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Order of Operations

The order of operations outlines the order by which certain operatives should take place in a mathematical expression for it to remain true

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Evaluate Numerical and Variable Expressions Using the Order of Operations

License: CC BY-NC 3.0

Jeb enters his name into a draw to win a handheld game system from his local electronics megastore. A week later he visits the store for the draw and his name is picked! In order to claim the prize, he must correctly answer a skill testing question.

The question is:\begin{align*}\frac{4}{5}\left[30-\left(4\times2 -3\right)\right]\end{align*}

Jeb has to answer the skill testing question without the use of technology. How can Jeb answer this question correctly to claim the prize?

In this concept, you will learn to use the order of operations to solve numerical and variable expressions.

Order of Operations

In mathematics, you will often hear the word evaluate. Before you begin, it is important for you to understand what the word evaluate means. When you evaluate a mathematical sentence, you figure out the value of the number sentence. Often times you think of evaluating as solving, and it can be that, but more specifically, evaluating is figuring out the value of a sentence.

In mathematics, you can evaluate different types of number sentences. Sometimes you will be working with equations and other times you will be working with expressions. First you need to know the difference between an equation and an expression.

An equation is a statement that two mathematical expressions have the same value. An equation has an equal sign such that the quantity on the left side of the equal sign is equal to the quantity on the right side of the equal sign. This means both sides of the equation stand for the same number. \begin{align*}5x=30\end{align*} is an equation. It states that \begin{align*}"5x" \ \text{and}\ "30"\end{align*} have the same value.

An expression is a general term in mathematics for a group of numbers, symbols and variables representing numbers and operations. You evaluate an expression to figure out the value of the mathematical statement itself, you are not trying to make one side equal another, as with an equation. \begin{align*}3a-2b+8\end{align*} is an expression. There is no equal sign.

Let’s start with an example of evaluating expressions.

Two eighth grade math students evaluated the expression \begin{align*}2+3\times 4\div 2\end{align*}. Macy’s answer was ten and Cole’s answer was eight. The students you are asked to write their step by step solutions on the board.

Macy‘s Solution: 

\begin{align*}& =2+3\times 4\div 2 \\ &= 5\times 4 \div 2 \\ &= 20\div 2\\ &= 10\end{align*}

Cole‘s Solution:

\begin{align*}&= 2+3\times 4\div 2\\ &= 2+ 12 \div 2\\ &= 2+ 6 \\ &= 8\end{align*} 

It appears that each student performed the indicated operations in different orders. Of course, there cannot be two correct solutions for the same expression. There is actually a specific order in which operations must be performed.

The order of operations is a rule that tells you which operation you need to perform and the order in which it must be done to achieve the correct answer. The order of operations is often called PEMDAS and each of the letters represents one part of the rule. P: parenthesis and grouping symbols; E: exponents; M: multiplication; D: division; A: addition; S: subtraction. MD (multiplication and division) are performed in the order they appear in the expression from left to right. AS (addition and subtraction) are also performed in the order they appear in the expression from left to right.

Looking at the two solutions for evaluating the expression \begin{align*}2+3\times 4\div 2,\end{align*} who has the correct answer?


\begin{align*}&\text{First}: 2 +3 =5\\ &\text{Next}: 5 \times 4=20\\ &\text{Then}: 20\div 2=10\end{align*} 


\begin{align*}&\text{First}: 3\times 4=12 \\ &\text{Next}: 12\div 2=6 \\ &\text{Then}: 6+2=8\end{align*}

Macy evaluated the expression by performing the addition first, followed by multiplication and division. Cole performed the multiplication and division first, then addition. Macy simply completed the operations as they appeared from left to right. Cole completed the multiplication and division as they appeared from left to right and then performed the addition as his final step. Cole used the order of operations rule, PEMDAS, and his answer is correct.

This is called evaluating or simplifying a numerical expression. A numerical expression is an expression made up only of numbers and operations.

In addition to numbers, expressions can also have letters. The letters in an expression are called variables. These variables represent an unknown quantity. When an expression is written with a variable in it, you call it a variable expression.

A variable expression is evaluated using the order of operations in the same way as a numerical expression is evaluated. In this Concept there will be a value given for the variable and you will substitute it into the variable expression before evaluating the expression.

Evaluate the variable expression:

\begin{align*}60\div 2 \cdot 2a+16-4 \ \text{when} \ a=5.\end{align*}

First, substitute the given value for \begin{align*}a\end{align*} in the expression:

\begin{align*}60\div 2 \cdot 2\left(5\right)+16-4\end{align*}

Remember that, used this way, parenthesis mean you are multiplying 2 times (5).

