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Order of Operations

In mathematics, an expression of symbols intended to represent a numerical value must follow commonly accepted and unambiguous rules. We have to do certain operations before doing other operations, like multiplication should be done before addition.

For example, the rule for evaluate 2 + 3 × 4 in mathematics and in most computer languages is to firslty do the multiplication, so the correct answer is 14. Which have their own rules, may be used to avoid confusion, the convention needed to clarify which operator should be applied first is known as an order of operations.

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What is Order of Operations

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Here is math order of operations elaborated for your better understanding. In long math problems with +, - , x , % , (), and exponents in them, you have to know what and which to do first. Without follow the same rules, you may get different answers. You can easily remember the silly sentence, Please Excuse My Dear Aunt Sally, you can memorize the order of operations, and you must follow.

"P" denotes Parentheses.

"E" represents Exponents

"M" stands for Multiply

"D" represents Division.

"A" means Addition.

"S" refers to Subtract.

When an expression with more than one operator comes then we have to follow the above order to obtain the correct answer. It is to be noted that the division and multiplication operations have same order, which means that any of them can be done first. Similarly, the order of subtraction and addition are the same, i.e. any one of them can be performed first. The answer will be the same.

The important thing is that first parentheses (if any) has to be simplified, then exponents, afterwards multiplication or division and lastly addition or subtraction.

Now we will see the examples of how to solve different expressions using the concept of order of operators.

Order of Operations Definition

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Basically, Order of operation for an arithmetic expression provides is very basic knowledge for the order of calculating different arithmetic operations. The order of operation actually defines few rules explaining in which order to solve an arithmetic expression. These rules are denoted in short by PEMDAS. Here, each letter represents an arithmetic operation. PEMDAS denotes the order of performing operations. Arithmetic operations involve parenthesis, exponents, addition(+), subtraction(-), multiplication(*) and division(÷).

How to do Order of Operations

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Below you could see the steps for how to do order of operations

Step 1: First of all, perform the operations inside parenthesis, if any.

Step 2: Then, perform the operations of exponents if they are available.

Step 3: Now multiplication and division operations are done starting from left side to right.

Step 4: Then at the end, addition and subtraction operations are completed from left to right.

Order of Operations Problems

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Below are order of operations examples:

Solved Examples

Question 1: Solve $\frac{14}{ 7}$(7- 6) + 2 - 2 using order of operations.
Solution:
 
Eliminate parentheses. 7- 6 =1

$\frac{14}{7}$x1+ 2 -2

Eliminate exponents. We do not have any exponents.

$\frac{14}{7}$x1+ 2 -2

Eliminate Multiply and divide after that in order from left to right

$\frac{14}{7}$=2 then 2x1+ 2 -2

2 x 1 = 2 then 2+2-2

Last, we add and subtract in order from left to right.

2 + 2 =4 then 4 - 2

4 - 2 = 2

Answer is 2

 

Question 2: Solve $\frac{12}{4}$ + 42 x 3 - 1 using order of operations.
Solution:
 
Eliminate parentheses. Here we don't have parentheses

$\frac{12}{4}$ + 42 x 3 - 1

Eliminate exponents. 42 = 16

$\frac{12}{4}$ + 16 x 3 - 1

Eliminate Multiply and divide after that in order from left to right

$\frac{12}{4}$=3 then 3 + 16 x 3 -1

16 x 3 = 48 then 3 + 48 -1

Last, we add and subtract in order from left to right.

3 + 48 = 51 then 51-1

51-1 = 50

Answer is 50

 

Question 3: Solve $\frac{(6 + 5)}{11}$ +7- 23 × 3 +1 using order of operations.
Solution:
 
Eliminate parentheses. 6 + 5 = 11

$\frac{11}{11}$ + 7- 23 × 3 +1

Eliminate exponents. 23 = 8

$\frac{11}{11}$ +7- 8 × 3 +1

Eliminate Multiply and divide after that in order from left to right

$\frac{11}{11}$=1 then 1 + 7- 8 × 3 +1

8 x 3 = 24 then 1+7-24+1

Last, we add and subtract in order from left to right.

