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Core Mathematics

Katherine Yoshiwara

Section 4.5 Order of Operations

  • Which Operations Come First
  • Combined Operations
  • Parentheses
  • Fraction Bars
  • Using a Calculator

Subsection 4.5.1 Which Operations Come First

You would like to bake a cake. Here are the steps:

Baking a Cake.

  1. Preheat the oven
  2. Mix the batter.
  3. Pour the batter into the cake pan.
  4. Put the cake pan in the oven.
  5. Bake until done, then remove from the oven.
  6. Frost the cake.
What if you perform those steps out of order? Your cake might not be a success! The order in which we perform a sequence of steps makes a difference to the outcome.
Algebraic expressions often involve more than one operation. Just as with cake baking, the order in which we perform those operations makes a difference to the answer. You probably know the following rule.

Rule 1.

Perform additions and subtractions in order from left to right.

Example 4.5.1.

Simplify \(~50 - 20 - 15 + 10\)

Solution.

We perform the operations in the order they occur, from left to right.
\begin{align*} \blert{50 - 30} - 15 + 10 \amp = \blert{20} - 15 + 10 \amp \amp \blert{\text{Subtract}~~ 50-30.}\\ \blert{20 - 15} + 10 \amp = \blert{5} + 10 \amp \amp \blert{\text{Subtract}~~ 20-15.}\\ \blert{5 + 10} \amp = 15 \amp \amp \blert{\text{Add}~~ 5+10.} \end{align*}

Checkpoint 4.5.2.

Simplify \(~18 - 10 + 6 - 3\text{.}\) Show your steps.
Answer.
\begin{align*} 18 - 10 + 6 - 3 \amp = 8 + 6 - 3\\ \amp = 14 - 3 = 11 \end{align*}

Subsection 4.5.2 Combined Operations

How do we simplify the expression
\begin{equation*} 5 + 2 \cdot 4 \end{equation*}
Should we add \(5 + 2\) first, and then multiply by 4, or should we multiply \(2 \cdot 4\) first, and then add 5?
\(\blert{\text{Option 1}}\)
\begin{align*} \alert{5~} \amp \alert{~+ 2} \cdot 4 \hphantom{000} \text{Add first.}\\ \amp = \alert{7} \cdot 4 \hphantom{000} \text{Then multiply.}\\ \amp = 28 \end{align*}
\(\blert{\text{Option 2}}\)
\begin{align*} 5 \amp + \alert{2 \cdot 4} \amp \amp \text{Multiply first.}\\ \amp = 5 + \alert{8} \amp \amp \text{Then add.}\\ \amp = 13 \end{align*}
The two options give us different answers, so they cannot both be right.

Note 4.5.3.

You might decide to perform the operations from left to right in the order they appear, but this causes problems. (For instance, we would like \(5 + 2 \cdot 4\) and \(2 \cdot 4 + 5\) to have the same answer.)
Instead, people who use mathematics have agreed upon the following rule.

Rule 2.

Perform all multiplications and divisions before additions and subtractions.
By following this rule, everyone means the same thing when they write an algebraic expression, and everyone gets the same answers. So Option 2 above is correct:
\begin{align*} 5 \amp + \blert{2 \cdot 4} \amp \amp \text{Multiply first.}\\ \amp = 5 + \blert{8} \amp \amp \text{Then add.}\\ \amp = 13 \end{align*}

Example 4.5.4.

Use Rule 2 to simplify each expression.
  1. \begin{align*} 25 \amp - \blert{6 \cdot 3} \amp \amp \blert{\text{Multiply first.}}\\ \amp = 25 - 18 \amp \amp \blert{\text{Subtract.}}\\ \amp = 7 \end{align*}
  2. \begin{align*} \blert{48~} \amp \blert{~\div 10} - 2 \amp \amp \blert{\text{Divide first.}}\\ \amp = 4.8 - 2 \amp \amp \blert{\text{Subtract.}}\\ \amp = 2.8 \end{align*}

Checkpoint 4.5.5.

Simplify each expression. Show your work.
  1. \(\displaystyle 22 - 2 \cdot 7\)
  2. \(\displaystyle 8 + 6 \div 2\)
Answer.
  1. \begin{align*} 22 - 2 \cdot 7 \amp = 22 - 14\\ \amp = 8 \end{align*}
  2. \begin{align*} 8 + 6 \div 2 \amp = 8 + 3\\ \amp = 11 \end{align*}
We can combine these two rules into a set of guidelines.

Guidelines for Simplifying Expressions.

  • First, perform all multiplications and divisions in order from left to right.
  • Next, perform all additions and subtractions in order from left to right.

Example 4.5.6.