Now apply the order of operations (PEMDAS) and continue to evaluate the expression.

First, divide: \begin{align*}60 \div 2=30\end{align*}and write the new expression.

\begin{align*}30 \cdot 2(5)+16-4\end{align*}

Next, multiply: \begin{align*}30 \cdot 2 = 60,\end{align*} since this is the first multiplication from left to right.

\begin{align*}60 (5) +16-4\end{align*}

Next, multiply: \begin{align*}60 (5)=300\end{align*} and write the new expression.


Next, add: \begin{align*}300+16=316\end{align*} and write the new expression.


Then, subtract: \begin{align*}316−4=312\end{align*}

The answer is 312.

Now let’s add in the grouping symbols. The grouping symbols that you will be working with are brackets [  ] and parenthesis (  ). According to the order of operations (PEMDAS), you perform all operations inside the grouping symbols BEFORE any other operation in the list.

Evaluate the numerical expression

\begin{align*} 7+4 \left(15 \div 5 \right) - 6.\end{align*}

First, perform the operation in the parenthesis: \begin{align*}15\div 5=3,\end{align*} and write the new expression. 


Next, multiply \begin{align*}4\left(3\right)=12 \end{align*} to clear the parenthesis and write the new expression.

\begin{align*}7+12-6\end{align*}Next, add:\begin{align*}7+12=19\end{align*} and write the new expression. 


Then, subtract: \begin{align*}19−6=13\end{align*}

The answer is 13.

Brackets can be used to group more than one operation. When you see a set of brackets, remember that brackets are a way of grouping numbers and operations with other groups already in parenthesis. Always evaluate grouping symbols from the innermost to the outermost.

Evaluate the numerical expression.

\begin{align*}6+\left[5+\left( 4 \times 6 \right)\right]-17.\end{align*}

First, evaluate the innermost grouping symbols: the parenthesis.

Multiply:\begin{align*}4\times 6=24\end{align*} and write the new expression.

\begin{align*}6+\left[5 + 24 \right]-17.\end{align*}

Next, perform the operation inside the brackets.

Add:\begin{align*}5+24=29\end{align*} and write the new expression. 


Next, add:\begin{align*}6+29=35\end{align*} and write the new expression. 

\begin{align*}35-17\end{align*}Then, subtract: \begin{align*}35−17=18\end{align*} 

The answer is 18.


Example 1

Earlier, you were given a problem about Jeb and his (almost) prize.

Remember, when solving a problem like this skill testing question, you need to complete the indicated operations in the order of PEMDAS.

\begin{align*}\frac{4}{5}\left[30 - \left(4 \times 2 -3 \right)\right]\end{align*}

Notice there are parenthesis within the brackets. You need to evaluate from the inside out. So start with the parenthesis. According to PEMDAS multiplication comes before subtraction, and this is just as true within a group as without.

First, multiply: \begin{align*}4 \times 2 =8\end{align*} and write the new expression. 

\begin{align*}\frac{4}{5}\left[30 - \left( 8 - 3 \right)\right]\end{align*}Next, subtract: \begin{align*}8-3=5\end{align*}  and write the new expression.

\begin{align*}\frac{4}{5}\left[30 - 5 \right]\end{align*}

Next, subtract: \begin{align*}30-5=25\end{align*}  and write the new expression. 


Next, multiply:\begin{align*}\frac{4}{5} \times \frac{25}{1}=\frac{100}{5}\end{align*} and write the new expression.


Then, divide: \begin{align*}\frac{100}{5}=20\end{align*}

The answer is 20.

Jeb needs to answer 20 to claim his prize.

Example 2

Evaluate the expression \begin{align*}3+9 \cdot 2 \div 3 + 8\end{align*}  

First, multiply \begin{align*}9 \cdot 2=18,\end{align*} and write down the new expression. 

\begin{align*}3+18\div 3+8\end{align*}

Next, divide \begin{align*}18\div 3=6\end{align*} and write down the new expression. 


Next add \begin{align*}3+6=9\end{align*} and write down the new expression. 

\begin{align*}9+8\end{align*}Then, add \begin{align*}9+8=17.\end{align*}

The answer is 17.