1+7=8 then 8-24+1

8-24= -16 then -16+1

-16+1 = -15

Answer is -15

 

Order of Operations with Exponents

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The order of operation with exponents can be done in the following way.
When there is exponentiation in the arithmetic operation, one is required to solve it after opening the parenthesis, as the rule of PEMDAS says. For example :
13 - 3$^{3}$ + 2 (5 + 1)
We shall first solve parenthesis.
= 13 - 3$^{3}$ + 2 (6)
= 13 - 3$^{3}$ + 12
Now, solve exponent
= 13 - 27 + 12
= 25 - 27 = -2

In case, when exponent is inside the parenthesis, we follow PEMDAS inside it too. For example :
5 + 3 - 4(2$^{2}$ + 3 - 7)
Here, inside parenthesis, first we solve exponent
= 5 + 3 - 4(4 + 3 - 7)
= 5 + 3 - 4(0)
= 5 + 3 = 8

Solved Examples

Question 1: Solve the arithmetic expression 32 x 43 using order of operations.
Solution:
There is no parenthesis so we go to next operation.

Exponents are here. So we have to calculate each exponent operation.

32 = 3 x 3 = 9

43 = 4 x 4 x 4 = 64

So the expression is reduced to 9 x 64 .

Then we go to next operation.

Multiplication of these two numbers are 9 x 64 = 576.

The answer is 576 . 


Question 2: Solve the arithmetic expression: 27 - 256 ÷ 43 using order of operations.
Solution:
There is no parenthesis so we go to next operation.

Exponent is going to calculate.

43 = 4 x 4 x 4 = 64

So the expression is reduced to 27 - 256 ÷ 64

Now we are going to solve division operation.

256 ÷ 64 = 4

So the expression is reduced to 27 - 4.

We are going to solve subtraction operation.

27 - 4 = 23

The answer is 23. 


Order of Operations Word Problems

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Below you could see order of operations word problem

Solved Examples

Question 1: Steven charges 20 Dollar per square foot to lay a floor. If a square-shaped hallway is 9 feet along one side, and the customer has a coupon for 30 Dollar off the total, then how much will the floor cost?
Solution:
If one side of hallway is 9 feet then area of the hallway is (9 feet)2 .

The arithmetic expression is

  $20 x 92 - $30

using order of operations,

We are going to solve exponent operation.

 92 = 9 x 9 = 81

So the expression is reduced to Dollar 20 x 81 -  Dollar 30

We are going to solve multiplication operation.

20 x 81 = 1620

So the expression is reduced to Dollar 1620 - $30

We are going to solve subtraction operation.

1620 - 30 = $ 1590

The cost of floor is $ 1590 . 


Question 2: Five years ago, Sam’s age was half of the age he will be in 10 years. How old is he now?
Solution:
Step 1: Let x be Sam’s age now. Look at the problem and put the significant expressions above it.

Five years ago x-5,

Sam’s age was half of the age $(\frac{1}{2})$ he will be in 10 years.

Step 2: copy out the equation in above word problem.

Note: we need to apply order of operations.

x - 5 = $\frac{1}{2}$ (x + 10)

x - 5 = $\frac{1}{2}$ x + 5

Isolate variable x

x - $\frac{1}{2}$ x = 5 + 5

$\frac{1}{2}$ x = 10

x = 20

Answer: Sam is now 20 years old.


Order of Operations Practice Problems

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Below you could see order of operations practice problems

Practice Problems

Question 1: Solve the arithmetic expression: 9 x (5 + 3)2 - 144 using order of operations.
Question 2: Solve the arithmetic expression: 5 × (4 × 6) ÷ 6 + 52 - 3 using order of operations.
Question 3: Solve the arithmetic expression: 6 × (5 × 7) ÷ 7 + 62 - 4 using order of operations.
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