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Section 1.5 Order of Operations

Subsection Addition and Multiplication

If you add together three or more numbers, such as

\begin{equation*} 2+5+8 \end{equation*}

it doesn't matter which addition you do first; you will get the same answer either way.

Example 1.33.
  1. A sum of three or more terms can be added in any order. In the sums below, the parentheses tell us which part of the expression to simplify first.
    \begin{equation*} \begin{aligned} (\blert{2} \amp \blert{+5})+8 \amp \text{and}~~~~~~~~~~2 \amp + (\blert{5+8})\\ \amp = \blert{7}+8=15 \amp \amp = 2+\blert{13}=15 \end{aligned} \end{equation*}
  2. Similarly, a product of three factors can be multiplied in any order. Thus
    \begin{equation*} \begin{aligned} (\blert{3} \amp \blert{\cdot 2}) \cdot 4 \amp \text{and}~~~~~~~~~~3 \cdot \amp + (\blert{2 \cdot 4})\\ \amp = \blert{6}\cdot 4=24 \amp \amp = 3 \cdot \blert{8}=24 \end{aligned} \end{equation*}

These two facts illustrate the associative laws for addition and multiplication.

Associative Law for Addition.

If \(a, b\) and \(c\) are any numbers, then

\begin{equation*} \blert{(a+b)+c = a+(b+c)} \end{equation*}
Associative Law for Multiplication.

If \(a, b\) and \(c\) are any numbers, then

\begin{equation*} \blert{(a \cdot b) \cdot c = a \cdot (b \cdot c)} \end{equation*}

Subsection Subtraction and Division

What about a string of subtractions or a string of divisions, such as

\begin{equation*} 20-8-5~~~~~~~~~~\text{or}~~~~~~~~~~36 \div 6 \div 2 \end{equation*}

In these calculations, we get different answers, depending on which operations we perform first, as you can see in Example 1.34.

Example 1.34.
  1. Subtraction:
    \begin{equation*} (\blert{20-8})-5=\blert{12}-5=7 ~~~~~~\text{but}~~~~~~ 20-(\blert{8-5})=20-\blert(3)=17 \end{equation*}
  2. Division:
    \begin{equation*} (\blert{36 \div 6}) \div 2 =\blert{6}\div 2=3 ~~~~~~\text{but}~~~~~~ 36 \div(\blert{6 \div 2})=36-\blert(3)=12 \end{equation*}

The associative laws do not hold for subtraction or division.

So, if there are no parentheses in the expression, how do we know which operations to perform first?

  1. In a string of additions and subtractions, we perform the operations in order from left to right.
  2. Similarly, we perform multiplications and divisions in order from left to right.
Example 1.35.

Simplify each expression.

  1. \(20-8-5\)
  2. \(36 \div 6 \div 2\)
Solution

Perform the operations in order from left to right.

  1. \(\blert{20-8}-5=\blert{12}-5=7\)
  2. \(\blert{36 \div 6} \div 2 =\blert{6}\div 2=3\)

Reading Questions Reading Questions

1.

What do the associative laws tell us?

2.

Which two operations are not associative?

3.

In what order should we perform a string of additions and subtractions?

Subsection Combined Operations

How should we simplify the expression \(4+6 \cdot 2\) ?

If we do the addition first, we get

\begin{equation*} (\blert{4+6}) \cdot 2 = \blert{10} \cdot 2 = 20 \end{equation*}

If we do the multiplication first, we get

\begin{equation*} 4+(\blert{6 \cdot 2}) = 4 + \blert{12} = 48 \end{equation*}

Which one is correct? In order to avoid confusion, we make the following agreement.

Always perform multiplications and divisions before additions and subtractions.

Example 1.36.

The correct way to simplify the expression \(4+6 \cdot 2\) is

\begin{equation*} \begin{aligned} 4 \amp + \blert{6 \cdot 2} \amp \amp \blert{\text{Multiply first.}}\\ \amp = 4 + 12 \amp \amp \blert{\text{Then add.}}\\ \amp = 16 \end{aligned} \end{equation*}
Look Closer.

