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What Is It?

The Order of Operations

When children initially learn addition, subtraction, multiplication, and division, they begin by performing operations on two numbers. But what happens when an expression requires multiple operations? Over time, mathematicians have developed a set of rules called the order of operations to determine which operation to do first. The rules are:

  1. Multiply and divide from left to right.
  2. Add and subtract from left to right.

When simplifying an expression such as
12 divided by 4 + 5 times 3 – 6, you must first compute 12 divided by 4 since order of operations requires you to begin on the left side of the expression and do multiplication or division (whichever comes first) before addition or subtraction. After 12 divided by 4 is calculated, you continue moving from left to right while searching for multiplication or division. So, the netimest step is to compute 5 times 3. Now that all multiplication and division has been completed, you would continue by adding or subtracting (whichever comes first) from left to right. The steps are shown below.

12divided by 4 + 5 times 3 – 6
3 + 5 times 3 – 6 (since 12 divided by 4 = 3)
3 + 15 – 6 (since 5 times 3 = 15)
18 – 6 (since 3 + 15 = 18)
12 (since 18 – 6 = 12)

Grouping symbols such as parentheses ( ), brackets [ ], or braces { }, allow you to determine the order by which particular operations are performed. Look at this expression:

6 + 4 times 7 – 3
6 + 28 – 3 (since 4 times 7 = 28)
34 – 3 (since 6 + 28 = 34)
31 (since 34 – 3 = 31)

What happens if you insert parentheses into the expression? Parentheses allow you to determine what operation is performed first. The order of operations states that operations inside parentheses are performed before operations outside the parentheses. What happens if we put parentheses around 6 + 4?

(6 + 4) times 7 – 3
10 times 7 – 3 (6 + 4 = 10, which is done first because it's inside the parentheses)
70 – 3 (normal order of operations resumes, and 10 times 7 = 70)
67 (since 70 – 3 = 67)

Notice that the answer is significantly different than before. What if we put parentheses around 7 – 3?
6 + 4 times (7 – 3)
6 + 4 times 4 (This time, 7 – 3 is in parentheses, so we do that first.)
6 + 16 (Once there are no parentheses left, we proceed with multiplication before addition.)
22 (since 6 + 16 = 22)

This set of parentheses yields yet another answer. So, when parentheses are involved, the rules for order of operations are:

  1. Do operations in parentheses.
  2. Multiply and divide from left to right.
  3. Add and subtract from left to right.
 

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