
7 with a denominator
(bottom number) of 28.
In order to make the denominator 28, we must multiply the
original denominator of 7 by 4 because 7 4 = 28.
Remembering that to leave the value of the fraction unchanged
we must multiply both numerator and denominator by the same
amount, then
2
7 =
2  4
7  4
=
8
28
Example: Reduce
210
336 to its lowest terms.
210
336 =
105
168 (by dividing top and bottom by 2)
=
35
=
5
Hence,
210
336 reduced to its lowest terms is
5
8 .
Types of Fractions
If the numerator of a fraction is less than its denominator, the
fraction is called a proper fraction. Thus,
2
3 ,
5
8 and
3
4 are all
proper fractions. Note that a proper fraction has a value which is
less than 1.
If the numerator of a fraction is greater than its denominator, the
fraction is called an improper fraction or a top heavy fraction.
Thus
5
4 ,
3
2 and
9
7 are improper fractions. Note that all improper
fractions have a value which is greater than 1.
Every improper fraction can be expressed as a whole number
and a proper fraction. These are sometimes called mixed
numbers. Thus, 1
1
2 , 5
1
3 and 9
3
4 are all mixed numbers. In
order to convert an improper fraction into a mixed number it
must be remembered that:
number
bottom
number
top
number
bottom
number
top



1.1 ‐ 15 Mar, 2014
Example: Express
15
8 as a mixed number.
15
8 = 1
7
8 (Because 15 ÷ 8 = 1 and remainder 7).
From the above example we see that we convert an improper
fraction into a mixed number by dividing the bottom number into
the top number. Notice that the remainder becomes the
numerator in the fractional part of the mixed number. To change
a mixed number into an improper fraction we multiply the whole
number by the denominator of the fractional part. To this we add
the numerator of the fractional part and this sum then becomes
the numerator of the improper fraction. Its denominator is the
same as the bottom number of the fractional part of the mixed
number.
Example: Express 3
5
8 as a top heavy (improper) fraction.
3
5
8 =
( )
8  3 + 5
8 =
24 + 5
8 =
29
8
Lowest Common Denominator
When we wish to compare the values of two or more fractions
the easiest way is to express the fractions with the same bottom
number. This common denominator should be the L.C.M. of the
denominators of the fractions to be compared and it is called the
lowest common denominator.
Example: Arrange the fractions
3
4 ,
5
8 ,
7
10 and
11
20 in order of
size starting with the smallest.
The lowest common denominator of 4, 8, 10 and 20 is 40.
Expressing each of the given fractions with a bottom number of
40 gives:
3
4 =
3  10
4  10
=
30
40
5
8 =
5  5
8  5
=
25
40
7
10 =
7  4
10  4
=
28
40
11
20 =
11  2
20  2
=
22
40
Therefore the order is
22
40,
25
40,
28
40,
30
40 or
11
20,
5
8,
7
10 and
3
4 .
Addition of Fractions
The steps when adding fractions are as follows:
1. Find the lowest common denominator of the fractions to
be added.
2. Express each of the fractions with this common
denominator.
3. Add the numerators of the new fractions to give the
numerator of the answer. The denominator of the
answer is the lowest common denominator found in first
step.

1.1 ‐ 16 Mar, 2014
Example: Find the sum of
2
7 and
3
4 .
First find the lowest common denominator (this is the L.C.M. of
7 and 4). It is 28. Now express
2
7 and
3
4 with a bottom number
of 28.
2
7 =
2  4
7  4
=
8
28
3
4 =
3  7
4  7
=
21
28
Adding the top numbers of the new fractions:
2
7 +
3
4 =
8
28 +
21
28 =
29
28 = 1
1
28
A better way of setting out the work is as follows:
2
7 +
3
4 =
2  4 + 3  7
28 =
8 + 21
28 =
29
28 = 1
1
28
Example:Simplify
3
4 +
2
3 +
7
10 .
The L.C.M. of the bottom numbers 4, 3 and 10 is 60.
3
4 +
2
3 +
7
10 =
3  15 + 2  20 + 7  6
60
=
45 + 40 + 42
60
=
127
60 = 2
7
60
Example: Add together 5
1
2, 2
2
3 and 3
2
5
First add the whole numbers together, 5 + 2 + 3 = 10. Then
add the fractional parts in the usual way. The L.C.M. of 2, 3 and
5 is 30.
5
1
2 + 2
2
3 + 3
2
5 = 10 +
15  1 + 10  2 + 6  2
30
= 10 +
15 + 20 + 12
30
= 10 +
47
30 = 10 + 1
17
30
= 11
17
30
Subtraction of Fractions
The method is similar to that in addition. Find the common
denominator of the fractions and after expressing each fraction
with this common denominator, subtract.
Example: Simplify
5
8 -
2
5
The L.C.M. of the bottom numbers is 40.
5
8 -
2
5 =
5  5 - 8  2
40 =
25 - 16
40 =
9
40

