How to Solve Problems using The Order of Operations

In the Language of Algebra, which is Signed Numbers (Positive and Negative Numbers) there are four Basic Operations: (1) Addition, (2) Subtraction, (3) Multiplication, (4) Division. The Order in which these Operations are performed is VERY IMPORTANT!!! For Example...1 + 2 x 3 can be performed, or solved, in two different ways. The First way is to Add 1 with 2 and then Multiply by 3. The answer will be 9. The Second way is to first Multiply 2 x 3 and then Add that Product to 1. The answer will be 7. Which answer is the correct answer? Even if you should use a Calculator to solve this problem, one Calculator could give you the answer 9, and the other Calculator could give you the answer 7. The difference between the two Calculators is that one will have the Order of Operations programmed into it, the other will not. This Article will show you how the Order of Operations should be done.

Things You'll Need

• Pencil
• Paper

Instructions

1. Let us Solve the following problem:
   1 + 2 - 3 x 4 ÷ 5
   What do you think is the correct answer to this problem?

2. The Order of Operations states that if there is a Problem that has all of the following Operations; Parentheses, Exponents, Multiplication, Division, Addition and Subtraction, then the Operations must be performed in the order in which they are stated.

3. There are exceptions to this rule. Multiplication and Division should be performed in the order that they are written from left to right. That is, if we have the following problems...
   20 ÷ 10 x 5, Division is done First and Multiplication Second, so the Solution to this problem is,...20 ÷ 10 = 2. Then 2 x 5 = 10. Similarly, in the problem,
   20 x 10 ÷ 5 ,Multiplication is done First and Division second, so the Solution to this problem is,... 20 x 10 = 200. Then, 200 ÷ 5 = 40.
   As you read from left to right, if the Operation of Division comes before Multiplication, then Division is done first and Multiplication after. And if the Operation of Multiplication comes first, then Multiplication is done first and Division after.

4. Similarly, Addition and Subtraction should be performed in the Order that they are written from left to right. That is, if we are given the following problems...
   10 - 5 + 2 and 10 + 5 - 2 , in order to solve them correctly the following steps need to be done.
   As you read from left to right, if the Operation of Subtraction comes before Addition, then Subtraction is done first and Addition after. And if the Operation of Addition comes first, then Addition is done first and Subtraction after.

5. So the solutions to the problems, 10 - 5 + 2 and 10 + 5 - 2 , are...
   7 and 13 respectively, where
   10 - 5 = 5 and 5 + 2 = 7, and 10 + 5 = 15 and 15 - 2 = 13

6. 

   Now we solve the problem in Step 1. Please click on the Image on the left to get a better understanding.

7. 

   If the problem was written with parentheses in the following way
   (1 + 2 -3) x 4 ÷ 5
   What do you think is the correct answer to this problem?
   Please click on the Image on the left to get a better understanding.

8. If the problem was written with parentheses and exponents in the follwing way
   (1 + 2 -3) x 4 ÷ 5^2
   We First perform the operations that are inside the parentheses that is
   First we Add 1 + 2 = 3, so the problem is now (3 - 3) x 4 ÷ 5^2
   we now perform the Subtraction 3 - 3 = 0, so the problem is now
   0 x 4 ÷ 5^2 . We now perform the operation of Exponents, that is
   5^2 = 25, so the problem is now,... 0 x 4 ÷ 25 . Finally, we Multiply
   0 x 4 = 0, so the problem is now,... 0 ÷ 25 = 0 .
   So the Answer to the problem (1 + 2 - 3) x 4 ÷ 5^2 = 0

Tips & Warnings

• The Acronym P E M D A S stands for "Please Excuse My Dear Aunt Sally" which stands for the Order in which the Operations should be done. That is,

• Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.

• Don't forget the exception to the rule between Multiplication and Division and Addition and Subtraction, whichever comes first as you read from right to left is done first.

• ENJOY!!!