Department of DECISION SCIENCES
Elementary
Quantitative Methods
Only study guide for
QMI1500
University of South Africa
Pretoria
,
,COMPONENT 1
Numbers and working with numbers
On completion of this component you should have obtained basic numeracy skills.
CONTENTS
Study unit 1.1 Priorities and laws of operations
Study unit 1.2 Variables
Study unit 1.3 Fractions
Study unit 1.4 Powers and roots
Study unit 1.5 Ratios, proportions and percentages
Study unit 1.6 Signs, notations and counting rules
Study unit 1.7 Units and measures
1
, QMI1500/1
Study unit 1.1 Priorities and laws of operations
Learning objectives: On completion of this study unit you should
• know and be able to apply the priority rules in solving problems
• know and be able to apply the laws of operations in solving problems
1.1.1 Priorities
What are the values of the expressions
2 × 3 + 4 × 5 and 8 ÷ 2 − 6 ÷ 3?
I hope you are able to answer 26 and 2 respectively, without using your calculator. How do you
derive these results? In the first case, 2 × 3 is equal to 6, 4 × 5 is 20 and 6 + 20 is 26. In the
second case 8 ÷ 2 is equal to 4, 6 ÷ 3 is 2 and 4 − 2 is 2. Elementary!
Yes. It is elementary, but why?
It is essential that when we see an expression we all understand the same thing and obtain the
same result or else all our numbers and operations will lead to nonsense. Therefore, we all agree
to perform the operations in a specific order.
Looking at the above two expressions and their accepted results, it is evident that multiplication
takes precedence over addition, and division takes precedence over subtraction. But what about
more complex expressions like
2 × 3 − 4 × 5 + 6 ÷ 3?
Here, too, multiplication and division take precedence over addition and subtraction, but whether
you do the multiplication before the division and the subtraction before the addition, or vice versa,
does not matter. Check for yourself.
The commonly accepted convention is thus: multiplication and division have a higher priority
than addition and subtraction.
You may wonder whether there are operations that have a higher priority than multiplication
and division. Yes, there are. Specifically, the operation “change sign”, which changes a posi-
tive number to a negative number and vice versa, has the highest priority of all. Next comes
exponentiation in its various forms. The accepted priority level is the following:
Highest priority
Change sign
Exponentiation in all its forms
Multiplication and division
Addition and subtraction
Lowest priority
When you have more than one operation with the same priority, you just operate from left to
right. For instance,
15 ÷ 3 × 4
is not 15 ÷ 12, but is rather 5 × 4. This gives
5 × 4 = 20
because, going from left to right, you get to the division first.
2
Elementary
Quantitative Methods
Only study guide for
QMI1500
University of South Africa
Pretoria
,
,COMPONENT 1
Numbers and working with numbers
On completion of this component you should have obtained basic numeracy skills.
CONTENTS
Study unit 1.1 Priorities and laws of operations
Study unit 1.2 Variables
Study unit 1.3 Fractions
Study unit 1.4 Powers and roots
Study unit 1.5 Ratios, proportions and percentages
Study unit 1.6 Signs, notations and counting rules
Study unit 1.7 Units and measures
1
, QMI1500/1
Study unit 1.1 Priorities and laws of operations
Learning objectives: On completion of this study unit you should
• know and be able to apply the priority rules in solving problems
• know and be able to apply the laws of operations in solving problems
1.1.1 Priorities
What are the values of the expressions
2 × 3 + 4 × 5 and 8 ÷ 2 − 6 ÷ 3?
I hope you are able to answer 26 and 2 respectively, without using your calculator. How do you
derive these results? In the first case, 2 × 3 is equal to 6, 4 × 5 is 20 and 6 + 20 is 26. In the
second case 8 ÷ 2 is equal to 4, 6 ÷ 3 is 2 and 4 − 2 is 2. Elementary!
Yes. It is elementary, but why?
It is essential that when we see an expression we all understand the same thing and obtain the
same result or else all our numbers and operations will lead to nonsense. Therefore, we all agree
to perform the operations in a specific order.
Looking at the above two expressions and their accepted results, it is evident that multiplication
takes precedence over addition, and division takes precedence over subtraction. But what about
more complex expressions like
2 × 3 − 4 × 5 + 6 ÷ 3?
Here, too, multiplication and division take precedence over addition and subtraction, but whether
you do the multiplication before the division and the subtraction before the addition, or vice versa,
does not matter. Check for yourself.
The commonly accepted convention is thus: multiplication and division have a higher priority
than addition and subtraction.
You may wonder whether there are operations that have a higher priority than multiplication
and division. Yes, there are. Specifically, the operation “change sign”, which changes a posi-
tive number to a negative number and vice versa, has the highest priority of all. Next comes
exponentiation in its various forms. The accepted priority level is the following:
Highest priority
Change sign
Exponentiation in all its forms
Multiplication and division
Addition and subtraction
Lowest priority
When you have more than one operation with the same priority, you just operate from left to
right. For instance,
15 ÷ 3 × 4
is not 15 ÷ 12, but is rather 5 × 4. This gives
5 × 4 = 20
because, going from left to right, you get to the division first.
2
