Homework Help: the Order of Operations in Mathematics

Mathematics, like other sciences, has an order on how to do things. When one is solving an equation (or a mathematical statement with an "equals" sign in it), there is a certain required order on how to do it. As a math teacher, I saw many students that were able to solve parts of their equations but because they failed to follow the order of operations, they had the wrong answer.

For example, in the equation

2 + 3 x 5

If you work from left to right, you would multiply 2 +3 and get 5, and then multiply that times 5 and get 25. However, if you multiplied 3 x 5 first, you'd have 15 and then added to 2, would be 17. Two DIFFERENT answers!

That is why mathematicians developed a mathematical order of operations. The order of operations states that we do equations in this order:

1.) All operations within PARENTHESES.

2.) All operations with EXPONENTS.

3.) MULTIPLICATION and DIVISION, regardless of which of those two is first, working from left to right.

4.) ADDITION and SUBTRACTION, regardless of which of those two is first, working from left to right.

To remember this, many teachers use the sentence, "Please Excuse My Dear Aunt Sally" and using the first letter of each word, we are reminded of Parentheses (Please), Exponents (Excuse), Multiplication and Division (My Dear) and then Addition and Subtraction (Aunt Sally). The sentence makes it easier to remember the order.

Now, let's look at some equations solved following the Order of Operations.

Note, 2^3 means two to the third power, or two with an exponent of 3. I will use this notation since my keyboard has no way to make the little number of an exponent.)

Note: The / sign will mean divided by, so 12 / 3 means twelve divided by three.

In the equation:

(3 x 4) + (12 / 4) -2 + 1

First we would solve what is inside our PARENTHESES.

(3 x 4) + (12 / 4) -2 + 1 = 12 + 3 - 2 + 1

Now we would solve the addition and subtraction from left to right.

(3 x 4) + (12 / 4) -2 + 1 = 12 + 3 - 2 + 1
= 15 - 2 + 1
= 13 +1
= 14

Now, let's look at another equation that includes an exponent.

3^2 - 4 -3 + (12 / 6)

We would first solve our exponent, and know that 3 ^ 2 means 3 squared, or 3 to the second power, or 3 x 3)

3^2 - 4 -3 + (12 / 6) = 9 - 4 - 3 + (12 / 6)

Now we solve what is inside our parentheses.

3^2 - 4 -3 + (12 / 6) = 9 -4 -3 + (12 / 6)

= 9 - 4 - 3 + 2

Now we solve all addition and subtraction moving from left to right.

3^2 - 4 -3 + (12 / 6) = 9 - 4 - 3 + (12 / 6)

= 9 - 4 - 3 + 2

= 5 - 3 + 2

= 2 + 2

= 4

The Order of Operations is always followed in mathematics, whether you are just solving basic equations or algebraic equations.

Now you try one.

(14 / 2) - 2^ 2 + 4

Get your answer before reading further.

Did you get the answer 7?

Let's look at the solution:

(14 / 2) - 2^ 2 + 4 = (14 / 2) - 4 + 4

= 7 -4 + 4

= 3 + 4

= 7

Keep practicing until you automatically follow the Order of Operations. A professor of mine in college went to a large department store to find a summer job during her summer break. Even though she had a master's degree in Mathematics, she wanted a job wrapping presents because it was something she really enjoyed. She filled out an application and they told her she would have to take a basic skills test. She turned in her answers to some basic math problems, and was told she failed the test! When she looked at their answer key, she realized they had solved every equation from left to right without following the Order of Operations. She pointed out the mistake and was hired immediately. Needless to say, they changed their answer key!