1.1 ‐ 17 Mar, 2014
When mixed numbers have to be subtracted the best way is to
turn the mixed numbers into improper fractions and then
proceed in the way shown in Example 9.
Example: Simplify 3
7
10 - 2
3
4
3
7
10 - 2
3
4 =
37
10 -
11
4 =
37  2 - 11  5
20
=
74 - 55
20 =
19
20
Multiplication of Fractions
When multiplying together two or more fractions we first multiply
all the numerators together and then we multiply all the
denominators together. Mixed numbers must always be
converted into improper fractions before multiplication. Similarly,
if the answer is an improper fraction, this can be finally
converted into the corresponding mixed number.
Sometimes in calculations with fractions the word 'of' appears. It
should always be taken as meaning multiply. Thus:
4
5 of 20 =
4
5
/1 
20
/ 4
1 =
4  4
1  1
=
16
1 = 16
Example: Simplify
5
8 
3
7
5
8 
3
7 =
5  3
8  7
=
15
56
Example: Simplify
2
5  3
2
3
15
7
1
15
22
3
5
11
2
3
11
5
2
3
2
3
5
2








Cancellation of Fractions
Example: Simplify
4
35
8
7
20
16


10
9
4
10
49
2
1
5
7
7
1
4
5
3
8
7
5
2
6
1
2
7
1
5
1
2


















Division of Fractions
To divide by a fraction, all we have to do is to invert it (i.e. turn it
upside down) and multiply. Thus:
3
5 ÷
2
7 =
3
5 
7
2 =
3  7
5  2
=
21
10 = 2
1
10

1.1 ‐ 18 Mar, 2014
Example: Divide 1
4
5 by 2
1
3
1
4
5 ÷ 2
1
3 =
9
5 ÷
7
3 =
9
5 
3
7 =
27
35
Operations with Fractions
The sequence of operations when dealing with fractions is the
same as those used with whole numbers. They are, in order:
1st Work out brackets;
2nd Multiply and divide;
3rd Add and subtract.
Example: Simplify
1
5 ÷




1
3 ÷
1
2
1
5 ÷




1
3 ÷
1
2 =
1
5 ÷




1
3 
2
1
=
1
5 ÷
2
3 =
1
5 
3
2 =
3
10
Example: Simplify
2
4
5 + 1
1
4
3
3
5
-
5
16
With problems of this kind it is best to work in stages as shown
below.
2
4
5 + 1
1
4 = 3
16 + 5
20 = 3
21
20 = 4
1
20
4
1
20
3
3
5
=
81
20 ÷
18
5 =
81
20 
5
18 =
9
8
9
8 -
5
16 =
18 - 5
16 =
13
16

1.1 ‐ 19 Mar, 2014
Practice Questions
1. Write down the following fractions with the denominator
(bottom number) stated.
(a)
3
4 with denominator 28
(b)
3
(c)
5
(d)
1
2. Reduce the following fractions to their lowest terms:
(a)
8
16 (b)
15
25 (c)
210
294
3. Express each of the following as a mixed number:
(a)
7
2 (b)
22
10 (c)
21
8
4. Express each of the following as top heavy (improper)
fractions:
(a) 2
3
8 (b) 8
2
3 (c)4
3
7
5. Arrange the following sets of fractions in order of size,
beginning with the smallest:
(a)
2
1
,
6
5
,
3
2
,
12
7
(b)
4
3
,
8
5
,
5
3
,
20
13
6. Arrange the following sets of fractions in order of size,
beginning with the smallest:
(a)
2
1
,
6
5
,
3
2
,
12
7
(b)
4
3
,
8
5
,
5
3
,
20
13
7. Simplify:
(a)
1
2 +
1
3 (b)
1
8 +
2
3 +
3
5
(c)1
3
8 + 3
9
16 (d) 4
1
2 + 3
5
6 + 2
1
3
(e)
3
4  1
7
9 (f) 5
1
5 
10
13
(g) 1
5
8 
7
26 (h)
5
7 of 140
(i) 3
1
15 ÷ 2
5
9 (j) 3
3
14 ÷




