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Level 7

1.02 Order of operations with whole numbers

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Order of operations with whole numbers

The order of operations is the way we understand and remember how to evaluate expressions involving two or more binary operations.

This convention is used so that everyone can agree about how to write and perform mathematics.

The order is as follows:

  • Evaluate whatever is contained in brackets.
  • Evaluate any multiplication or division, reading from left to right.
  • Evaluate any addition or subtraction, reading from left to right.

There's no standout reason why this should be the order but it is the one everybody uses. Before it becomes natural, this is something you will have to learn by heart.

Examples

Example 1

Consider the expression 6+16\div4.

a

Which operation should we perform first?

A
Add 6 and 16.
B
Divide 16 by 4.
Worked Solution
Create a strategy

Division comes before addition.

Apply the idea

We should divide 16 by 4 first before performing addition, option B.

b

Now, evaluate 6+16\div4.

Worked Solution
Create a strategy

Perform division first before addition.

Apply the idea
\displaystyle 6+16\div4\displaystyle =\displaystyle 6+4Divide 16 by 4
\displaystyle =\displaystyle 10Evaluate the addition

Example 2

Consider the expression 4+7\times3.

Which of the following expressions gives the same value as the given expression? Select all that apply.

A
7\times3+4
B
3\times4+7
C
(4+7)\times3
D
4+(7\times3)
Worked Solution
Create a strategy

Evaluate the expression in each option and compare it to the value of the given expression.

Apply the idea

Evaluate the given expression:

\displaystyle 4+7\times3\displaystyle =\displaystyle 4+21Evaluate the multiplication
\displaystyle =\displaystyle 25Evaluate the addition

Option A:

\displaystyle 7\times3+4\displaystyle =\displaystyle 21+4Evaluate the multiplication
\displaystyle =\displaystyle 25Evaluate the addition

Option B:

\displaystyle 3\times4+7\displaystyle =\displaystyle 12+7Evaluate the multiplication
\displaystyle =\displaystyle 19Evaluate the addition

Option C:

\displaystyle (4+7)\times3\displaystyle =\displaystyle 11\times3Evaluate the brackets
\displaystyle =\displaystyle 33Evaluate the multiplication

Option D:

\displaystyle 4+(7\times3)\displaystyle =\displaystyle 4+21Evaluate the brackets
\displaystyle =\displaystyle 25Evaluate the addition

The expressions that give the same value as 4+7\times3 are option A: 7\times3+4 and option D: 4+(7\times3).

Example 3

Consider the expression 4+18-5\times2.

a

Which of the following is the value of the given expression?

A
12
B
30
C
34
Worked Solution
Create a strategy

Evaluate the expression following the correct order of operations.

Apply the idea
\displaystyle 4+18-5\times2\displaystyle =\displaystyle 4+18-10Evaluate the multiplication
\displaystyle =\displaystyle 22-10Evaluate the addition
\displaystyle =\displaystyle 12Evaluate the subtraction

4+18-5\times2=12, option A.

b

Without changing the order of the numbers and operations, write the expression which would evaluate to 30.

Worked Solution
Create a strategy

We can use brackets to change the order the operations are done.

Apply the idea

Put 18-5 inside a bracket so that subtraction will be evaluated first followed by multiplication then addition.

\displaystyle 4+(18-5)\times2\displaystyle =\displaystyle 4+13\times2Evaluate the subtraction inside the bracket
\displaystyle =\displaystyle 4+26Evaluate the multiplication
\displaystyle =\displaystyle 30Evaluate the addition

So the expression is 4+(18-5)\times2.

c

Without changing the order of the numbers and operations, write the expression which would evaluate to 34.

Worked Solution
Create a strategy

We can use brackets to change the order the operations are done.

Apply the idea

Put 4+18-5 inside a bracket so that addition and subtraction will be evaluated first followed by multiplication.

\displaystyle (4+18-5)\times2\displaystyle =\displaystyle (22-5)\times2Evaluate the addition inside the bracket
\displaystyle =\displaystyle 17\times2Evaluate the subtraction inside the bracket
\displaystyle =\displaystyle 34Evaluate the multiplication

So the expression is (4+18-5)\times2.

Idea summary

The order of operation is as follows:

  • Evaluate whatever is contained in brackets.
  • Evaluate any multiplication or division, reading from left to right.
  • Evaluate any addition or subtraction, reading from left to right.

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