# Order of Operations : A Step-by-Step Approach to Building Blocks of Algebra

## Introduction: Define the Order of Operations

The steps or approach we use to solve a problem, such as addition, subtraction, multiplication, and division, are referred to as the order of operations. Mathematical equations are made simpler by the order of operations.

The order of operations is like grammatical rules in the language of mathematics. It shows how to read an equation to get the true intent. We might each receive a different response if this order weren’t followed.

## PEMDAS Vs BODMAS

 S.No. PEMDAS BODMAS PEMDAS stand for,P- ParenthesesE- ExponentsM- MultiplicationD- DivisionA- AdditionS- Subtraction BODMAS stand for,B- BracketsO- Order of Powers or RootsD- DivisionM- MultiplicationA- AdditionS- Subtraction The word PEMDAS is primarily used in the United States. BODMAS is the term used in India and the United Kingdom.

## Understanding the Rules of the Order of Operations

### Rules of the Order of Operations are:

1. Rule 1: Simplify any and all parentheses or brackets. Grouping operations are resolved from the inside out. The round (), curly {}, and box [] brackets must first be resolved before attempting to solve the parenthesis. The parenthesis’ order of operations must be adhered to.
2. Rule 2: Simplify all the Parenthesis and Exponents xn. After resolving the numbers included in parenthesis, locate and resolve any phrase that contains exponents.
3. Rule 3: Simplify all Multiplication  (×) and Division (÷), when simplifying move from left to right.
4. Fourth, Subtraction (-) and Addition (+) (-), when simplifying move from left to right.

## Order of Operations Examples

1. Solve 4 – 5 ÷ (8 – 3) × 2 + 5 using PEMDAS

Solution

Step 1 – Parentheses:

4 – 5 ÷ (8 – 3) × 2 + 5

= 4 – 5 ÷ 5 × 2 + 5

Step 2 – Division:

4 – 5 ÷ 5 × 2 + 5

= 4 – 1 × 2 + 5

Step 3 – Multiplication:

4 – 1 × 2 + 5

= 4 – 2 + 5

Step 4 – Subtraction:

4 – 2 + 5

= 2 + 5

2 + 5

2. Solve 3 x (2 + 4) + 52 using BODMAS

Solution:

Step 1 – Brackets:

3 x (2 + 4) + 5 2

= 3 x (6) + 5 2

Step 2 – Order of Powers or Roots:

3 x (6) + 5 insert raise to the power 2

= 3 x (6) + 25

Step 3 – Division: no division sign here, no division required

Step 4 – Multiplication:

3 x (6) + 25

= 3 x 6 + 25

= 18 + 25

18 + 25

= 43

Step 6 – Subtraction: no subtraction sign here, no subtraction is required

## Common Mistakes When Applying the Order of Operations and How to Avoid Them

### 1. Working Only from Left to Right

No matter what, students always work from left to right. For the past four to five years, students have been pushing their minds to think from left to right. It’s crucial to explicitly teach kids the guidelines to follow in order to overcome this habit.

### 2. Always Adding or Multiplying

The desire to perform one procedure before another is the following error. When performing addition, subtraction, or multiplication/division, students now prefer to work from left to right.Teaching kids the terms “multiplication division left to right” and “addition subtraction left to right” is essential.

### 3. Utilizing Numbers to Double

Sometimes students will carry over the number they just used into the subsequent section. Students may combine the numbers to form a single lengthy number. Students must mark out any numbers and operations they correctly solve in order to solve this issue.

### 4. Not Adhering to the Parentheses’ Order of Operations

When there are numerous operations inside those parentheses, students frequently default to solving the problem from left to right because we teach them to solve the parts of the problem enclosed in parenthesis first. It’s crucial to instill in them the knowledge that inside parenthesis, the rules for addition, subtraction, and multiplication still hold true.

### 5. Multiplying the Exponent by the Base

Students multiply the base number by the exponent in this typical order of operations error instead of multiplying the number by itself. It’s crucial to comprehend the fundamentals of exponent problem solving.

### 6. Not Correctly Rewriting all of the Problem’s Components

Students will properly solve a section of the problem before mistakenly copying it to the subsequent layer. Students must be very cautious in writing all the numbers as well as symbols and eliminating numbers and symbols as you solve them.

## Tips & Tricks for Memorizing & Understanding the Order Of Operations

1. Making an acronym, which is done by combining letters to form a word, is one strategy. For instance, the acronym BODMAS is used to aid math students in recalling the order of operations they ought to employ while solving a problem. The letters in the acronym BODMAS stand for brackets, order of powers or roots, division, multiplication, addition, and subtraction. Thus, if a student is solving a problem, knowing the acronym BODMAS will assist the learner to remember that multiplication comes first in the equation before the addition is done.

2. Making an acrostic, which is a sentence where each word’s first letter stands in for a concept you need to remember, is another method. “Please Excuse My Dear Aunt Sally” is an acronym for the word PEMDAS.  When figuring out how to solve arithmetic issues, this can be helpful.

3. Say the details out loud. According to a study at the University of Waterloo, reading information aloud to ourselves increases our memory retention compared to reading information silently. This impact can be attributed to the fact that words spoken aloud stand out to our brains more than words said silently.

4. Before attempting to memorize the material, completely comprehend it. When using rote learning, content is challenging to memorize.

5. Our ability to retain knowledge is greatly improved when we write it out. By doing this, we are compelled to assess and organize the fresh data. The new information is strengthened in our memory through this process. It wouldn’t be feasible to write out everything, therefore, just the essential facts, formulas, definitions, etc. that you must memorize should be written down.

## Practice Problems on Order of Operations

Solution:

= (18 ÷ 3) × (-2)

= 6 × (-2)

Solution:

= (17 – 6 ÷ 2) + 4 × 3

= (17 – 3) + 4 × 3

= (14) + 4 × 3

= (14) + 12

= 14 + 12

Solution:

= -1 × [(3 – 4 × 7) ÷ 5] – 2 × 24 ÷ 6

= -1 × [(3 – 28) ÷ 5] – 2 × 24 ÷ 6

= -1 × [(- 25) ÷ 5] – 2 × 24 ÷ 6

= -1 × [-5] – 2 × 24 ÷ 6

= -1 × -5 – 2 × 4

= -1 × -5 – 8

= 5 – 8

#### FAQ's

There are two exceptions:

1. Perform the division first and the multiplication second if the phrase does not contain an addition or subtraction sign.
2. If the statement’s addition and subtraction are followed by no signs of multiplication or division, perform the subtraction first.

The word PEMDAS is primarily used in the US, although we refer to it as BODMAS in India and the UK. However, there is no distinction between them. Both rules follow the same order of operations for brackets, ordering, addition, subtraction, multiplication, and division.

The highest precedence is given to parentheses, followed by exponentiation. Multiplication and division have the same precedence, which is higher than the precedences of addition and subtraction.

## Conclusion

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