Download BBA120 Business Mathematics

SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT
BBA120 BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS
Lecture 2
Moses Mwale
e-mail: [email protected]
BBA 120 Business Mathematics
Contents
Unit 2: Equations and Inequalities
3
2.0 Arithmetic Operations................................................................................................. 3
2.1. Addition and subtraction .................................................................................. 3
Examples ............................................................................................... 4
2.2 Multiplication and Division............................................................................... 4
Examples ............................................................................................... 5
2.3 Order of Operations ........................................................................................... 5
Examples ............................................................................................... 5
2.4 Fractions ............................................................................................................ 6
2.4.1 Addition and Subtraction of Fractions .................................................. 7
Example ................................................................................................. 7
2.4.2 Multiplication and Division of fractions ............................................... 7
Examples ............................................................................................... 8
2.5 Decimal Representation of Numbers ................................................................ 8
2.5.1 Converting Decimals to Fractions ........................................................ 9
2.5.2 Converting repeating decimals to fractions .......................................... 9
Example #1: ......................................................................................... 10
Example #2: ......................................................................................... 11
2.5.2 Significant Figures .............................................................................. 12
Example ....................................................................................................... 13
BBA 120 Business Mathematics
Unit 2: Equations and Inequalities
2.0 Arithmetic Operations
Arithmetic or arithmetic’s is
the oldest and most elementary
branch of mathematics, used by
almost everyone, for tasks
ranging from simple day-to-day
counting to advanced science
and business calculations. It
involves the study of quantity,
especially as the result of
operations that combine
numbers. In common usage, it refers to the simpler properties when using
the traditional operations of addition, subtraction, multiplication and
division with smaller values of numbers.
2.1. Addition and subtraction
Adding: If all the signs are the same, simply add all the terms and give
the answer with the common overall sign.
Subtracting: When subtracting any two numbers or two similar terms,
give the answer with the sign of the largest number or term.
If a and b are any two numbers, then we have the following rules
𝑎 + (−𝑏) = 𝑎 − 𝑏,
𝑎 − (+𝑏) = 𝑎 − 𝑏,
𝑎 − (−𝑏) = 𝑎 + 𝑏.
Thus we can regard −(−𝑏) as equal to +𝑏.
We consider a few examples:
4 + (−1) = 4 − 1 = 3,
and
3 − (−2) = 3 + 2 = 5.
The last example makes sense if we regard 3 − (−2) as the difference
between 3 and −2 on the number line.
Note that a − b will be negative if and only if a < b. For example,
−2 − (−1) = −2 + 1 = −1 < 0.
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Unit 2: Equations and Inequalities
Examples
Add/subtract with numbers, mostly
Add/subtract with variable terms
5 + 8 + 3 = 16
5𝑥 + 8𝑥 + 3𝑥 = 16𝑥
5 + 8 + 3 + 𝑣 = 16 + 𝑦
5𝑥 + 8𝑥 + 3𝑥 + 𝑣 = 16𝑥 + 𝑦
(The y-term is different, so it cannot be
5𝑥𝑣 + 8𝑥𝑣 + 3𝑥𝑣 + 𝑣 = 16𝑥𝑣 + 𝑦
added to the others)
2.2 Multiplication and Division
Multiplication
If a and b are any two positive numbers, then we have the following rules
for multiplying positive and negative numbers:
𝑎 × (−𝑏) = −(𝑎 × 𝑏),
Remember
(−𝑎) × 𝑏 = −(𝑎 × 𝑏),
It is useful to remember
that a minus sign is a -1,
(−𝑎) × (−𝑏) = 𝑎 × 𝑏.
so -5 is read as −1 × 5
Also
0 × (any real number) = 0
0 ÷ (any real number) = 0
But you cannot divide by
0
Brackets are used for
grouping terms together
in maths for:
(i) Clarity
(ii) Indicating the order in
which a series of
operations should be
carried out
So multiplication of two numbers of the same sign gives a positive
number, while multiplication of two numbers of different signs gives a
negative number.
For example, to calculate 2 × (−5), we multiply 2 by 5 and then place a
minus sign before the answer. Thus, 2 × (−5) = −10.
These multiplication rules give, for example,
(−2) × (−3) = 6, (−4) × 5 = −20, 7 × (−5) = −35.
