Order of Operations

Updated July 31, 2023 | Infoplease Staff

When you have a math problem that involves more than one operation—for example, addition and subtraction, or subtraction and multiplication—which do you do first?

Example #1: 6 – 3 x 2 = ?

• Do you do the subtraction first (6 – 3 = 3) and then the multiplication (3 x 2 = 6)?
• Or do you start with the multiplication (3 x 2 = 6) and then subtract (6 – 6 = 0)?

PEMDAS

In cases like these, we follow the order of operations. The order in which operations should be done is abbreviated as PEMDAS:

1. Parentheses
2. Exponents
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

(One way to memorize this is to think of the phrase Please Excuse My Dear Aunt Sally.)

• In the above example, we're dealing with multiplication and subtraction. Multiplication comes a step before Subtraction, so first we multiply 3 x 2, and then subtract the sum from 6, leaving 0.

Example #2: 30 ÷ 5 x 2 + 1 = ?

• There are no Parentheses.
• There are no Exponents.
• We start with the Multiplication and Division, working from left to right.
  
  NOTE: Even though Multiplication comes before Division in PEMDAS, the two are done in the same step, from left to right. Addition and Subtraction are also done in the same step.
• 30 ÷ 5 = 6, leaving us with 6 x 2 + 1 = ?
• 6 x 2 = 12, leaving us with 12 + 1 = ?
• We then do the Addition: 12 + 1 = 13

Note that if we'd done the multiplication before the division, we'd have ended up with the wrong answer:

• 5 x 2 = 10, leaving 30 ÷ 10 + 1 = ?
• 30 ÷ 10 = 3, leaving 3 + 1 = ?
• 3 + 1 = 4 (off by 9!)

One last example for advanced students, using all six operations:

Example #3: 5 + (4 – 2)² x 3 ÷ 6 – 1 = ?

• Start with the Parentheses: 4 – 2 = 2. (Even though subtraction is usually done in the last step, because it's in parentheses, we do this first.) That leaves 5 + 2² x 3 ÷ 6 – 1 = ?
• Then Exponents: 2² = 4. We now have 5 + 4 x 3 ÷ 6 – 1= ?
• Then Multiplication and Division, starting from the left: 4 x 3 = 12, leaving us with 5 + 12 ÷ 6 – 1 = ?
• Then moving to the right: 12 ÷ 6 = 2, making the problem 5 + 2 – 1 = ?
• Then Addition and Subtraction, starting from the left: 5 + 2 = 7, leaving 7 – 1 = ?
• Finally, moving to the right: 7 – 1 = 6

(For more practice, try our Operation Order game!)