GED Mathematical Reasoning: Order Of Operations

When a math problem contains more than one operation, the answer may depend on the order in which each operation is performed.

Consider the example: 2 + 3 \times 5

  • This statement contains two operations: addition and multiplication.
  • If we add first: 2 + 3 \cdot 5 = 5 \cdot 5 = 25
  • If we multiply first: 2 + 3 \cdot 5 = 2 + 15 = 17
  • 25 and 17 are very different answers, so which one is correct?

Realizing that this could pose a problem and be very confusing, mathematicians have agreed upon the order in which math problems will be worked when there is more than one operation. This agreed-upon order is called: The Order of Operations.

Here is the order in which we are to solve this problem and any problem that contains more than one operation.

  • First, we need to work any grouping symbols that enclose operations. Grouping symbols usually come in the form of parenthesis or brackets. The division bar can also be thought of as a grouping symbol in that we will simplify the numerator and denominator separately before dividing. If more than one set of grouping symbols is present, work from the inside, out.
  • Second, exponents and roots should be simplified.
  • Third, multiplication and division happen in the same step working from left to right, meaning – whichever comes first.
  • Finally, we will work any addition and subtraction from left to right, meaning – whichever comes first.

There are two popular tools used to commit the order of operations to memory.

  • One is the sentence: Please Excuse My Dear Aunt Sally.

math 172

  • The other tool is simply the make-believe word: PEMDAS, made up of the same letters.
  • Although some students find these tools to be helpful, one drawback is that they fail to emphasize some very important details related to the order of operations. For instance, you need to keep in mind that multiplication and division are worked in the same step – from left to right. Multiplication does not always come before division. The same is true for addition and subtraction.

 

Example 1

Solve: 20 \div 4 + {(17 - 13)}^2 \cdot (-2)

 

Notice that the statement in this first example contains many operations. There are parentheses and exponents, as well as the operations division, addition, and multiplication.

In example one, we see two sets of parentheses.  Only the first set, though, encloses an operation, which is subtraction. The second set of parenthesis is used simply to distinguish the -2. So our first step, according to the order of operations, is to do the operation in parenthesis and simplify 17 minus 13, which is equal to 4. I’m going to keep the parenthesis around the 4, which is a good habit to get into for more complicated problems.

20 \div 4 + {(17 - 13)}^2 \cdot (-2)
20 \div 4 + {(4)}^2 \cdot (-2)

Now that we have simplified all grouping symbols that enclose an operation, we may move on to the next step which is to simplify anything involving an exponent or root.This means our next step will be to simplify ‘4 squared,’ which is equal to 16.

20 \div 4 + 16 \cdot (-2)

Now we are down to the operations division, addition and multiplication. The order of operations tells us that multiplication and division happen in the next step, and to do them in order from left to right. Since the division operation comes first, we’ll work that next. 20 divided by 4 is equal to 5.

5 + 16 \cdot (-2)

Then, we’ll multiply. 16 multiplied by -2 is equal to -32. I’ll keep parenthesis around the -32 emphasize that it’s negative.

5 + (-32)

Finally, we’re down to only one operation, which is addition. When we add 5 plus -32, the answer is -27. Therefore the final answer is -27.

 

Example 2

Simplify: (-4)(-6) - \frac{{(4 + 2)}^2}{3}

 

In example one, the word “solve” was used. Here, the word is “simplify”. They both mean the same thing in terms of order of operation exercises- that we need to follow the order of operations to solve for or simplify to the answer.

In this example, the operations include multiplication (between the -4 and -6), subtraction and division. We also have a set of grouping symbols enclosing an operation, an exponent, and a division bracket. There’s a lot going on in this problem!

Let’s take things one step at a time, following the order of operations, starting with the parenthesis containing ‘4 plus 2.’ ‘4 plus 2’ is equal to 6.

(-4)(-6) - \frac{{(4 + 2)}^2}{3}
(-4)(-6) - \frac{{(6)}^2}{3}

Next, we’ll simplify the exponent portion ‘6 squared,’ which is equal to 36.

(-4)(-6) - \frac{36}{3}

At this point, we’re left with multiplication, subtraction, and division. Multiplication and division are always worked before addition and subtraction and since the multiplication operation comes first, we’ll simplify ‘-4 times -6’ which is equal to positive 24. Next, let’s divide 36 by 3, which is equal to 12

24 - \frac{36}{3}

[24 - 12

Finally, we’ll subtract 24 minus 12, which is equal to 12. Therefore, the final answer is 12.

Once you commit the order of operations to memory, order of operation exercises are not difficult. However, it is easy to become over-confident and try to do more than one step at a time, which can cause an incorrect answer. I encourage you to take each problem one step at a time, as I am doing in these examples. You will be less likely to make a mistake. And if you do make a mistake, having each step written out will make it easier for you to go back, look at your work, and troubleshoot what went wrong.

 

Example 3

Evaluate the expression: -2\left[{(5 - 3)}^3 + 7\right]

 

Another word you might see related to an order of operations problem is the word “evaluate,” as in example 3. Just like the words ‘solve’ and ‘simplify,’ the word ‘evaluate’ implies that we need to follow the order of operations to find the answer.

Beginning with the grouping symbols, notice that we have a set of parenthesis enclosed within a set of brackets. Sometimes brackets or squiggly brackets are used in problems containing more than one set of grouping symbols. This is to help avoid confusion. In this case, we will work from the inside, out – meaning we will work the inner-most set of grouping symbols first, following the order of operations, and work our way out.

So first, we’ll subtract ‘5 minus 3,’ which is equal to 2.

-2\left[{(5 - 3)}^3 + 7\right]
-2\left[{(2)}^3 + 7\right]

Now that we’ve done the operation inside the parenthesis, focus only on what is inside the brackets. According to the order of operations, we should work the exponent next.

‘2 raised to the power of 3’ is equal to 8.

-2\left[8 + 7\right]

Next we will continue working inside the brackets and add the 8 and 7. 8 plus 7 is equal to 15.

-2\left[15\right]

Brackets mean the same thing as parenthesis, they just looks a little different. Here, in our final step, they indicate multiplication. So as our final step, we’ll multiply -2 times 15, which is equal to -30.

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