Division
The same rules hold for division because it is the same sort of operation
as multiplication, since
𝑎
1
=𝑎×
𝑏
𝑏
So the division of a number by another of the same sign gives a positive
number, while division of a number by another of the opposite sign gives
a negative number.
BBA 120 Business Mathematics
Examples
a) 5 × 7 = 35
b) −5 × −7 = 35
c) −5 × 7 = −35
d) (−15) ÷ (−3) = 5
e) (−16) ÷ 2 = −8
f)
1
2 ÷ (−4) = − 2
g) (−𝑥)(−𝑦) = 𝑥𝑦
h) 2(𝑥 + 4) = 2𝑥 + 8
i)
(𝑥 − 3)(𝑥 + 4) = 𝑥(𝑥 + 4) − 3(𝑥 + 4)
= 𝑥 2 + 4𝑥 − 3𝑥 − 12
= 𝑥 2 + 𝑥 − 12
2.3 Order of Operations
The order in which operations in an arithmetical expression are
performed is important. Consider the calculation
12 + 8 ÷ 4.
Different answers are obtained depending on the order in which the
operations are executed.
If we first add together 12 and 8 and then divide by 4, the result is 5.
However, if we first divide 8 by 4 to give 2 and then add this to 12, the
result is 14.
Note: Other texts
use the Acronyms
BODMAS or
PEMDAS.
Therefore, the order in which the mathematical operations are performed
is important and the convention is as follows:
Brackets, exponents, division, multiplication, addition, and subtraction.
This convention has the acronym BEDMAS.
However, the main point to remember is that if you want a calculation to
be done in a particular order, you should use brackets to avoid any
ambiguity.
Examples
Evaluate the expressions
a) 3 ∗ ( 5 + 8 ) − 22 ÷ 4 + 3
Brackets or Parenthesis first: 5 + 8 = 13
3 ∗ 13 − 22 ÷ 4 + 3
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Unit 2: Equations and Inequalities
Exponent next: square the 2 or 22 = 4
3 ∗ 13 − 4 ÷ 4 + 3
Multiplication and Division next (3 ∗ 13) (4 ÷ 4)
left to right:
39 − 1 + 3
Addition and Subtraction next
left to right:
39 − 1 + 3 = 41
b) 23 × 3 + (5 − 1).
c) 4 – 3[4 – 2(6 – 3)] ÷ 2.
2.4 Fractions
𝑎
A fraction is a number that expresses part of a whole. It takes the form 𝑏
where a and b are any integers except that b ≠ 0.
Note that in the
𝑎
fraction 𝑏
a can be greater
than b
The integers a and b are known as the numerator and denominator of
the fraction, respectively.
Examples of statements that use fractions are,


3
5
1
3
of students in a lecture may be female or
of a person’s income may be taxed by the government.
Fractions may be simplified to obtain what is known as a reduced
fraction or a fraction in its lowest terms. This is achieved by identifying
any common factors in the numerator and denominator and then
cancelling those factors by dividing both numerator and denominator by
them.
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For example, consider the simplification of the fraction 45. Both the
numerator and denominator have 9 as a common factor since 27 = 9 × 3
and 45 = 9 × 5 and therefore it can be cancelled:
27 3 × 9
3
=
=
45 9 × 5
5
We say that
fraction.
27
45
3
and 5 are equivalent fractions and that
3
5
is a reduced
Comparing two fractions
To compare the relative sizes of two fractions and also to add or subtract
two fractions, we express them in terms of a common denominator.
The common denominator is a number that each of the denominators of
the respective fractions divides, i.e., each is a factor of the common
denominator.
BBA 120 Business Mathematics
Least Common
Multiple (LCM)
The least common
multiple (also called
the lowest common
multiple or smallest
common multiple) of
two integers a and b,
usually denoted by
LCM(a, b), is the
smallest positive
integer that is divisible
by both a and b.
If either a or b is 0,
LCM(a, b) is defined
to be zero.
For example the LCM
of 4 and 6 is 12
Suppose we wish to determine which is the greater of the two fractions
4
5
and 11.
9
The common denominator is 9 × 11 = 99. Each of the denominators (9
and 11) of the two fractions divides 99.
We follow a similar procedure when we want to add or subtract two
fractions.
