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 Order of Operations ( or PEMDAS Rule )

 

The key concept behind the order of operations is to perform arithmetic operators in the "right" order or sequence. Let's take a look at how Rob and Patty tried to simplify a given numerical expression by applying the order of operations.

 

6 + 4 x 3 −10 ÷ 5 = ?

 

dog

      Rob

= 6 + 4 x 3 −10 ÷ 5

=10 x 3 − 10 ÷ 5

= 30 − 10 ÷ 5

= 20 ÷ 5

= 4  wrong

pig

   Patty

= 6 + 4 x 310 ÷ 5

= 6 + 1210 ÷ 5

= 6 + 1210 ÷ 5

= 6 + 122

=16  check

What is Rob's mistake?

  • He carelessly simplified the numerical expressions by applying arithmetic operations from left to right.
 

Patty got the correct answer because she properly applied the rules on order of operations.

  • She performed multiplication and division first before addition and subtraction.

 

history order of operations

To avoid different answers like what happened to Rob and Patty, mathematicians decided to agree on certain rules and procedures to follow in simplifying or calculating numerical expressions. That day, the Order of Operations or PEMDAS rule was created...

 

ORDER OF OPERATIONS

 

                             order of operations animated

 

The order of operations or PEMDAS is simply a rule that prioritizes the sequence of operations starting from the most important to the least important. 

Step 1: Do as much as you can to simplify everything inside the parenthesis first

Step 2: Simplify every exponential number in the numerical expression

Step 3: Multiply and divide whichever comes first, from left to right

Step 4: Add and subtract whichever comes first, from left to right   

 

  Let's take a look at some examples...

Example 1: Simplify 5 ÷ 5 + 3 − 6 x 2  using the rules for order of operations

 

= 5 ÷ 5 + 3 − 6 x 2 Divide
=1 + 3 − 6 x 2 Multiply
=1 + 3 − 12 Add
= 4 12 Subtract
= 8 check mark Final answer

 

Example 2: Simplify  3 x 7 − 11 + 15 ÷ 3 using the rules for order of operations

 

= 3 x 7 − 11 + 15 ÷3 Multiply
= 21− 11 + 15 ÷3 Divide
= 21− 11 + 5 Subtract
= 10 + 5 Add
= 15 check mark Final answer

 

The next examples will now involve parentheses. Remember that you have to simplify everything inside the parenthesis first before going forward. The rules for order of operations apply the same way inside the parenthesis.

 

Example 3: Simplify  25 − (7 − 12 ÷ 6 ) x 4  using the rules for order of operations

 

= 25 − (7 − 12 ÷ 6) x 4 Simplify first everything inside the parenthesis
= 25 − (7 − 12 ÷ 6) x 4 Divide
= 25 − (7 −2) x 4 Subtract
= 25 − (5) x 4 Multiply
= 25 −20 Subtract
= 5 check mark Final answer

 

Example 4:  Simplify  5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 using PEMDAS rule.

 

= 5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 Simplify first the expressions inside the parenthesis
= 5(4 + 3 x 2) − (9 −28 ÷ 7 ) ÷ 5 Multiply, Divide
= 5(4 + 6) − (9 − 4 ) ÷ 5 Add, Subtract
= 5(10) − (5) ÷ 5 Multiply
= 50(5) ÷ 5 Divide
= 501 Subtract
= 49 check mark Final answer

 

The final examples will involve exponents. Be careful in each step because they are so many things going on. As long as you remain focus in following the rules governing the order of operations, it shouldn't be that difficult! Here we go...

 

Example 5: Simplify 24−5(10−42 ÷ 2) + (30−33) using PEMDAS rule.
= 24− 5 (10 − 4 2 ÷ 2 ) + (30− 33 ) Simplify first the expressions inside the parenthesis, and the exponential numbers.
= 24− 5 (10 − 42 ÷ 2 ) + (30− 33 ) Exponents
= 16 − 5 (10 −16÷ 2 ) + (30−27) Divide
= 16 − 5 (10 − 8) + (30−27) Subtract
= 16 − 5 (2) + (3) Multiply
= 16 − 10 + (3) Subtract
= 6 + (3) Add
= 9 check mark Final answer

 

Example 6: Simplify (32 − 3 3  ÷ 9 x 10)− 42 ÷ 8 + 32  using PEMDAS rule.
=(32 − 3 3  ÷ 9 x 10 )− 42 ÷ 8 + 32 Simplify first the expressions inside the parenthesis, and numbers with exponents
=(32 − 3 3  ÷ 9 x 10 )42 ÷ 8 + 32 Exponents
=(32 − 27 ÷ 9 x 10 )−16÷ 8 + 9 Divide
=(32 − 3 x 10 )−16÷ 8 + 9 Multiply
=(32 −30)−16÷ 8 + 9 Subtract
=(2)5 −16÷ 8 + 9 Exponent
=32 −16÷ 8 + 9 Divide
=32 −2+ 9 Subtract
= 30+ 9 Add
= 39 check mark Final answer

 


 

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About This Topic: Order of Operations

 

Practice Problems with Answers
Worksheet 1 Worksheet 2

 

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