T. Henson
Math 65 Elementary Algebra
ORDER OF OPERATIONS
Pink Elephants aRe Marching Down Algebra Street
OBJECTIVE: Explain the reasons behind the Order of Operations.
The general rule is to work from left to right, the same order in which we read a sentence,
and to perform operations in this order:
Pink
Elephants aRe
Marching Down
Algebra Street
The Order of Operations: PERMDAS
Perform operations with Parentheses and other grouping
P
symbols first.
Next
evaluate Exponents and Roots.
ER
MD After taking care of parentheses, exponents and roots, do
any Multiplications and Divisions as they occur in order
from left to right.
Lastly, perform Additions and Subtractions as they occur
AS
in order from left to right.
INTRODUCTION: The Order of Operations is a collection of rules for evaluating
numerical expressions. We need rules for performing operations in a prescribed manner;
otherwise, there would be many different answers to a single problem. Consider the
following situation:
Example: Two second-grade classes, their teachers and their teacher’s aides are going
on a field trip to the Planetarium. There are 22 students in one class and 27 in the other.
The teachers want to order a snack for the students and they want there to be two cheese
and cracker packets in each snack. The teachers also want one packet each for
themselves and their aides. How many packets should be ordered?
Solution: We would reason like this: First we need to find the total number of students
by adding 22 and 27. Next we would multiply that result by 2 since we want two packets
for each student. Finally add 4 (for the two teachers and their two aides). We could
write a mathematical expression that symbolizes this procedure:
4 + 2(22 + 27)
The order in which we perform the operations to arrive at the correct total number of
packets is precisely what the Order of Operations tells us we should do. First add 22 and
27, as the sum 22+27 is in parentheses. Next, multiply by 2 since multiplication is done
before addition. Lastly, add 4. Here is the work:
4 + 2 ( 22 + 27 ) = 4 + 2 ( 49 )
= 4 + 98
= 102
The parentheses play a very important role in putting this expression together. If the
parentheses were removed we would have
Math 65 Elementary Algebra
Order of Operations: PERMDAS
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4 + 2 ⋅ 22 + 27
If this expression were simplified using the Order of Operations, we would obtain
4 + 2 ⋅ 22 + 27 = 4 + 44 + 27
= 48 + 27
= 75
There are going to be some unhappy second-graders, because there won’t be enough
snacks to give two to every child!
A WORD OF CAUTION: A common mistake when simplifying expressions is to add
before multiplying. What would happen if this mistake were made when simplifying the
original expression?
4 + 2 ( 22 + 27 ) = 4 + 2 ( 49 )
= 6 ( 49 )
incorrectly added 4 and 2
= 294
In this case, if the teachers do not follow the order of operations they will have too many
snack packets.
TRY THIS practice problem: A man finds three envelopes on the ground, each
containing $10. There are no identifying marks on the envelopes, so the man keeps the
money he found. Amazingly, the next day he finds another envelope, this one containing
$15. Feeling lucky, he bets all the found money with his bookie and doubles his cash.
Write a mathematical expression which represents how much money the man has and
simplify it. (Answer on page 4.)
FAQ: The rules for the order in which we perform operations follow naturally from the
way we would arrive at the answer to a problem. But there are also mathematical reasons
for doing operations in PERMDAS order.
1. Why do we work from left to right? All of mathematics is based on assumptions;
fundamental concepts on which everyone agrees. The rules for the Order of
Operations were developed by mathematicians who spoke languages that were read
from left to right; this is why we work from left to right.
2. Why are multiplications and divisions done before additions and subtractions?
To answer this question we have to think about what the processes of multiplication
and division really meant, to understand why they are performed first. This rule is
followed because, in whole number arithmetic, multiplication and division are
shortcuts for repeated addition and subtraction.
Example One. For whole numbers, multiplication can be thought of repeated
addition. If we want to add five 3’s, we write 3 + 3 + 3 + 3 + 3 . The multiplication
5 ⋅ 3 means “five 3’s” and is a shorter way of saying 3 + 3 + 3 + 3 + 3 . When we
see a problem such as 17 + 5 ⋅ 3 we need to keep in mind that what this expression
really represents is the statement 17 + 3 + 3 + 3 + 3 + 3 . If we think of the problem
this way, we can see why it would be wrong to add 17 and 5 before multiplying
Math 65 Elementary Algebra
Order of Operations: PERMDAS
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by 3. The Order of Operations allows us to use a shortcut (multiplying 5 times 3)
instead of having to write out the sum of five 3’s.
