Order of Operations
Children should be led to make their
own investigations, and to draw upon
their own inferences. They should be
told as little as possible which can
produce unlimited learning potential.
Herbert Spencer
Intellectual Moral and Physical
1864
ORDER OF OPERATIONS
How to do a math problem
with more than one operation in
the correct order.
Order of Operations

Problem:
Evaluate the following
arithmetic expression:
3+4x2

Solution:
Student 1
3+4x2
=
7x2
=
14
Student 2
3+4x2
3+8
11
Order of Operations

It seems that each student interpreted the
problem differently, resulting in two different
answers.
 Student 1 performed the operation of addition
first, then multiplication
 Student 2 performed multiplication first, then
addition.
Order of Operations

When performing arithmetic operations
there can be only one correct answer. We
need a set of rules in order to avoid this
kind of confusion. Mathematicians have
devised a standard order of operations for
calculations involving more than one
arithmetic operation.
Order of Operations
Rule 1:
First perform any calculations
inside parentheses.
Rule 2:
Next perform all multiplications and
divisions, working from left to right.
Rule 3:
Lastly, perform all additions and
subtractions, working from left to
right.
Example
Expression
6+7x8
=
=
=
Evaluation
6+7x8
6 + 56
62
Operation
Multiplication
Addition
16 ÷ 8 - 2
16 ÷ 8 - 2
2-2
0
Division
Subtraction
(25 - 11) x 3
14 x 3
42
Parentheses
Multiplication
=
=
=
(25 - 11) x 3 =
=
=
Time to do some computing!
Evaluate using the order of operations.
3 + 6 x (5 + 4) ÷ 3 - 7
Solution:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
=
=
=
=
=
3+6x9÷3-7
3 + 54 ÷ 3 - 7
3 + 18 - 7
21 - 7
14
Parentheses
Multiplication
Division
Addition
Subtraction
Examples
2) 8 – 3 • 2 + 7
8 - 6 +7
2 + 7
9
3) 39 ÷ (9 +
4)
39 ÷ 13
3
1) 5 + (12 – 3)
5+ 9
14
Fractions
Evaluate the arithmetic expression below:
This problem includes a fraction bar, which means we
must divide the numerator by the denominator. However,
we must first perform all calculations above and below
the fraction bar BEFORE dividing.
The fraction bar can act as a grouping symbol
Fractions
Thus
Evaluating this expression, we get:
Write an arithmetic expression
Mr. Smith charged Jill $32 for parts and $15 per
hour for labor to repair her bicycle. If he spent 3
hours repairing her bike, how much does Jill owe
him?
Solution:
32 + 3 x 15
32 + 3 x 15
=
= 32 + 45 = 77
Jill owes Mr. Smith $77.
Add Parentheses to Obtain Result
7−1+6=0
7 − (1 + 6) = 0
3+8÷2=7
3 + (8 ÷ 2) = 7
1 + 2 × 5 + 6 = 21
(1 + 2) × 5 + 6 = 21
8+3−7=4
8+3−7=4
2 + 2 × 5 ÷ 3 − 1 = 10
(2 + 2) × 5 ÷ (3 − 1) =
10
Summary
When evaluating arithmetic expressions, the order
of operations is:
•Simplify all operations inside parentheses.
•Perform all multiplications and divisions,
working from left to right.
•Perform all additions and subtractions, working
from left to right.
If a problem includes a fraction bar, perform all calculations above and
below the fraction bar before dividing the numerator by the
denominator.