Order of Operations


Arithmetic Expression

Given an arithmetic expression without parentheses  we perform operations moving from left to right in the following order:

1.   Multiply or Divide
2.   Add or Subtract

Example:

Convert the following expression into a number.

 2×4-6+8÷2

Solution:

  1. Perform multiplication and division.

 

    8-6+4
  1. Perform addition and subtraction
 2+


 6

Click on the following interactive demo to create more examples.






Expressions with non-nested parentheses

If an expression has non-nested parentheses, this is, there is just one set of parentheses. The  following steps must be used.

1.
Calculate the value of the expression within the parentheses.
2.
Multiply and Divide.
3.
Add and Subtract.

 

Example:

Convert the following expression into a number.

  32-(5-68+15)-3×(5-12+40÷8)

Solution:

1.
Calculate the value of the expression within the parentheses.

  32-(5-68+15)-3×(5-12+40÷8)


 32-(-48)-3×(5-12+5)
 
 32-(-48)-3×(-2)

 

2.
Multiply and Divide.
 32+48+6

 

3.
Add and Subtract.
 86

 


Click on the following interactive demo to create more examples.






Expressions with Nested Parentheses

If a set of parentheses lies within another set of parentheses, we eliminate the interior parentheses first we repeat the same process for the expression within parentheses until we can convert the expression into a number.

Example 1.

Convert the following expression into a number.

 12-4×(5-(11-6×2)-(-7))


Solution:

1.
Obtain the value of the expression within interior parentheses.
 12 - 4×(5 - (11 - 12) + 7))


  12 - 4×(5 - ( - 1) + 7))


  12 - 4×(5 + 1 + 7)


2.
Calculate the value of the expression within exterior parentheses.
  12 - 4×(5 + 1 + 7)


  12 - 4×13
 

 

Example 2.

Convert the following expression into a number.

-14 - (5×(7-6 ÷ 3+(-6-5+3) ) -3×(5 -(11×2-17) - (15-7)))

Solution:

1.
Calculate the value of the expression within internal parentheses.
-14 -(5×(7-6 ÷ 3+(-6-5+3) ) -3×(5-(11×2-17) - (15-7)))

-14 - (5×(7-6 ÷ 3+(-8) ) -3×(5 -(22-17) - (8)))
-14 - (5×(7-2-8 ) -3×(5 -5 - 8))

 

2.
Calculate the value of the expression within the next set of parentheses.
-14 - (5×(7-2-8 ) -3×(5 -5 - 8))
 
-14 - (5×(-3 ) -3×(- 8))


-14 - (-15+24)

 

3.
Calculate the value of the expression within the next set of parentheses.
-14 - (-15+24)


-14 - (9)


-23


Sometimes there are exercises where we found square brackets [ ] or round bracket { } instead of parentheses ( ), but the way to solve these exercises is the same just replace the brackets for parenthesis .