Use the guidelines above to simplify each expression.
  1. \(\displaystyle 18 + 3(12) - 24\)
  2. \(\displaystyle 5 + 5 \cdot 5 - 5 \div 5\)

Solution.

  1. We start with multiplications and divisions, working from left to right. It is helpful to underline all the multiplication and division operations before beginning.
    \begin{align*} 18 \amp + \blert{\underline{3(12)}} - 24 \amp \amp \blert{\text{Multiply 3 times 12.}}\\ \amp = \blert{18 + 36} - 24 \amp \amp \blert{\text{Add 18 plus 36.}}\\ \amp = 54 - 24 \amp \amp \blert{\text{Subtract 24 from 54.}} \end{align*}
  2. First, underline all the multiplication and division operations.
    \begin{align*} 5 \amp + \blert{\underline{5 \cdot 5}} - \blert{\underline{5 \div 5}} \amp \amp \blert{\text{Multiply and divide.}}\\ \amp = \blert{5 + 25} - 1 \amp \amp \blert{\text{Add and subtract from left to right.}}\\ \amp = \blert{30 - 1} = 29 \end{align*}

Checkpoint 4.5.7.

Simplify each expression. Show your work.
  1. \(\displaystyle 10 + 10 \div 2 \cdot 8\)
  2. \(\displaystyle 26 - 6 \div 3 + 4 \cdot 7\)
Answer.
  1. \begin{align*} 10 + 10 \div 2 \cdot 8 \amp = 10 + 5 \cdot 8\\ \amp = 10 + 40 = 50 \end{align*}
  2. \begin{align*} 26 - 6 \div 3 + 4 \cdot 7 = \amp = 26 - 2 + 28\\ \amp = 24 + 28 = 52 \end{align*}

Subsection 4.5.3 Parentheses

The guidelines we have already established are not sufficient for every situation. How can we write the following phrase in mathematical symbols?
"Add 2 and 5, then multiply the sum by 3."
If you do the computations as stated, you’ll see that the result is 21. But, as you can see, neither of the expressions below gives the right answer.
\(\blert{\text{Option 1}}\)
\begin{align*} 2 \amp + \alert{5 \cdot 3} \amp \amp \text{Multiply first.}\\ \amp = 2 + \alert{15} \amp \amp \text{Then add.}\\ \amp = 17 \end{align*}
\(\blert{\text{Option 2}}\)
\begin{align*} \alert{3} \amp \cdot \alert{2} + 5 \amp \amp \text{Multiply first.}\\ \amp = \alert{6} + 5 \amp \amp \text{Then add.}\\ \amp = 11 \end{align*}
According to our guidelines, multiplication comes before addition. We need an override mechanism to use when we really want addition to come first. For this purpose, we use parentheses :
\begin{equation*} \blert{3 (2 + 5)} \end{equation*}
The parentheses are a "grouping symbol;" they tell us to perform any operations inside first. Thus,
\begin{equation*} \blert{3 (2 + 5) = 3(7) = 21} \end{equation*}

Rule 3.

Perform any operations inside parentheses first.

Example 4.5.8.

Simplify each expression.
  1. \(\displaystyle 18 - 6(9 - 7)\)
  2. \(\displaystyle 24 \div (6 - 3) \cdot 4\)

Solution.

  1. \begin{align*} 18 \amp - 6(\blert{9 - 7}) \amp \amp \blert{\text{Subtract inside parentheses first.}}\\ \amp = 18 - \blert{6(2)} \amp \amp \blert{\text{Next, multiply.}}\\ \amp = 18 - 12 = 6 \end{align*}
  2. \begin{align*} 24 \amp - \blert{(6 - 3)} \cdot 4 \amp \amp \blert{\text{Subtract inside parentheses first.}}\\ \amp = \blert{24 \div 3} \cdot 4 \amp \amp \blert{\text{Multiply and divide in order.}}\\ \amp = 8 \cdot 4 = 32 \end{align*}

Checkpoint 4.5.9.

Simplify each expression. Show your work.
  1. \(\displaystyle 6 \div (12 - 10) \cdot 3\)
  2. \(\displaystyle 7 + 3(5 + 2)\)
Answer.
  1. \begin{align*} 6 \div (12 - 10) \cdot 3 \amp = 6 \div 2 \cdot 3\\ \amp = 3 \cdot 3 = 9 \end{align*}
  2. \begin{align*} 7 + 3(5 + 2) \amp = 7 + 3(7) \\ \amp = 7 + 21 = 28 \end{align*}

Subsection 4.5.4 Fraction Bars

A fraction bar is another kind of grouping symbol. Any operations that appear above or below a fraction bar should be completed first. For example, in the expression
\begin{equation*} \dfrac{16 + 14}{8 - 2} \end{equation*}
we compute the sum \(16+14\) and the difference \(8-2\) before dividing:
\begin{equation*} \dfrac{16 + 14}{8 - 2} = \dfrac{30}{6} = 5 \end{equation*}

Rule 4.