Correct Order of Operations Solution
Incorrect Solution
17 + 5 ⋅ 3 = 22 ⋅ 3
17 + 5 ⋅ 3 = 17 + 15
= 66
= 32
Example Two. For whole numbers, division can be thought of as repeated
subtraction. That is, the division problem 18 ÷ 3 can be solved by repeatedly
taking away 3:
18 − 3 = 15
15 − 3 = 12
12 − 3 = 9
9−3 = 6
6−3 = 3
3−3 = 0
How many times did we subtract 3? The answer is 6, so 18 ÷ 3 = 6 . We can think
of the expression 18 ÷ 3 as asking the question, “How many times can we take 3
away from 18?” We have to know the answer to that question before we can
perform other operations. When we see a problem such as this
10 + 18 ÷ 3
we have to determine the answer to the division question before we can add 10.
Correct Order of Operations Solution
Incorrect Solution
10 + 18 ÷ 3 = 10 + 6
10 + 18 ÷ 3 = 28 ÷ 3
= 16
1
= 9
3
3. Why are Exponents and Roots evaluated before the other operations? The
reason again lies in what these operations mean and because these operations act on a
single number rather than a pair of numbers.
Just as multiplication is used as a short cut for repeated addition, exponent
notation is used to denote repeated multiplication. For instance, 34 means
3 ⋅ 3 ⋅ 3 ⋅ 3 ; that is, three multiplied by itself four times. So 5 ⋅ 34 means 5 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 .
A common error when working problems such as 5 ⋅ 34 is to multiply the 5 and 3
4
4
together first, yielding 15 . Can you see why this would be wrong? 15 means
15 ⋅15 ⋅15 ⋅15 and is not the same as 5 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 .
The symbol
a is read “the square root of a”; it is used when you want to ask the
question, “What number do you square to get a?” For instance, 16 = 4 because
42 = 16 . Whenever we see a square root, we must find the answer to this
question before we can use the number to perform other operations.
Math 65 Elementary Algebra
Order of Operations: PERMDAS
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4. Why do we use Parentheses? Think back to the example of the two second grade
classes. To figure out how many cheese-and-cracker packets were needed we first
added together the number of students in the two classes, then multiplied by two. The
order in which we performed those operations was contrary to the prescribed order of
operations. To indicate that we want to add before multiplying, we enclose the
addition operation in parentheses and write
2 ( 22 + 27 )
This expression would have a very different meaning without the parentheses. We
use parentheses to indicate that we wish to override the Order of Operations.
Sometimes w also use the square bracket [ and ] or the curly braces { and } to
enclose operations. These three types of symbols are called grouping symbols.
When one type of grouping symbol is enclosed by another type, the rule is to work
from the inside out. For example in the following expression
6 ⎡⎣5 + 3 ( 7 − 4 ) ⎤⎦
7 − 4 is to be
the parentheses are used to indicate that the subtraction
done before the multiplication by 3. Because the parentheses pair occurs inside the
square bracket pair, we would begin by simplifying inside the parentheses, then
perform the indicated operations inside the square brackets, before multiplying by 6.
Here is the solution:
6 ⎡⎣5 + 3 ( 7 − 4 ) ⎤⎦
= 6 ⎡⎣5 + 3 ( 3) ⎤⎦
= 6 [5 + 9]
= 6 [14]
= 84
Subtracted 4 from 7, to get 3. The parentheses around
the 3 are there to indicate multiplication.
Multiplied 3 times 3, to get 9.
Added the 5 and 9, to get 14. The square brackets
around the 14 are there to indicate multiplication.
Multiplied 6 times 14, to get 84.
Answer to Practise Problem: 2 ( 3 ⋅10 + 15 )
Math 65 Elementary Algebra
Order of Operations: PERMDAS
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