In longer expressions, it can be helpful to group the expression into its terms before beginning. Terms are expressions separated by addition or subtraction symbols. We simplify each term before combining them.

Example 1.37.

Simplify \(6 + 2 \cdot 5 - 12 \div 3 \cdot 2\)

Solution

We start by underlining each term of the expression separately. Then we simplify each term.

\begin{equation*} \begin{aligned} \underline{6} \amp + \underline{2 \cdot 5} - \underline{12 \div 3 \cdot 2} \amp \amp \blert{\text{Perform multiplications and divisions from left to right.}}\\ \amp = 6 + 10 - 8 \amp \amp \blert{\text{Perform additions and subtractions from left to right.}}\\ \amp = 16 - 8 = 8 \end{aligned} \end{equation*}
Caution 1.38.
  1. In Example 1.37, we do not start by adding \(6+2\text{.}\) We must perform multiplications before additions.
  2. We simplify \(12 \div 3 \cdot 2\) by performing the operations from left to right:
    \begin{equation*} \blert{12 \div 3} \cdot 2 = \blert{4} \cdot 2 = 8 \end{equation*}

Subsection Parentheses

What if the addition should come first in a particular calculation? In that case, we use parentheses to enclose the sum, like this:

\begin{equation*} (4+6) \cdot 2 \end{equation*}

Perform any operations inside parentheses first.

Example 1.39.

Simplify.

  1. \begin{equation*} \begin{aligned} 2 + \amp 5 (\blert{7 - 3}) \amp \amp \blert{\text{Subtract inside parentheses.}}\\ \amp = 2 + \blert{5(4)} \amp \amp \blert{\text{Multiply.}}\\ \amp = 2+20 = 22 \amp \amp \blert{\text{Add.}} \end{aligned} \end{equation*}
  2. \begin{equation*} \begin{aligned} 6(\amp 10- \blert{2\cdot 4}) \div 4 \amp \amp \blert{\text{Multiply inside parentheses.}}\\ \amp = 6(\blert{10-8}) \div 4 \amp \amp \blert{\text{Subtract inside parentheses.}}\\ \amp = \blert{6(2)} \div 4 \amp \amp \blert{\text{Multiply and divide in order from left to right.}}\\ \amp = 12 \div 4 = 3 \end{aligned} \end{equation*}

Reading Questions Reading Questions

4.

Is it true that multiplications should be performed before divisions? Why or why not?

5.

What do parentheses tell us?

6.

If there are no parentheses, in what order should we perform multiplications and divisions?

Subsection Fraction Bars

Like parentheses, a fraction bar is a grouping device.

Expressions that appear above or below a fraction bar should be simplified first.

Example 1.40.

Simplify \(\dfrac{24+6}{12+6}\)

Solution

We begin by computing the sums above and below the fraction bar. Then we can reduce the fraction. Thus,

\begin{equation*} \dfrac{24+6}{12+6} = \dfrac{30}{18}=\dfrac{5}{3} \end{equation*}
Caution 1.41.

Do not be tempted to "cancel" the terms in Example 1.40. For example, it would be incorrect to write

\begin{equation*} \dfrac{24+6}{12+6} = \dfrac{\Ccancel[blue]{24} +6}{\Ccancel[blue]{12} +6} = \dfrac{2+6}{1+6}~~~~~~~~\alert {\leftarrow \text{Incorrect!}} \end{equation*}

According to the order of operations, we must simplify the numerator and denominator first, before dividing.

Reading Questions Reading Questions

7.

Besides parentheses, what other symbol serves as a grouping device?

8.

When adding fractions, which should you do first: reduce each fraction, or find an LCD?

Subsection Summary

Combining all our guidelines for simplifying expressions, we state the rules for the order of operations.