1
1
49 
7
10
(k) 3
2
3 ÷




2
3 +
4
5

1.1 ‐ 20 Mar, 2014
DECIMALS
The Decimal System
The decimal system is an extension of our ordinary number
system. When we write the number 666 we mean 600 + 60 + 6.
Reading from left to right each figure 6 is ten times the value of
the next one.
We now have to decide how to deal with fractional quantities,
that is, quantities whose values are less than one. If we regard
666.666 as meaning 600 + 60 + 6 +
6
10 +
6
100 +
6
1000 then
the dot, called the decimal point, separates the whole numbers
from the fractional parts. Notice that with the fractional or
decimal parts, e.g. 0.666, each figure 6 is ten times the value of
the following one, reading from left to right. Thus
6
10 is ten times
as great as
6
100 , and
6
100 is ten times as great as
6
1000 , and
so on.
Decimals then are fractions, which have denominators of 10,
100, 1000 and so on, according to the position of the figure after
the decimal point.
If we have to write six hundred and five we write 605; the zero
keeps the place for the missing tens. In the same way if we
want to write
3
10 +
5
1000 we write 0.305; the zero keeps the
place for the missing hundredths. Also
6
100 +
7
1000 would be
written
0.067; the zero in this case keeps the place for the missing
tenths.
When there are no whole numbers it is usual to insert a zero in
front of the decimal point so that, for instance, .35 would be
written 0.35.
Fraction to Decimal Conversion
We found, when doing fractions, that the line separating the
numerator and the denominator of a fraction takes the place of
a division sign. Thus
17
80 is the same as 17 ÷ 80.
Therefore to convert a fraction into a decimal we divide the
denominator into the numerator.
Example: Convert
27
32 to decimal number.
27
32 = 27 ÷ 32

1.1 ‐ 21 Mar, 2014
Sometimes a fraction will not divide out exactly. For example,
1
3
= 0.333…
Conversion of Decimals to Fractions
We know that decimals are fractions with denominators 10, 100,
1000, etc. Using this fact we can always convert a decimal to a
fraction.
Example: Convert 0.32 to a fraction.
0.32 =
32
100 =
8
25
Operations of Decimal Numbers
The basic operations (that is addition, subtraction, multiplication
and division) of decimals is same as of integers. However, in
addition and subtraction of decimals, the decimal points of each
number are aligned vertically. For the multiplication and division
of decimal numbers, the numbers are first converted into
fractions and then simplified, and then the result is finally
converted into decimal number.
Powers of Ten
Decimal numbers may be expressed in index form, using the
powers of ten. For example:
1,000,000 = 1 × 106
100,000 = 1 × 105
10,000 = 1 × 104
1000 = 1 × 103
100 = 1 × 102
10 = 1 × 101
0 = 1 × 100
1/10 = 0.1 = 1 × 10-1
1/100 = 0.01 = 1 × 10-2
1/1000 = 0.001 = 1 × 10-3
1/10,000 = 0.0001 = 1 × 10-4
1/100,000 = 0.00001 = 1 × 10-5
1/1,000,000 = 0.000001 = 1 × 10-6

1.1 ‐ 22 Mar, 2014
We show the number one million (1,000,000) as 1×106, i.e. 1
multiplied by 10, six times. The exponent (index) of 10 is 6, thus
the number is in exponent or exponential form.
Note that we multiply all the numbers, represented in this
manner by the number 1. This is because we are representing
one million, one hundred thousand, one tenth, etc.
When representing decimal numbers in index (exponent) form,
the multiplier is always a number which is ≥1.0 or <10; i.e. a
number greater than or equal to (≥1.0) one or less than (<10)
ten.
So, for example, the decimal number is 8762.0 = 8.762 × 103 in
index form. Note that with this number, greater than 1.0, we
displace the decimal point three (3) places to the left; i.e. three
powers of ten. Numbers rearranged in this way, using powers of
ten, are said to be in index form or exponent form or standard
form.
Now consider the decimal number 0.000245? In order to obtain
a multiplier that is greater than or equal to one and less than 10,
we need to displace the decimal point four (4) places to the
right. Note that the zero in front of the decimal point is placed
there to indicate that a whole number has not been omitted.
Therefore, the number in index form now becomes 2.45 × 10−4.
Notice that for numbers less than 1.0, we use a negative index.
In other words, all decimal fractions represented in index form
have a negative index and all numbers greater than 1.0,
represented in this way, have a positive index.