Example 3

Evaluate the variable expression \begin{align*}6y+3-\left(2 \times 4 \right) \ \text{when} \ y=15.\end{align*}

First, substitute \begin{align*}y=15\end{align*} into the expression.

\begin{align*}6 \left(15\right)+3-\left(2 \times 4 \right)\end{align*}

Next, perform the operation inside the parenthesis.

Multiply: \begin{align*}2 \times 4 =8\end{align*} and write the new expression. 

\begin{align*}6\left(15\right)+3-8 \end{align*}

Next, multiply: \begin{align*}6\left(15\right)=90\end{align*} to clear the parenthesis and write the new expression. 


Next add: \begin{align*}90+3=93\end{align*} and write the new expression.

\begin{align*}93-8\end{align*}Then, subtract:

\begin{align*}93-8=85\end{align*}The answer is 85.

Example 4

Evaluate the numerical expression.

\begin{align*}7+4\left[4+\left(3 \times 2\right)\right]-5\end{align*}

Remember to evaluate the grouping symbols from the innermost to the outermost.

First, perform the operation inside the parenthesis.

Multiply: \begin{align*}3 \times 2=6\end{align*} and write the new expression.


Next, perform the operation inside the brackets.

Add: \begin{align*}4+6=10\end{align*} and write the new expression. 


Then multiply \begin{align*}4 \times 10 = 40.\end{align*} 


Next, add: \begin{align*}7+40=47\end{align*} and write the new expression. 

\begin{align*}47-5\end{align*}Finally, subtract 


The answer is 42.

Example 5

Evaluate the variable expression.

\begin{align*}14 \times 2 \div 7 + 3b -4 \ \text{when} \ b =12.\end{align*} 

First substitute \begin{align*}b=12\end{align*}  into the expression.

\begin{align*}14 \times 2 \div 7+ 3\left(12\right)-4\end{align*}

First, multiply:\begin{align*}14 \times 2 =28\end{align*} and write the new expression. 

\begin{align*}28 \div 7 + 3(12) - 4\end{align*}

Next, divide:\begin{align*}28 \div 7 =4\end{align*} and write the new expression. 


Then, multiply again: \begin{align*}3 \times 12 = 36,\end{align*} and write another new expression.


Next, add:\begin{align*}4+36=40\end{align*} and write the new expression. 


Finally, subtract : \begin{align*}40-4=36\end{align*}

The answer is 36.


Evaluate each numerical expression using the order of operations.

1. \begin{align*}4+5\times 2-3\end{align*} 

2.\begin{align*}6+6 \times 3 \div 2-7\end{align*}

3. \begin{align*}5+5 \times 8\div 2+6\end{align*}

4.\begin{align*}13-3 \times 2+8-2\end{align*}

5.\begin{align*}17-5 \times 3+8 \div 2\end{align*}

6.\begin{align*}9+4 \times 2+7-1\end{align*}

7.\begin{align*}8+5 \times 6+2 \times 4-3\end{align*} 

8.\begin{align*}19+2 \times 4-3·2+10\end{align*}

9. \begin{align*}12+4 \times 4 \div 8-3\end{align*}

10. \begin{align*}12 \times 2 + 6 \div 2 - 12 \end{align*}

Evaluate each variable expression. Remember to use PEMDAS when necessary.

11.\begin{align*}4y+6-2, \text{when} \ y=6\end{align*} 

12.\begin{align*}9+3x-5+2, \text{when} \ x=8\end{align*} 

13.\begin{align*}6y + 2y -5, \text{when} \ y =3\end{align*}

14.\begin{align*}8+3y-5, \text{when} \ y =4 \end{align*}

15.\begin{align*}7x-2 \times 3 \div 3 + 12, \text{when} \ x=5\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.4. 

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Brackets Brackets [ ], are symbols that are used to group numbers in mathematics. Brackets are the 'second level' of grouping symbols, used to enclose items already in parentheses.
Equation An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
Evaluate To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Expression An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Grouping Symbols Grouping symbols are parentheses or brackets used to group numbers and operations.
nested parentheses Nested parentheses describe groups of terms inside of other groups. By convention, nested parentheses may be identified with other grouping symbols, such as the braces "{}" and brackets "[]" in the expression \{ 3 + [ 2 - ( 5 + 4 ) ] \}. Always evaluate parentheses from the innermost set outward.
Numerical expression A numerical expression is a group of numbers and operations used to represent a quantity.
Parentheses Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.
PEMDAS PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.
Variable Expression A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.

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  1. [1]^ License: CC BY-NC 3.0

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