Order of Operations.
  1. First, perform any operations that appear inside parentheses, or above or below a fraction bar.
  2. Next, perform all multiplications and divisions in order from left to right.
  3. Finally, perform all additions and subtractions in order from left to right.

Subsection Algebraic Expressions

The order of operations applies to variables as well as constants. For example, the expression

\begin{equation*} 5+2x \end{equation*}

tells us to multiply \(x\) by 2, then add 5 to the result. Thus, to evaluate the expression for \(x=\alert{8}\) for example, we write

\begin{equation*} \begin{aligned} \text{For }x=\alert{8},~~~~~~~~~~5+2x \amp = 5+2(\alert{8}) \amp \amp \blert{\text{Multiply first.}}\\ \amp = 5+16=21 \end{aligned} \end{equation*}

In the Example below, note how parentheses change the meaning of an expression.

Example 1.42.

Write algebraic expressions for each of the following phrases. Then evaluate each phrase for \(x=2\) and \(y=6\text{.}\)

  1. Three times the sum of \(x\) and \(y\)
  2. The sum of \(3x\) and \(y\)
Solution
  1. \begin{equation*} \begin{aligned} \amp 3(x+y) \\ \amp 3(\alert{2}+\alert{6})=3(8)=24 \end{aligned} \end{equation*}
  2. \begin{equation*} \begin{aligned} \amp 3x+y \\ \amp 3(\alert{2})+\alert{6}=6+6=12 \end{aligned} \end{equation*}

Subsection Skills Warm-Up

Exercises Exercises

Perform the operations on fractions as indicated.

1.
  1. \(\dfrac{5}{4} + \dfrac{3}{4}\)
  2. \(\dfrac{5}{4} - \dfrac{3}{4}\)
  3. \(\dfrac{5}{4} \cdot \dfrac{3}{4}\)
  4. \(\dfrac{5}{4} \div \dfrac{3}{4}\)
2.
  1. \(\dfrac{1}{2} + \dfrac{1}{6}\)
  2. \(\dfrac{1}{2} - \dfrac{1}{6}\)
  3. \(\dfrac{1}{2} \cdot \dfrac{1}{6}\)
  4. \(\dfrac{1}{2} \div \dfrac{1}{6}\)
3.
  1. \(\dfrac{2}{3} + \dfrac{1}{4}\)
  2. \(\dfrac{2}{3} - \dfrac{1}{4}\)
  3. \(\dfrac{2}{3} \cdot \dfrac{1}{4}\)
  4. \(\dfrac{2}{3} \div \dfrac{1}{4}\)

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 1.5

For Problems 1–12, simplify each expression by following the order of operations.

1.

\(2+4(3)\)

2.

\(15-\dfrac{3}{4}(16)\)

3.

\(6 \div \dfrac{1}{4} \cdot 3\)

4.

\(3+3(2+3)\)

5.

\(\dfrac{1}{3} \cdot 12 - 3(\dfrac{5}{6})\)

6.

\(2+3 \cdot 8-6+3\)

7.

\(\dfrac{3(8)}{12}-\dfrac{6+4}{5}\)

8.

\(28+6 \div 2 - 2(5+3 \cdot 2)\)

9.

\(3[3(3+2)-8]-17\)

10.

\(\dfrac{3(3)+5}{6-2(2)}+\dfrac{2(5)-4}{9-4-2}\)

11.

\(7[15-24 \div 2]-9[5(4+2)-4(6+1)]\)

12.

\(20-4(9-3 \cdot 2)+8-5 \cdot 3\)

For Problems 13–16, use your calculator to simplify each expression. Round your answers to three decimal places if necessary.

13.

\(\dfrac{6.4+3.5}{3.6(3.2)}\)

14.

\(\dfrac{14.6-6.8}{7.3+3.4}\)

15.

\(\dfrac{26.2-9.1}{8.4 \div 7.7}+ 5.1(6.9-1.6)\)

16.