1.1 ‐ 23 Mar, 2014
Estimation Techniques
In most areas of engineering, there is little need to work to so
many places of decimals. If we have so many decimal places
accuracy in a number, this is unlikely to be needed, unless we
are dealing with a subject like rocket science or astrophysics!
So this leads us into the very important skill of being able to
provide approximations or estimates to a stated degree of
accuracy.
Example: For the numbers (a) 8762.87412355 and (b)
0.0000000234876;
1. Convert these numbers into standard form with
three decimal place accuracy.
2. Write down these numbers in decimal form, correct
to two significant figures.
(i)(a) By converting the given number in the standard form we
get 8.76287412355 × 103. Now looking at the decimal places
for the stated accuracy we must consider the first four places
8.7628 and since the last significant figure is 8, in this case
(greater than 5) we round up to give the required answer as
8.763 × 103.
(b) 0.0000000234876 = 2.34876 × 10−8 and now following the
same argument as above, this number is to three decimal
places = 2.349 × 10−8.
(ii)(a) For the number 8762.87412355, the two required
significant figures are to the left of the decimal place. So we are
concerned with the whole number 8762 and the first two figures
are of primary concern again to find our approximation we need
to first consider the three figures 876, again since 6 is above
halfway between 1 and 10, then we round up to give the
required answer 8800. Note that we had to add two zeros to the
left of the decimal point. This should be obvious when you
consider that all we have been asked to do is approximate the
number 8762 to within two significant figures.
(b) For the number 0.0000000234876 the significant figures are
any integers to theright of the decimal point and the zeros. So,
in this case, the number to the required number of significant
figures is 0.000000023.

1.1 ‐ 24 Mar, 2014
Practice Questions
1. Read off as decimals:
(a)
7
10 (b)
3
100
(c)
3
10 +
7
100 (d)
1
100 +
7
1000
(e)
5
10 +
8
100 +
9
1000
2. Convert the following to decimals correcting the
answers, where necessary, to 4 decimal places:
(a)
1
4 (b)
1
2 (c)1
5
6
(d) 2
7
16 (e)
11
16
3. Convert the following to fractions in their lowest terms:
(a) 0.2 (b) 0.312 5 (c) 0.007 5
(d) 0.45 (e) 2.55 (f) 2.125
4. Evaluate the following:
(a) 2.375 + 0.625 (b) 12.48 - 8.36
(c) 3.196 + 2.475 + 18.369 (d) 2.42  8
(e) 3.35  2.5 (f) 2.05 ÷ 1.5
5. Express the following numbers in normal decimal
notation:
(a) 3 × 10−1 + 5 × 10−2 + 8 × 10−2
(b) 5 × 103 + 81 − 100
6. Express the following numbers in standard form:
(a) 318.62; (b) 0.00004702;
(c) 51,292,000,000; (d) −0.00041045
7. Round-off the following numbers correct to three
significant figures:
(a) 2.713; (b) 0.0001267; (c) 5.435 × 104

1.1 ‐ 25 Mar, 2014
WEIGHTS, MEASURES AND CONVERSION FACTORS
The International System of Units
Together with major metric countries, Britain has adapted the
International System of Units known worldwide as the S.I.
(System International) Units. The effect of this system is to
introduce standard units for many of the quantities for which a
multitude of units exist as present.
S I Base Units:
Quantity Unit Symbol
Length metre m
Mass kilogram kg
Time second s
Electric Current Ampere A
Thermodynamic Temperature Kelvin K
Plain Angle Radians Rad
Luminous Intensity candela cd.
Factors of Multiples & Sub-multiples:
Multiple Prefix Symbol
106 Mega M
103 kilo k
10-3 milli m
10-6 micro µ
10-9 nano n
10-12 pico p
Space & Time:
Area square metre m2
Volume cubic metre m3
Velocity metre per second m/s
Acceleration metre per second squared m/s2
Angular Velocity radian per second rad/s
Angular Acceleration radian per second squared rad/s2
Frequency Hertz Hz
Mechanics:
Density Kilogram per cubic metre kg/m3
Momentum Kilogram metre per second kg m/s
Force Newton N =
kg.m/s2
Torque or Moment Newton metre N m
Energy, work Joule J = N m
Power Watt W = J/s
Pressure & Stress Newton per square metre
or Pascal
N/m2 = Pa

