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Order of operations


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Numeracy Basics

Order of Operations

Introduction

This topic will cover how to:

  • use the rules of precedence (BIMDAS) to solve problems involving multiple operations and positive numbers;
  • add, substract, multiply and divide expressions with negative numbers; and
  • use the rules of precedence (BIMDAS) to solve problems involving multiple operations and negative numbers.
  • use the rules of precedence (BIMDAS) to solve problems involving multiple operations and positive numbers;
  • add, subtract, multiply and divide expressions with negative numbers; and
  • use the rules of precedence (BIMDAS) to solve problems involving multiple operations and negative numbers.

Problems Involving Multiple Operations

Suppose you are asked to solve the following problem:

4+3(6-5)-(-3+50/5)x-2

This problem requires you to have an understanding of the following ideas, facts and rules about numbers:

  • what order to do each part of the problem; and
  • what to do about negative numbers

The following topic will explain these concepts, and by the time you can complete it you should be able to solve such problems with ease!

4 + 3(6 - 5) - (-3 + 50 ÷ 5) x -2

Order of Mathematical Operations

When a problem involves many different operations like the one specified, we need to know what order to do them in.

Consider the following example:

9 + 3 x 6 – 4 ÷ 2

Depending on what order you add, multiply, substract and divide, some of the possible solutions to this problem could be:

9 + 3 x 6 - 4 ÷ 2 = 34?

9 + 3 x 6 - 4 ÷ 2 = 11.5?

9 + 3 x 6 - 4 ÷ 2 = 12?

9 + 3 x 6 - 4 ÷ 2 = 25?

9 + 3 x 6 - 4 ÷ 2 = 70?

Clearly, we need a rule telling us which of these is the right answer.

9 + 3 x 6 - 4 ÷ 2 = 34?
9 + 3 x 6 - 4 ÷ 2 = 11.5?
9 + 3 x 6 - 4 ÷ 2 = 12?
9 + 3 x 6 - 4 ÷ 2 = 25?
9 + 3 x 6 - 4 ÷ 2 = 70?

Order of Mathematical Operations: BIMDAS

The rule we use for this is known by the acronym BIMDAS

This stands for:

Hence we work out our problem as follows:

B - Brackets . So evaluate whatever is in brackets first.
I - Indices . Next we work out any indices (not discussed here).
MD - Multiplication and Division . Next we work out any multiplication or division. Note that these operations are considered equal, so we work from left to right and do whichever comes first.
AS - Addition and Substraction . Finally, we work out nay addition and substraction. Note again that these operations are considered equal, so we work from from left to right and do whichever comes first.

9 + 3 x 6 - 4 ÷ 2

= 9 + 18 - 4 ÷ 2 (evaluate 3 x 6)

= 9 + 18 - 2 (evaluate 4 ÷ 2)

= 27 - 2 (evaluate 9 + 18)

= 25 (evaluate 27 - 2)

Examples: BIMDAS

3+4 x 2-7 = 3 + 8 - 7

(25 + 3 x 5) ÷ 8

40 - 12 x (7 - 5) ÷ 8

= 3 + 8 - 7

= 11 - 7

= 4

= (25 + 15) / 8

= 40 / 8

= 5

= 40 - 12 x 2 ÷ 8

= 40 - 24 ÷ 8

= 40 - 3

= 37

Activity 1: Practice questions

Working with Negative Numbers

Adding a negative number is the same as substracting a positive number. For example:

4 + (-5) = 4 - 5 = -1

Substracting a negative number is the same as adding a positive number. For example:

4 - (-5) = 4 + 5 = 9

If you multiply or divide two numbers with the same sign , the answer is positive. For example:

5(3) = 5x3 = 15
-5(-3) = -5 x -3 = 15
6 / 3 = 6 / 3 = 2
-6/-3 = -6 / -3 = 2

If you multiply or divide two numbers with the different signs, the answer is negative. For example:

5(-3) = 5x-3 = -15
-5(3) = -5 x 3 = -15
6 / -3 = 6 / -3 = -2
-5/3 = -5 / 3 = -2

Examples: Working with Negative Numbers

Simplify the following expressions involving addition and substraction of negative numbers:

a. 4 + (-8) = 4 - 8 = - 4

b. 13 - (-9) = 13 + 9 = 22

c. -3 + (-4) = -3 - 4 = -7

Simplify the following expressions involving multiplication and division of negative numbers:

a. -4(-8) = -4 x -8 = 32

b. 8(-9) = 8 x -9 = -72

c. -12 ÷ 4 = -12 ÷ 4 = -3

d. -8 ÷ -2 = -8 ÷ -2 = 4

= 4 - 8
= -4
= 13 + 9
= 22
= -3 - 4
= -7
= -4 x -8
= 32
= 8 x -9
= -72
= -12 ÷ 4
= -3
= -8 ÷ -2
= 4

Activity 2: Practice questions

Activity: BIMDAS and Negative Numbers

1. 10 - (4 - 5) = 10 - (-1) = 10 + 1 =11


2. 10 - (4 + 5) = 10 - 9 = 1


3. 10(4 - 5) = 10(-1) = -10


4. -10(4 - 5) = -10(-1) = 10

= 10 - (-1)

= 10 + 1

= 11

= 10 - 9

= 1

= 10(-1)

= -10

= -10(-1)

= 10

Solving Problems with Multiple Operations

Now that we have learnt about BIMDAS and negative numbers, we can return to our original problem:

4 + 3(6 - 5) - (-3 + 50 ÷ 5) x -2

You should now be able to solve this problem as follows:

4 + 3(6 - 5) - (-3 + 50 ÷ 5) x - 2

= 4 + 3(6 - 5) - (-3 + 10) x - 2

= 4 + 3(1) - (-3 + 10) x - 2

= 4 + 3(1) - 7 x - 2

= 4 + 3 - 7 x - 2

= 4 + 3 - (-14)

= 7 - (-14)

= 7 + 14

= 21

End of Topic

Congratulations, you have complete this topic.

You should now have a better understanding of Order of Operations

 


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