\(\dfrac{1728(847-603)}{216(98-38)}+ 6(876-514)\)

For Problems 17–22, evaluate each pair of expressions mentally.

17.
  1. \(8+2 \cdot 5\)
  2. \((8+2) \cdot 5\)
18.
  1. \(\dfrac{24}{2+6}\)
  2. \(\dfrac{24}{2}+6\)
19.
  1. \((9-4)-3\)
  2. \(9-(4-3)\)
20.
  1. \(6 \cdot 8 - 6\)
  2. \(6(8-6)\)
21.
  1. \(\dfrac{36}{6(3)}\)
  2. \(\dfrac{36}{6}(3)\)
22.
  1. \((30-5)(5)\)
  2. \(30-(5)(5)\)

For Problems 23–24, fill in the table to evaluate the expression in two steps.

23.

\(5z-3\)

\(z\) \(5z\) \(5z-3\)
\(2\) \(\hphantom{00}\) \(\hphantom{00}\)
\(4\) \(\hphantom{00}\) \(\hphantom{00}\)
\(5\) \(\hphantom{00}\) \(\hphantom{00}\)

24.

\(2(12+Q)\)

\(Q\) \(12+Q\) \(2(12+Q)\)
\(0\) \(\hphantom{00}\) \(\hphantom{00}\)
\(4\) \(\hphantom{00}\) \(\hphantom{00}\)
\(8\) \(\hphantom{00}\) \(\hphantom{00}\)

For Problems 25–30, evaluate for the given values.

25.

\(2y+x~~~~~~~~\)for \(x=8\) and \(y=9\)

26.

\(4a+3b~~~~~~~~\)for \(a=8\) and \(b=7\)

27.

\(\dfrac{a}{b}-\dfrac{b}{a}~~~~~~~~\)for \(a=8\) and \(b=6\)

28.

\(\dfrac{24-2x}{2+y}-\dfrac{4x+1}{3y}~~~~~~~~\)for \(x=4\) and \(y=6\)

29.

\(\dfrac{a}{1-r}~~~~~~~~\)for \(a=6\) and \(r=0.2 \)

30.

\(mx+b~~~~~~~~\)for \(m=\dfrac{3}{5},~x=\dfrac{2}{3}\) and \(b=\dfrac{9}{10}\)

31.

Consider the expression \(20-2 \cdot 8 + 1\text{.}\) How would you change the expression if you wanted:

  1. The subtraction performed first?
  2. The addition performed first?
  3. The multiplication performed first?
32.

Write an algebraic expression for the following instructions:

"Multiply the sum of 5 and 7 by 4, then divide by the difference of 10 and 8."

33.

Write two algebraic expressions for "the difference of 20 and 8, divided by the sum of 6 and 4 and 10":

  1. Using a fraction bar
  2. Using a division symbol
34.

Without performing the calculations, write down the steps you would use to simplify the expression: \(~~825-32(12)\div 4+2\)

For Problems 35–38, choose the correct algebraic expression for each English phrase.

\begin{align*} \amp \dfrac{m}{12}-3 \amp\amp\amp \amp \dfrac{m-3}{12}\\ \amp \dfrac{12}{m-3} \amp\amp\amp\amp 3-\dfrac{12}{m} \end{align*}
35.

Twelve divided by 3 less than \(m\text{.}\)

36.

Three less than the quotient of \(m\) divided by 12.

37.

The quotient of 3 less than \(m\) divided by 12.

38.

Subtract from 3 the quotient of 12 and \(m\text{.}\)

39.

Find and correct the error in the calculation:

\begin{equation*} \begin{aligned} (5+4)\amp -3(8-3 \cdot 2)\\ \amp = 9-3(8-6)\\ \amp = 6(2)=12 \end{aligned} \end{equation*}
38.

Think of the region shown below as a rectangle with a smaller rectangle removed, and write an expression for its area. Then simplify your expression to find the area. (The measurements given are in centimeters.)

rectangle with rectangular hole