1.1 ‐ 26 Mar, 2014
Heat:
Celsius
temperature
Degrees Celsius °C
Expressing SI Units
The symbol for SI units and the conventions which govern their
use should be strictly followed.
 Use the correct symbols used in the foregoing lists.
 Never use a prefix without a unit either in writing or
speech, e.g. kilogram or kilometre not kilo, or millimetre
or millilitre not mil.
 Always put a zero before a decimal quantity less than
Unit, e.g. 0.705 m.
 When two units are multiplied together use a small
space between the symbols as the multiplier, e.g.
o Kilogram metresquared kg m2.
o Newton metre N m.
 When dividing, use an oblique stroke to separate the
numerator and denominator.
 metre per second m/s
 Joule per second J/s
 Use a space as a thousands marker not the comma.
The comma is used as a decimal marker in most
countries using the metric system and its use as a
thousand marker will cause confusion. Up to four figures
may be blocked together but five or more figures should
be grouped in threes, e.g.
1000 mm = 1 m
1 000 000 J = 1 MJ (MegaJoule)
0.000 000 001 s = 1ns (nanosecond).
 Leave a small space between figures and symbols.
Conversion Factors
The units which it is thought most likely you will be required to
know are set out below with appropriate conversion factors.
To go from the first quantity into the second multiply by the
number given.
Inches Millimetres 25.4
m Inches 39.37
Knot km/hr 1.852
Pounds Kilograms 0.4536
Kilograms Pounds 2.205
Imp. Galls Liters 4.546
bar p.s.i. 14.5
p.s.i. Pa (Pascal) 6895
bar Pa 105
N/m2 Pa 1
1bf N (Newton) 4.45
horsepower W (Watt) 746
B.T.U. KJ 1.055
ft 1bf J (Joule) 1.356

1.1 ‐ 27 Mar, 2014
RATIO AND PROPORTION
Ratio
A ratio is a comparison between two similar quantities. If the
length of a certain aircraft is 20 metres and a model of it is 1
metre long then the length of the model is
1
20 th of the length of
the aircraft. In making the model the dimensions of the aircraft
are all reduced in the ratio of 1 to 20. The ratio 1 to 20 is
usually written 1 : 20.
As indicated above a ratio may be expressed as a fraction and
all ratios may be looked upon as fractions. Thus the ratio 2 : 5 =
2
5 . The two terms of a ratio may be multiplied or divided without
altering the value of the ratio. Hence 6:36 = 1:6 =
1
6 . Again,
1:5 = 0.20.
Before a ratio can be stated the units must be the same. We
can state the ratio between 7 paisa and Rs.2 provided both
sums of money are brought to the same units. Thus if we
convert Rs.2 to 200 paisa the ratio between the two amounts of
money is 7 : 200.
Example: Express the ratio 20p to Rs.4 in its simplest form.
Rs.4 = 4  100p = 400p
20 : 400 =
20
400 =
1
20
Example: Two lengths are in the ratio 8:5. If the first length is
120 meters, what is the second length?
The second length =
5
8 of the first length
=
5
8 120 = 75 meters.

1.1 ‐ 28 Mar, 2014
Proportional Parts
The following diagram shows a line AB whose length is 16
centimeters divided into two parts in the ratio 3 : 5. As can be
seen in the diagram the line has been divided into a total of 8
parts.
The length AC contains 3 parts and the length BC contains 5
parts.
Each part is
16
8 = 2 centimeters long; hence AC is
3 2 = 6 centimeters long, and BC is
5 2 = 10 centimeters long.
We could tackle the problem in this way:
Total number of parts = 3 + 5 = 8 parts.
Length of each parts =
16
8 = 2 centimeters.
Length of AC = 3  2 = 6 centimeters.
Length of BC = 5  2 = 10 centimeters.
Example: Divide Rs.1100 into two parts in the ratio 7:3.
Total number of parts = 7 + 3 = 10
Amount of each part =
1100
10 = Rs.110
Amount of first part = 7  110 = Rs.770
Amount of second part = 3  110 = Rs.330
Example: An aircraft carries 2880 liters of fuel distributed in
three tanks in the ratio 3 : 5 : 4. Find the quantity in each
tank.
Total number of parts = 3 + 5 + 4 = 12.
Amount of each part =
2880
12 = 240 liters.
Amount of 3 parts = 3  240 = 720 liters.
The three tanks contain 720, 1200 and 960 liters.
Direct Proportion
Two quantities are said to vary directly, or be in direct
proportion, if they increase or decrease at the same rate. Thus
the quantity of fuel used and the distance travelled by an aircraft
are in direct proportion.