2.4.1 Addition and Subtraction of Fractions
To add or subtract fractions we use the following method;
Step 1: Take a common denominator, that is, a number or term which is
divisible by the denominator of each fraction to be added or subtracted. A
safe bet is to use the product of all the individual denominators as the
common denominator.
Step 2: For each fraction, divide each denominator into the common
denominator, then multiply the answer by the numerator.
Step 3: Simplify your answer if possible.
Example
Simplify
a)
Greatest Common
Divisor (gcd)
The greatest common
divisor (gcd), also
known as the greatest
common factor (gcf),
or highest common
factor (hcf), of two or
more non-zero
integers, is the largest
positive integer that
divides the numbers
without a remainder.
For example, the GCD
of 8 and 12 is 4.
b)
c)
13
5
1
+3−5
− 16
24
7
𝑥
7
+
2
2𝑥
3
4
−
4𝑥
5
2.4.2 Multiplication and Division of fractions
To multiply together two fractions, we simply multiply the numerators
together and multiply the denominators together:
𝑎 c a × b ac
× =
=
𝑏 d c × b bd
To divide one fraction by another, we multiply by the reciprocal of the
𝑏
divisor where the reciprocal of the fraction a/b is defined to be provided
𝑎
a, b ≠ 0.
That is
𝑎 𝑐 a × d 𝑎𝑑
÷ =
=
𝑏 𝑑 c × b 𝑐𝑏
7
8
Unit 2: Equations and Inequalities
Examples
2
5
(2)(5)
10
a) ( ) ( ) =
=
3
7
(3)(7)
21
2
7
b) (− ) ( ) =
3
5
2
3
(−2)(7)
(3)(5)
2
=−
14
15
(3)(2)
6
1
c) 3 × = ( ) ( ) =
= =1
5
1
5
(1)(5)
5
5
3 (𝑥+3)
3(𝑥+3)
3𝑥+9
11
22
d) ( )
=
= 2
𝑥 (𝑥−5)
𝑥(𝑥−5)
𝑥 −5𝑥
e)
2
3
5
( )
11
f)
2𝑥
𝑥+𝑦
3𝑥
2(𝑥−𝑦)
( )
2
= (3) ( 5 ) = 15
=
=
2𝑥 2(𝑥−𝑦)
𝑥+𝑦
4𝑥(𝑥−𝑦)
3𝑥
4(𝑥−𝑦)
= 3(𝑥+𝑦)
3𝑥(𝑥+𝑦)
2.5 Decimal Representation of Numbers
A fraction or rational number may be converted to its equivalent decimal
representation by dividing the numerator by the denominator.
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For example, the decimal representation of is found by dividing 3 by 4
4
to give 0.75. This is an example of a terminating decimal since it ends
after a finite number of digits.
The following are examples of rational numbers that have a terminating
decimal representation:
BBA 120 Business Mathematics
1
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In some text
books, a bar
notation is used
instead of a dot
e.g.
52.71656565…
(65 repeating
infinitely often)
may be written
̅̅̅̅
52.7165
= 0.125,
and
3
25
= 0.12.
Some fractions do not possess a finite decimal representation – they go
1
on forever. The fraction 3 is one such example. Its decimal representation
is 0.3333... where the dots denote that the 3’s are repeated and we write
1
= 0. 3̇
3
where the dot over the number indicates that it is repeated indefinitely.
This is an example of a recurring decimal.
All rational numbers have a decimal representation that either terminates
or contains an infinitely repeated finite sequence of numbers. Another
example of a recurring decimal is the decimal representation of 1/13:
1
= 0.0769230769230 … = 0.07̇69230̇
13
where the dots indicate the first and last digits in the repeated sequence.
2.5.1 Converting Decimals to Fractions
To convert a decimal to a fraction, you simply have to remember that the
first digit after the decimal point is a tenth, the second a hundredth, and
so on.
For example,
0.2 =
2
1
=
10 5
and
0.375 =
375
3
=
1000 8
2.5.2 Converting repeating decimals to fractions
When converting repeating decimals to fractions, just follow the two
steps below carefully.
