Introduction to the order of operations: Definition, Rules, and Examples

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Introduction to the order of operations: Definition, Rules, and Examples

The order of operations is frequently used in mathematics to solve math expressions in a correct way. It is a technique that is used to find the correct sequence of math symbols. This term is used worldwide to get a similar result.

The order of operations can be used in various types of algebra. In this post, we will learn definite, rules, and examples.

What is the order of operations?

In mathematics, the order that is used to simplify the expressions in an order of arithmetic symbols is known as the order of operations. In simple words, the order of operations is an order in which you have to add, multiply, divide, open brackets & exponents, or divide to solve a math expression problem.

Rules of an order of operations

There are various rules of the order of operations.

  1. PEMDAS Rule
  2. BODMAS Rule

The PEMDAS rule is widely used in western countries while the BODMAS rule is frequently used in Asian countries. These rules are helpful in solving the math expression in a correct way. Let us briefly discuss these rules.

(i) PEMDAS Rule

The first rule of the order of operation is the PEMDAS rule. The PEMDAS is an abbreviation of all the mathematical symbols.Such as

  • P stands for Parentheses “()”
  • E stands for exponent “^”
  • M stands for multiplication “x”
  • D stands for division “/”
  • A stands for Addition “+”
  • S stands for subtraction “-”

PEMDAS is a generally used rule of the order of operations in which firstly, you have to solve the parenthesis of the math expression. After that calculate the exponential terms present in the math expression.

Then calculate the multiplication and division terms from left to right. In the end, calculate the addition and subtraction terms from left to right. If there is more than one parenthesis, exponents, or other symbols, then solve the left-most term first.

Use the PEMDAS calculator to get the result of math expression according to the above rule.

(ii) BODMAS Rule

The other rule of the order of operation is the BODMAS rule. The PEMDAS is an abbreviation of all the mathematical symbols. Such as

  • B stands for Brackets (brackets can be Parentheses “()”, curly brackets “{}”, or square brackets “[]”)
  • O stands for Order (exponent “^”)
  • D stands for division “/”
  • M stands for multiplication “x”
  • A stands for Addition “+”
  • S stands for subtraction “-”

In the BODMAS rule, first of all, calculate the bracket terms. After that calculate the order or exponential terms of the math expression. After solving the exponential terms, calculate the division and multiplication terms from left to right.

In the end, calculate the addition and subtraction terms from left to right. If there is more than one bracket, exponents, or other symbols, then solve the left-most term first. In Asian countries, this rule is frequently used.

How to solve the problems of the order of operations?

By using the PEMDAS or BODMAS rule, the problems of the order of operations can be solved easily. To solve the problems in the correct way, follow the below steps.

  1. Firstly, calculate the parentheses “()” or brackets present in the math expression.
  2. After that calculate the order or exponents terms.
  3. Calculate the multiplication and division terms from left to right.
  4. Calculate the addition and subtraction terms from left to right.
  5. Below are a few examples to understand the calculations accurately.

    Example 1: By PEMDAS rule

    Solve 8 * 5 + 62 – 32/2 + (12 - 8) – 6(1 * 2) + 7 by using the PEMDAS rule.

    Solution

    Step I: Firstly, write the mathematical expression.
    8 * 5 + 62 – 32/2 + (12 - 8) – 6(1 * 2) +7

    Step II: Calculate the parentheses.
    8 * 5 + 62 – 32/2 + (4) – 6(1 * 2) + 7
    8 * 5 + 62 – 32/2 + 4 – 6(2) + 7
    8 * 5 + 62 – 32/2 + 4 – 12 + 7

    Step III: Now solve the exponent terms.
    8 * 5 + 6 x 6 – 32/2 + 4 – 12 + 7
    8 * 5 + 36 – 3 x 3/2 + 4 – 12 + 7
    8 * 5 + 36 – 9/2 + 4 – 12 + 7

    Step IV: Solve the multiplication and the division terms from left to right.
    40 + 36 – 9/2 + 4 – 12 + 7
    40 + 36 – 4.5 + 4 – 12 + 7

    Step V: Now solve the addition and subtraction terms from left to right.
    76 – 4.5 + 4 – 12 + 7
    71.5 + 4 – 12 + 7
    75.5 – 12 + 7
    63.5 + 7
    70.5

    Step VI: Write the given math expression with the result.
    8 * 5 + 62 – 32/2 + (12 - 8) – 6(1 * 2) +7 = 70.5

    The order of operations calculator can also be used to avoid such a large number of steps in finding the result of math expression.

    Example 2: By the BODMAS rule

    Solve 20 + 3 + 23 – 9/3 + (22 * 12 + 8) – 6(3 * 2) + 10 by using the BODMAS rule.

    Solution

    Step 1: First of all, write the mathematical expression.
    20 + 3 + 23 – 9/3 + (22 * 12 + 8) – 6(3 * 2) + 10

    Step 2: Solve the brackets first.
    20 + 3 + 23 – 9/3 + (4 * 12 + 8) – 6(3 * 2) + 10
    20 + 3 + 23 – 9/3 + (36 + 8) – 6(3 * 2) + 10
    20 + 3 + 23 – 9/3 + (44) – 6(6) + 10
    20 + 3 + 23 – 9/3 + 44 – 36 + 10

    Step 3: Now solve the order or power terms.
    20 + 3 + 2 x 2 x 2 – 9/3 + 44 – 36 + 10
    20 + 3 + 8 – 9/3 + 44 – 36 + 10

    Step 4: Solve the division and the multiplication terms from left to right.
    20 + 3 + 8 – 3 + 44 – 36 + 10

    Step 5: Now solve the addition and subtraction terms from left to right.
    23 + 8 – 3 + 44 – 36 + 10
    31 – 3 + 44 – 36 + 10
    28 + 44 – 36 + 10
    72 – 36 + 10
    36 + 10
    46

    Step 6: Write the given math expression with the result.
    20 + 3 + 23 – 9/3 + (22 * 12 + 8) – 6(3 * 2) + 10 = 46

    Conclusion

    In this post, we have covered all the basics of the order of operations. Now you can solve any kind of math expression either by using the PEMDAS rule or BODMAS rule easily by following the above post.

                 





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