1.1 ‐ 29 Mar, 2014
In solving problems on direct proportion we can use either the
unitary method or the fractional method.
Example: If 25 kilograms of dry powder fire extinguishant cost
Rs.1700, how much does 8 kilograms cost?
1. Using the unitary method:
25 kilograms cost Rs.1700.
1 kilograms cost
1700
25 = Rs.68.
8 kilograms cost 8  68
= Rs.544.
2. Using the fractional method:
Cost of 8 kilograms.
=
8
25  1700 =
8  1700
25
= Rs.544
Inverse Proportion
Two quantities are said to vary inversely, or be in inverse
proportion, if one quantity increases on decreasing the other
quantity and vice versa. Suppose that 8 fitters working on an
aircraft 'C' check takes 10 days to complete it. If we double the
number of men then we should halve the time taken. If we halve
the number of men then the job will probably take twice as long.
This is an example of inverse proportion.
Example: 20 men working at a company produce 3000
components in 12 working days. How long will it take 15 men to
produce the 3000 components.
The number of men is reduced in the ratio
15
20 =
3
4 .
Since this is an example of inverse proportion the number of
days required must be increased in the ratio
4
3 .
Number of days required =
4
3  12.
= 16 days.

1.1 ‐ 30 Mar, 2014
Practice Questions
1. Express the following ratios as fractions in their lowest
terms:
(a) 8 : 3 (b) 9 : 15 (c) 12 : 4
2. Express the ratio of 30p to Rs.2 as a fraction in its
lowest terms.
3. Express the ratio Rs.5 : 80p as a fraction in its lowest
terms.
4. Divide Rs.800 in the ratio 5 : 3.
5. A sum of money is divided into two parts in the ratio 5 :
7. If the smaller amount is Rs.200, find the larger
amount.
6. A alloy consists of copper, zinc and tin in the ratios 2 : 3
: 5. Find the amount of each metal in 75 kilograms of
the alloy.
7. If 7 kilograms of silica gel cost Rs.280, how much do 12
kilograms cost?
8. If 40 rivets cost Rs.3500, how much does 1 cost? What
is the cost of 55 rivets?
9. An aircraft flies 2000 kilometres in 4 hours. How long
will it take to complete a journey of 3500 kilometres?
10. 10 men produce 500 composite panels in a week. How
long would it take 15 men to produce the same amount?
11. Two gear wheels mesh together. One has 40 teeth and
the other has 25 teeth. If the larger wheel makes 100
revolutions per minute how many revolutions per minute
does the smaller wheel make?
12. 4 men can do a piece of work in 30 hours. How many
men would be required to do the work in 6 hours?

1.1 ‐ 31 Mar, 2014
AVERAGES
To find the average of a set of quantities, add the quantities
together and divide by the number of quantities in the set.
Thus,
Average =
sum of the quantities
number of quantities
Example: A student falls asleep in every lesson, the following
number of times: 8, 20, 3, 0, 5, 9, 15 and 12. What is his
average per lesson?
Average score=
8 + 20 + 3 + 0 + 5 + 9 + 15 + 12
8
=
72
8 = 9
Example (Weighted Average): A light aircraft is loaded with 22
boxes. If nineboxes have a mass of 12 kg, eight boxes have a
mass of 14 kg and five boxes have a mass of 14.5 kg. What is
the total mass of the boxes and the average mass per box?
By finding the total mass of all 22 boxes, we can then find the
average mass per box. So we have:
9 × 12 = 108 kg
8 × 14 = 112 kg
5 × 15.5 = 77.5kg
Total mass = 297.5kg
Then average mass of all 22 boxes is 297.5 ÷ 22
= 13.52 kg
Practice Questions
1. Find the average of the following readings: 22.3 mm,
22.5 mm, 22.6 mm, 21.8 mm and 22.0 mm.
2. A train travels 300 km in 4 hours. What is its average
speed?
3. If a car travels for 5 hours at an average speed of 70
km/h how far has it gone?
4. If an aircraft flies a four-hour flight at the rate of 550
km/h and then two-hour flight at the rate of 450 km/h,
what is the average speed of the whole journey?

1.1 ‐ 32 Mar, 2014
PERCENTAGES
When comparing fractions it is often convenient to express them
with a denominator of a hundred. Thus:
1
2 =
50
100
2
5 =
40
100
Fractions with a denominator of 100 are called percentages.
Thus:
1
4 =
25
100 = 25 percent
3
10 =
30
100 = 30 percent
The sign % is usually used instead of the words per cent.
To convert a fraction into a percentage we multiply it by 100.
For example:
3
4 =
3
4  100 = 75
17
20 =
17
20  100 = 85
Decimal numbers may be converted into percentages by using
the same rule. Thus:
0.3 =
3
10 =
3
10  100 = 30%
The same rule result is produced if we omit the intermediate
step of turning 0.3 into vulgar fraction and just multiply 0.3 by
100. Thus:
0.3 = 0.3  100 = 30
To convert a percentage into a fraction we divide by 100. For
example,
45% =
45
100 = 0.45
3.9% =
3.9
100 = 0.039
Note that all we have done is to move the decimal point 2
places to the left.

