What is Order of Operations
Order of Operations refers to a set of rules used to determine the sequence in which mathematical operations should be performed in an expression or equation. The order of operations is also known as the “PEMDAS” rule, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
The order of operations can be remembered with the following acronym:
For example, consider the expression 6 + 3 * 2 – 4.
Using the order of operations, we would first perform the multiplication:
6 + (3 * 2) – 4 = 6 + 6 – 4
Then we would perform the addition and subtraction in order from left to right:
6 + 6 – 4 = 8
Therefore, the expression 6 + 3 * 2 – 4 simplifies to 8.
PEMDAS
- P – Parentheses first
- E – Exponents (ie Powers and Square Roots, etc.)
- MD – Multiplication and Division (left-to-right)
- AS – Addition and Subtraction (left-to-right)
Rules to follow when using the Order of Operations.
1. First, evaluate the operations within the parentheses.
2. Next, reduce the exponents and roots to its numerical form.
3. Afterwards, do all the multiplication and division from the left to right.
4. Finally, do the addition and subtraction to obtain the final answer.
Example
Let us look at how order of operation works.
3 x √25 – ( 4 +3 x 5)
According to the order of operations, we work out everything within the parentheses. In following rules (3) and (4) within the brackets, we have to multiply first before adding.
Now that the brackets have been removed, we can proceed on with the rest of the operations.
3 x √25 – 19
Next, we find the value of √25 .
3 x 5 – 19
Then, we multiply the terms and finally subtract to obtain the final answer of – 4.
Toppers Tips: Order of operations offer flexibility in writing mathematical expressions. Since addition and multiplication are commutative, it follows that 3 x 5 +2 can be written as 5 x 3 + 2 , 2 + 3 x 5 , 2 + 5 x 3 . It doesn’t change the mathematical concept behind the expression.
Why Follow the Order of Operations?
Following the order of operations is important because it ensures that mathematical expressions are evaluated in a consistent and unambiguous manner. If we didn’t have a set of rules for the order of operations, then different people could interpret the same expression differently, leading to inconsistent and potentially incorrect results.
By following the order of operations, we can be sure that mathematical expressions are evaluated correctly and that the result is the same for everyone who evaluates it. This is particularly important in fields such as mathematics, science, engineering, and finance where accurate calculations are critical.
Example: Here’s an example of how we can Calculate different answers if the correct order of operations is NOT followed:
Evaluate the arithmetic expression 4 + 6 x 3 . Below are the 2 students A and B, both Calculate in different order.
| Student A | Student B |
| 4 + 6 x 3
= 4 + 18 |
4 + 6 x 3
= 10 x 3 |
| = 22 | = 30 |
| Correct | Wrong |
As we can clearly see, both students got different answers to the same problem. Student A decided to multiply first before doing the addition whereas student B did the exact opposite. This is unacceptable as there can only be one correct answer in the evaluation of arithmetic expressions.
Solved Examples
Example 1: Simplify the expression 6 + 3 * 2 – 4
Step 1: P – Parentheses:In this example, there are no parentheses or brackets.
Step 2: E – There are no exponents.
Step 3 & 4: MD – Multiplication and Division: In this example, 3 * 2 is the only multiplication operation:
6 + 3 * 2 – 4
= 6 + 6 – 4 (3 * 2 equals 6)
= 8
Step 5 & 6: AS – Addition and Subtraction: In this example, we have:
6 + 3 * 2 – 4
= 6 + 6 – 4
= 8
Therefore, the simplified expression is 8.
Example 2: Simplify the expression 4 + 3 * 5 – 2^3
Step 1: P – Parentheses: There are no parentheses in this expression.
Step 2: E – Exponents: Evaluate 2^3:
4 + 3 * 5 – 2^3
= 4 + 3 * 5 – 8 (2^3 equals 8)
Step 3 &4: MD – Multiplication and Division: Perform the multiplication and division operations, working from left to right:
4 + 3 * 5 – 8
= 4 + 15 – 8 (3 * 5 equals 15)
Step 5 & 6: AS – Addition and Subtraction: Perform the addition and subtraction operations, working from left to right:
4 + 15 – 8
= 19 – 8
= 11
Therefore, the simplified expression is 11.
Example 3: Simplify the expression 12 – 4 * 2 + 6 / 3^2
Step 1: P – Parentheses: There are no parentheses in this expression.
Step 2: E – Exponents: Evaluate 3^2:
12 – 4 * 2 + 6 / 3^2
= 12 – 4 * 2 + 6 / 9 (3^2 equals 9)
Step 3 &4: MD – Multiplication and Division: Perform the multiplication and division operations, working from left to right:
12 – 4 * 2 + 6 / 9
= 12 – 8 + 6 / 9 (4 * 2 equals 8)
Step 5 & 6: AS – Addition and Subtraction: Perform the addition and subtraction operations, working from left to right:
12 – 8 + 6 / 9
= 4 + 6 / 9
= 4.67 (rounded to two decimal places)
Therefore, the simplified expression is 4.67.
Example 4: Simplify the expression 2 * (6 + 3) – 4 / 2^2
Step 1: P – Parentheses: Evaluate the operations inside the parentheses first:
2 * (6 + 3) – 4 / 2^2
= 2 * 9 – 4 / 2^2 (6 + 3 equals 9)
Step 2: E – Exponents: Evaluate 2^2:
2 * 9 – 4 / 2^2
= 2 * 9 – 4 / 4 (2^2 equals 4)
Step 3 &4: MD – Multiplication and Division: Perform the multiplication and division operations, working from left to right:
2 * 9 – 4 / 4
= 18 – 1 (4 / 4 equals 1)
Step 5 &6: AS – Addition and Subtraction: Perform the addition and subtraction operations, working from left to right:
18 – 1
= 17
Therefore, the simplified expression is 17.
Frequently Asked Questions
Q: What is the order of operations?
Ans: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed in an expression. The acronym PEMDAS (or sometimes BEDMAS or BODMAS) is often used to remember the order: Parentheses, Exponents, Multiplication and Division (working from left to right), and Addition and Subtraction (working from left to right).
Q: Why is the order of operations important?
Ans: Following the order of operations is important because it ensures that everyone evaluates expressions in the same way and gets the same answer. Without these rules, there could be ambiguity and confusion about how to interpret an expression.
Q: What happens if you don’t follow the order of operations?
Ans: If you don’t follow the order of operations, you might get a different answer than someone else who evaluated the same expression. Additionally, your answer might not be the correct solution to the problem you are trying to solve.
Q: Do you always have to follow the order of operations?
Ans: Yes
Q: Is PEMDAS the only way to remember the order of operations?
Ans: No, there are other acronyms that can be used to remember the order of operations, such as BEDMAS (Brackets, Exponents, Division and Multiplication (working from left to right), Addition and Subtraction (working from left to right)) or BODMAS (Brackets, Orders, Division and Multiplication (working from left to right), Addition and Subtraction (working from left to right)).