The Order of Operations: PEMDAS Rule
In Japan, the math problem 9 - 3 \div \frac{1}{3} + 1 went viral after a study found that only 60% of individuals in their 20’s were able to get the correct answer, lower than the 90% success rate in the 1980s.
This study reflects the reality around the world that more and more students have given less importance to mastering the order of operations. This has also resulted in more parents struggling to help at least their kids understand the basics of the order of operations. Individualized Instruction from tutoring services will lessen this kind of problem with parents.
To help students and parents, let’s explore the order of operations and how this basic math skill can change your math game.
PEMDAS Rule Explained
In a nutshell, operations in mathematics are something like adding, subtracting, squaring, etc. These operations cannot be done randomly when solving a numerical expression. They follow this basic order called PEMDAS:
P – parenthesis
E – xponent
M – ultiplication
D – ivision
A – ddition
S – ubtraction
- Perform the operations inside any parentheses or groupings first.
- After the groupings, simplify any expression with exponents.
- Perform multiplication/division, whichever comes first from left to right.
- Lastly, perform addition/subtraction, whichever comes first from left to right.
Let’s look at some examples to understand better how this rule works.
Example 1:
6 \div 3 \times 2
One notion about following the PEMDAS rule is to do the multiplication first all the time since it comes before division in the acronym. While it is possible to do that, it becomes a mistake when students are unfamiliar with the formal definition of division—multiplication by the inverse. So it’s safer to perform whichever comes first from left to right.
6 \div 3 \times 2
= 2 \times 2
= 4.
6 \div 3 \times 2
6 \times \frac {1}{3} \times 2
6 \times \frac {2}{3}
= 4.
Example 2:
4+5 \times 2 - 33 \div 3
It is tempting to start by adding 4 and 5, but that is a huge mistake. Remember to follow PEMDAS, doing multiplication/division first.
Here’s a cool trick. In this expression, you can perform both multiplication and division simultaneously! Why? Since they are separated by subtraction, they get the same rank. Don’t worry, and we’re not breaking any rule here. We still follow the PEMDAS rule by performing multiplication/division first.
4+5 \times 2 - 33 \div 3
= 4 + 10 - 11.
Just like for multiplication and division, it is possible to perform either of the two, regardless of which comes first, provided that you know the formal definition of subtraction—adding the inverse. For example, 5-3 means 5+(-3). So performing 10-6 will still yield the same answer. Do this if you are confident about the rules of integers. Otherwise, follow the standard PEMDAS rule.
4 + 10 - 11
= 4 - 11
= 3
4 + 10 - 11
= 4 - 1
= 3
Example 3:
20+ 154 \div [(9-6)^2 + (32-30) ] \times 7 - 99
Start by simplifying the grouping.
= 20 + 154 \div [3^2 + 2] \times 7 - 99
= 20 + 7 \times 154 \div [9 + 2] - 99
= 20 +154 \div 11 \times 7 - 99
Perform division/multiplication, then addition/subtraction.
= 20 + 14 \times 7 - 99
= 20 + 98 - 99
= 19
Applications of Order of Operations
Order of operations is one of the essential concepts you must master to succeed in all kinds of math. One common application is its use in real-world word problems.
Example 1: A telephone company charges $0.01 per minute for local calls and $0.05 per minute for calls to Canada. Last Friday, Jerry’s conversation with his mom over the telephone lasted for 2 minutes before he went to Canada. He paid $0.22 after they talked again when he arrived at his destination. For how many minutes did the international call last?
Cost of local call = 0.01 x 2
Cost of the international call = 0.22 – 0.01 x 2
Number of minutes of the international call = [0.22 – (0.01 x 2)] ÷ 0.05
[0.22 - (0.01 \times 2)] \div 0.05
= [0.22 - 0.2] \div 0.05
= 0.2 \div 0.05
= 4
Therefore, Jerry spent 4 minutes on the international call.
Example 2: Mady and Sarah went to a restaurant. Mady had a pancake and coffee, and Sarah had a slice of cake and tea. They decided to add up their bills and the tip and split the sum evenly. Sarah’s bill was twice as much as Mady’s, and they had $100 to pay the bill and the tip. If the tip amount was 5% of the total bill, how much was the change in terms of x, the amount of Mady’s bill?
Mady’s bill: x
Sarah’s bill: 2x
Total bill: 2x - x
Amount of tip: 0.05(2x+x)
Grand Total: 0.05(2x+x)+(2x+x)
Change: = 100-0.05(2x+x)+(2x+x)
100-0.05(2x+x)+(2x+x)
=100-1.05(2x+x)
=100-1.05(3x)
=100-3.15x
Therefore, their change is equal to 100-3.15x.
Order of operations is also important in solving higher mathematics degrees like Calculus.
Example 3:
Find \frac {d}{dx} (\frac {f}{g}) if f(x) = x^2 - 4 and g(x) = -x + 1
The derivative of the quotient of two functions, f, and g, is defined as
\frac {d}{dx} (\frac {f}{g}) = \frac {\frac{d}{dx}(f) \div g - f \div \frac{d}{dx}(g)}{g^2}
Then,
\frac {d}{dx} (\frac {f}{g}) = \frac {(2x)(-x+1)-(x^2-4)(-1)}{(-x+1)^2}
Apply the order of operations to simplify the derivative of \frac {f}{g} .
\frac {d}{dx} (\frac {f}{g}) = \frac {(-2x)^2+2)-(-x^2+4)}{(x^2-2x+1)}
\frac {d}{dx} (\frac {f}{g}) = \frac {-x^2+2x-4}{x^2-2x+1}
In the early stages of learning Math, students must first know the order of operations. And eventually, master its concepts to ace other harder math concepts and apply them in real-life situations.
So what’s the answer to 9 - 3 \div \frac {1}{3} + 1 ?
One mistake people made in solving this is inputting the problem in a calculator as 9 - 3 \div 1 \div 3 + 1 . The calculator will interpret the problem incorrectly because it will solve the first 31. Without the proper use of the calculator will lead to the wrong answer. Solving it manually is easy with the use of the PEMDAS Rule.
Let’s start with 3 \div \frac {1}{3} . Take the reciprocal of \frac {1}{3} and change division to multiplication.
9 - 3 \div \frac {1}{3} + 1
= 9 - 3 x 3 + 1
= 9 - 9 + 1
Then do subtraction/addition.
9 - 9 +1
= 0 +1
=1
Summary
- The order of operations is an essential math concept students should master to succeed in all kinds of math, from simple expression evaluation to real-life word problems.
- PEMDAS Rule stands for
P – arenthesis
E – xponent
M – ultiplication
D – ivision
A – ddition
S – ubtraction
- Perform the operations inside any parentheses or groupings first.
- After the groupings, simplify any expression with exponents.
- Perform multiplication/division, whichever comes first from left to right.
- Lastly, perform addition/subtraction, whichever comes first from left to right.
Be careful in performing the order of the operations. Unless you understand by heart the formal definitions of subtraction and division and the rules for integers, follow the standard PEMDAS Rule.
