Order of Operations

Consider the expression given as "4 + 2 × 3", the question that naturally arises is which way we have to solve this? As there are two approaches to solve this.

First can be 4 + 2×3 = (4 + 2)×3 = 6×3 = 18 or second could be multiply first 4 + 2×3 = 4 + (2×3) = 4 + 6 = 10

The question here is which answer is the right one?

It seems the answer depends on which way we look at the problem. But we don't have this kind of flexibility in mathematics. Math won't work if we can't be sure of the answer, or if the exact same expression can be calculated so that we can arrive at two or more different answers.

To eliminate this order of calculation confusion, we have some rules of precedence, establishing the "order of operations".

In Mathematics, we do basic operations like addition, subtraction, multiplication and division. These operations are performed by a certain rule or math order. PEMDAS rule is similar to the order of operations rules which is exactly equal to the BODMAS RULE. 

The order of operations math is the rules that state the sequence in which the multiple operations in an expression should be solved. 

PEMDAS Rule

According to the PEMDAS rule, the order of operation starts with the parentheses first or the calculation which is enclosed in brackets. Then the operation is performed on exponents (degree or square roots) and later we do operations on multiplication & division and at last addition and subtraction. The PEMDAS rules state that the order in which the operations in an expression should be solved are:

Mathematical Order of Operations

It is one of the ways to remember the order of the operations is PEMDAS, where each letter stands in mathematical order.

Order of operations math are given below in the expanded form:

P: P stands for parenthesis. First solve the calculation or equation which are present in the parentheses or brackets like small brackets( ), curly brackets{ } or big brackets [ ]. Priority is given to brackets first. 

E: E stands for exponential. These expressions should be calculated first before the operations of multiplication, division, addition and subtraction. Usually, they are expressed in power or roots, like 2² or \[\sqrt{4}\]

MD: MD stands for multiplication and division. Then perform multiplication or division from left to right order, whichever comes first in the equation.

AS: AS stands for Addition and Subtraction. At last, perform addition or subtraction whichever comes first moving from left to right order.

There is only an abbreviation difference between them.

PEMDAS Vs BODMAS

PEMDAS term is mainly used in the US but in India and the UK, we call it BODMAS. As there is no difference between them, the order of operations rules for brackets, orders, addition, subtraction, multiplication and division is the same for both the rules. The order of operation has given different names in different countries but the meaning the same for all.

Order of Operations Examples

We have to follow the rules of the order of operations to solve expressions so that everyone arrives at the same answer. 

Here we have provided an example of how we can get different answers if the correct order of operations is not followed.

Expression solved from left to right

6 x 3 + 4 x (9 ÷ 3)

= (6 x 3) + 4 x (9 ÷ 3)

= 18 + 4 x (9 ÷ 3)

= 22 x (9 ÷ 3)

= 198 ÷ 3

= 66

The answer we got here is not correct because here PEMDAS rule is not followed.

Expression solved using order of operations (PEMDAS)

6 x 3 + 4 x (9 ÷ 3)

= 6 x 3 + 4 x (9 ÷ 3)

= 6 x 3 + 4 x 3

= 18 + 4 x 3

= 18 + 12

= 30

This answer is correct. This way of solving is the correct method to solve.

Solved Examples:

Solve the given below problems by applying the PEMDAS rule

1. Solve 58 ÷ (4 x 5) + 3²

Sol: We will apply here the PEMDAS rule. According to that rule, we have to perform the operation which is in the parentheses so,

58 ÷ 20 + 3²

Now perform the exponent operation

= 58 ÷ 20 + 9

Now perform the division operation 

2.9+9

At last perform addition operation

=11.9

Hence the value of 58 ÷ (4 x 5) + 3² is 11.9

2. Simplify the Expression \[\sqrt{1 + 3}\] + 8

Sol: According to the BODMAS rule, first we need to calculate the exponent value

\[\sqrt{1 + 3}\] + 8

= \[\sqrt{4}\] + 8

Now add both the value

= 2+8=10

Conclusion: 

The order of operations gives us a consistent sequence for computation. Without the order of operations, we could come up with different answers to the same computation problem.

When we have to evaluate arithmetic expressions, the order of operations that should be followed are:

• First, simplify all operations inside parentheses.

• Then perform all multiplications and divisions, moving from left to right.

• After that perform all additions and subtractions, moving from left to right.

If the question contains a fraction bar, then we have to perform all calculations above and below the fraction bar before dividing the numerator by the denominator.