Step 1: Let x equal the repeating decimal you are trying to convert to a
fraction
Step 2: Examine the repeating decimal to find the repeating digit(s)
Step 3: Place the repeating digit(s) to the left of the decimal point
Step 4: Place the repeating digit(s) to the right of the decimal point
9
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Unit 2: Equations and Inequalities
Step 5: Subtract the left sides of the two equations. Then, subtract the
right sides of the two equations
As you subtract, just make sure that the difference is positive for both
sides
Now let's practice converting repeating decimals to fractions with two
good examples
Example #1:
What rational number or fraction is equal to 0.55555555555
Step 1: x = 0.5555555555
Step 2: After examination, the repeating digit is 5
Step 3: To place the repeating digit ( 5 ) to the left of the decimal point,
you need to move the decimal point 1 place to the right
Technically, moving a decimal point one place to the right is done by
multiplying the decimal number by 10.
When you multiply one side by a number, you have to multiply the other
side by the same number to keep the equation balanced
Thus, 10x = 5.555555555
Step 4: Place the repeating digit(s) to the right of the decimal point
Look at the equation in step 1 again. In this example, the repeating digit
is already to the right, so there is nothing else to do.
x = 0.5555555555
Step 5: Your two equations are:
10x = 5.555555555
x = 0.5555555555
10x - x = 5.555555555 − 0.555555555555
9x = 5
Divide both sides by 9
x = 5/9
BBA 120 Business Mathematics
Example #2:
What rational number or fraction is equal to 1.04242424242
Step 1: x = 1.04242424242
Step 2:After examination, the repeating digit is 42
Step 3:To place the repeating digit ( 42 ) to the left of the decimal point,
you need to move the decimal point 3 place to the right
Again, moving a decimal point three place to the right is done by
multiplying the decimal number by 1000.
When you multiply one side by a number, you have to multiply the other
side by the same number to keep the equation balanced
Thus, 1000x = 1042.42424242
Step 4:Place the repeating digit(s) to the right of the decimal point
In this example, the repeating digit is not immediately to the right of the
decimal point.
Look at the equation in step 1 one more time and you will see that there is
a zero between the repeating digit and the decimal point
To accomplish this, you have to move the decimal point 1 place to the
right
This is done by multiplying both sides by 10
10x = 10.4242424242
Step 5:Your two equations are:
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Unit 2: Equations and Inequalities
1000x = 1042.42424242
10x = 10.42424242
1000x - 10x = 1042.42424242 − 10.42424242
990x = 1032
Divide both sides by 990:
x = 1032/990
To master this lesson about converting repeating decimals to fractions,
you will need to study th2e two examples above carefully and practice
with other examples
2.5.2 Significant Figures
Sometimes we are asked to express a number correct to a certain number
of decimal places or a certain number of significant figures.
Suppose that we wish to write the number 23.541638 correct to two
decimal places. To do this, we truncate the part of the number following
the second digit after the decimal point:
23.54 | 1638.
Then, since the first neglected digit, 1 in this case, lies between 0 and 4,
then the truncated number, 23.54, is the required answer.
If we wish to write the same number correct to three decimal places, the
truncated number is
23.541 | 638,
and since the first neglected digit, 6 in this case, lies between 5 and 9,
then the last digit in the truncated number is rounded up by 1.
Therefore, the number 23.541638 is 23.542 correct to three decimal
places or, for short, ‘to three decimal places’.
To express a number to a certain number of significant figures, we
employ the same rounding strategy used to express numbers to a certain
number of decimal places but we start counting from the first non-zero
digit rather than from the first digit after the decimal point.
For example,
72,648 = 70,000 (correct to 1 significant figure)
= 73,000 (correct to 2 significant figures)
= 72,600 (correct to 3 significant figures)
BBA 120 Business Mathematics
= 72,650 (correct to 4 significant figures),
and
0.004286 = 0.004 (correct to 1 significant figure)
= 0.0043 (correct to 2 significant figures)
= 0.00429 (correct to 3 significant figures).
Note that 497 = 500 correct to 1 significant figure and also correct to 2
significant figures.
Example
a) A teacher may ask for an exact answer to the problem “What
is the length of the diagonal of a square whose sides have
length 2?” The exact answer is √8. An approximate answer
is 2.8284.
b) A somewhat more complicated example is 3227/555 =
5.8144144144…, here the decimal representation becomes
periodic at the second digit after the decimal point, repeating
the sequence of digits "144" indefinitely.
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