1.1 ‐ 33 Mar, 2014
Percentage of a Quantity
It is easy to find the percentage of a quantity if we first express
the percentage as a fraction.
Examples:
1. What is 10% of 40?
Expressing 10% as a fraction it is
10
100 and the problem then
becomes:
What is
10
100 of 40?
10% of 40 =
10
100  40 = 4
2. What is 25% of £50?
25% of £50 =
25
100  £50 = £12.50
3. 22% of a certain length is 55 cm. What is the complete
length?
1% of the length =
55
22 cm = 2.5 cm
Now the complete length will be 100%, hence:
Complete length = 100  2.5 cm = 250 cm
Alternatively,
22% of the length = 55 cm
Complete length =
100
22  55
=
100  55
22 = 250 cm
4. What percentage is 37 of 264? Give the answer correct
to 5 significant figures.
Percentage =
37
264  100
=
37  100
264
= 14.015%

1.1 ‐ 34 Mar, 2014
Practice Questions
1. Convert the following fractions to percentages:
(a)
7
10 (b)
4
5 (c)
11
20
2. Convert the following decimal numbers into
percentages:
(a) 0.7 (b) 0.68 (c) 0.819
3. Convert the following percentages into decimal fractions:
(a) 32% (b) 31.5% (c) 3.95%
4. What is:
(a) 20% of 50 (b) 12% of 20 (c) 3.7% of 68
5. What percentage is:
(a) 25 of 200 (b) 29 of 178 (c) 15 of 33
6. If 20% of a length is 23 cm, what is the complete length?

1.1 ‐ 35 Mar, 2014
AREAS AND VOLUMES
Areas
We are already familiar with the concept of length, e.g. the
distance between two points, we express length in some
chosen unit, e.g. in meters; but if we wish to fit a carpet to the
room floor, the length of the room is insufficient. We obviously
need to know the width as well. This 2-dimensional concept of
size is termed Area.
Consider a room 4m by 3m as shown above. Clearly it can be
divided up into 12 equal squares, each measuring 1m by 1m.
Each square has an area of 1 square meter. Hence, the total
area is 12 square meters (usually written as 12m2 for
convenience). So, to calculate the area of a rectangle, multiply
length of 1 side by the length of the other side.
Note: 4 m x 3 m = 12 m2 (Don't forget the m2).
Example: An office 8.5m by 6.3m is to be fitted with a carpet, so
as to leave surround 600mm wide around the carpet. What is
the area of the surround?
With a problem like this, it is often helpful to sketch a diagram.
The area of the surround = office area - carpet area.
= (8.5 x 6.3) - (8.5 - 2 x 0.6) (6.3 - 2 x 0.6)
= 53.55 - (7.3) (5.1)
= 53.55 - 37.23
= 16.32 m2
Note that 600mm had to be converted to 0.6m. Don't forget to
include units in the answer e.g. m2.
The following table shows the formulae for the more common
shapes.

1.1 ‐ 36 Mar, 2014

1.1 ‐ 37 Mar, 2014
Example: The cross section of a block of metal is shown. Find
its area.
Area of trapezoid
= ½ x 40 x (30 + 50)
= ½ x 40 x 80
= 1600 mm²
Example: A hollow shaft has an outside diameter of 2.5cm.
Calculate the cross-sectional area of the shaft.
Area of cross-section
= area of outside circle – area of inside circle
=  x 1.626² -  x 1.25²
=  (1.625² - 1.25²)
= 3.142 x (2.640 – 1.563)
= 3.142 x 1.077
= 3.388 cm²

1.1 ‐ 38 Mar, 2014
Example: Calculate the length of arc of a circle whose radius is
8m and which subtends an angle of 56° at the centre.
Length of arc = 2r x
360
0

= 2 x  x 8 x
360
56
= 31.28 m²
Volumes
The concept and calculation of volume is the logical extension
of length and area.
Instead of squares, we now consider cubes. This is a three-
dimensional concept and the typical units of volume are cubic
meters (m3).
If we have a box, length 4m, width 3m and height 2m, we see
that the total volume = 24 cubic meters (24m3).
Each layer contains
4 x 3 = 12 cubes.
There are 2 layers.
Hence the volume is
12 x 2 = 24m3.
Basically, therefore, when calculating volume, it is necessary to
look for 3 dimensions, at 90º to each other, and then multiply
them together. For a box-type shape, multiplying length x width
x height = volume.
The volume of a solid figure is measured by seeing how many
cubic units it contains. A cubic meter is the volume inside a
cube which has a side of 1 meter. Similarly a cubic centimeter is
the volume inside a cube which has a side of 1 centimeter. The
standard abbreviations for units of volume are:
Cubic meter m³
Cubic centimeter cm³
Cubic millimeter mm³
Example: How many cubic centimeters are contained in 1 cubic
meter?
1m = 10² cm
1m³ = (10² cm)³ = 10
6
cm³
= 1 000 000 cm³
The following table gives volumes of some simple solids

1.1 ‐ 39 Mar, 2014
Practice Questions
1. The area of a rectangle is 220mm². If its width is 25mm
find its length.
2. A sheet metal plate has a length of 147.5mm and a
width of 86.5mm find its length to the nearest four
decimal places.
3. Find the area of a triangle whose base is 7.5cm and
whose altitude is 5.9cm.
4. Find the area of a trapezium whose parallel sides are
75mm and 82mm long respectively and whose vertical
height is 39mm.
5. The parallel sides of a trapezium are 12cm and 16cm
long. If its area is 220cm², what is its altitude?
6. Find the areas of the shaded portions in the diagram.

1.1 ‐ 40 Mar, 2014
7. Find the circumference of a circle whose radii are:
(a) 3.5mm (b) 13.8mm (c) 4.2cm
8. Find the diameter of a circle whose circumference is
34.4mm.
9. How many revolutions will a wheel make in travelling
2km if its diameter is 700mm?
10. If r is the radius and 0is the angle subtended at the
centre by an arc find the length of arc when: r = 2cm, 0
=30°
11. Convert the following volumes into the units stated:
(a) 5 m² into cm³ (b) 0.08 m³ into mm³
(c) 830 000 cm³ into m³ (d) 850 000 mm³ into m³
12. A steel ingot whose volume is 2 m³ is rolled into a plate
15mm thick and 1.75m wide. Calculate the length of the
plate in m.
13. A block of lead 2.0 m x 1m x 0.72m is hammered out to
make a square sheet 10mm thick. What are the
dimensions of the square?
14. The volume of a small cylinder is 180 cm³. If the radius
of the cross-section is 25mm, find its height.
15. A cone has a diameter of 28mm and a height of 66mm.
What is its volume?
16. Calculate the diameter of a cylinder whose height is the
same as its diameter and whose volume is 220 cm³.

1.1 ‐ 41 Mar, 2014
SQUARES, CUBES AND SQUARE & CUBE ROOTS
Squares
When a number is multiplied by itself the result is called the
square of the number. The square of 9 is 9 9 = 81. Instead
of writing 9  9, it is usual to write 92 which is read as the
square of 9. Thus:
122 = 12  12 = 144
(1.3)2 = 1.3  1.3 = 1.69
Example: Find (168.8)2
.
(168.8)2
= 168.8  168.8
= 28,480
Example: Find the value of
2
15
.
0
9
.
0






.
Cubes
When a number is multiplied by itself, i.e. 3 3 = 9, it is usual
to write it as 32 or 3 squared. We can take this a stage further
and multiply by another 3, i.e.
3  3 3 = 27, it is usual to write it as 33 or 3 cubed.
Square Roots
The square root of a number is the number whose square
equals the given number. Since 52
= 25, the square root of
25 = 5.
The sign is used to denote a square root and hence we write
Similarly, since 92
= 81, so .
Cube Roots
The cubed root of a number is the number which cubed equals
the number. For example, the cubed root of 64 = 4.
The sign is used to denote a cubed root and hence we write
.
Practice Questions
1. Find the square of the following numbers:
(a) 1.5 (b) 23 (c) 3.15
2. Find the cube of the following numbers:
(a) 7 (b) 1.5
3. Find the value of (3.142)2 correct to 2 places of decimal.
36
6
15
.
0
9
.
0 2
2








5
25 
9
81 
3
4
64
3


PIA B_Basic_Module 1.1 (MATHS) Final.pdf

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PIA B_Basic_Module 1.1 (MATHS) Final.pdf