Simplify any expressions above or below a fraction bar before dividing the bottom into the top.

Example 4.5.10.

Simplify according to the order of operations: \(~~16 - \dfrac{24}{2(4)}\)

Solution.

We start by simplifying the denominator of the fraction.
\begin{align*} 16 \amp - \dfrac{24}{\blert{2(4)}} \amp \amp \blert{\text{Multiply below the fraction bar.}}\\ \amp = 16 - \blert{\dfrac{24}{8}} \amp \amp \blert{\text{Divide.}}\\ \amp = 16 - 3 = 13 \end{align*}

Checkpoint 4.5.11.

Simplify \(~~5 + \dfrac{7 + 9}{2 \cdot 4}.~~\) Show your work.
Answer.
\begin{align*} 5 + \dfrac{7 + 9}{2 \cdot 4} \amp = 5 + \dfrac{16}{8}\\ \amp = 5 + 2 = 7 \end{align*}
We can combine all the rules in this section into a set of guidelines for simplifying expressions. These guidelines are called the order of operations.

Order of Operations.

  1. First, perform all operations 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 4.5.5 Using a Calculator

We can use either a fraction bar or a division symbol, \(\div\) to write a quotient. For example, 15 divided by 3 can be written either
\begin{equation*} \dfrac{15}{3} ~~~~~~ \text{or} ~~~~~~ 15 \div 3 \end{equation*}
But the calculator does not have a fraction bar, so we must use the \(\boxed{\div}\) key for division.

Fraction Bars.

If there are operations above or below the fraction bar, we must enclose those operations in parentheses.
For example, to simplify the expression
\begin{equation*} \dfrac{120}{4(5)} \hphantom{0000} \text{we enter} \hphantom{0000} 120 \hphantom{0} \boxed{\div} \hphantom{0} \boxed{(} \hphantom{0} 4 \hphantom{0} \boxed{\times} \hphantom{0} 5 \hphantom{0} \boxed{)} \hphantom{0} \boxed{=} \end{equation*}

Note 4.5.12.

If your calculator does not have parentheses, you must perform the calculation in steps, following the order of operations: first compute \(4 \times 5\) to get 20, then divide 120 by 20 to get 6.
What answer would you get if you entered this keying sequence:
\begin{equation*} 120 \hphantom{0} \boxed{\div} \hphantom{0} 4 \hphantom{0} \boxed{\times} \hphantom{0} 5 \hphantom{0} \boxed{=} \end{equation*}

Example 4.5.13.

Use a calculator to compute \(\dfrac{4 + 12}{80}\)

Solution.

Because the calculator does not have a fraction bar, we must enclose \(4 + 12\) in parentheses. We enter the expression as
\begin{equation*} \boxed{(} \hphantom{0} 4 \hphantom{0} \boxed{+} \hphantom{0} 12 \hphantom{0} \boxed{)} \hphantom{0} \boxed{\div} \hphantom{0} 80 \hphantom{0} \boxed{=} \end{equation*}
The calculator returns the answer, 0.2. Can you explain why the keying sequence
\begin{equation*} 4 \hphantom{0} \boxed{+} \hphantom{0} 12 \hphantom{0} \boxed{\div} \hphantom{0} 80 \hphantom{0} \boxed{=} \end{equation*}
gives an incorrect answer?

Checkpoint 4.5.14.

Use a calculator to compute \(\dfrac{12.6+7.8}{8.3-7.8}\)
Answer.
\(40.8\)

Subsection 4.5.6 Exponents and Roots

Square roots and exponents occupy the same position in the order of operations: after parentheses but before multiplications and divisions. Here is an amended set of guidelines for the order of operations.

Order of Operations.

  1. Perform any operations inside parentheses, or above or below a fraction bar, or inside a radical.
  2. Compute all powers and roots.
  3. Perform all multiplications and divisions in order from left to right.
  4. Perform all additions and subtractions in order from left to right.

Example 4.5.15.

Simplify.
  1. \(\displaystyle 4 + 3\sqrt{49}\)
  2. \(\displaystyle \dfrac{6 + \sqrt{64}}{9-\sqrt{4}}\)

Solution.

  1. \begin{align*} 4 \amp + 3\sqrt{49} \amp \amp \blert{\text{Compute the square root.}}\\ \amp = 4 + 3(7) \amp \amp \blert{\text{Multiply.}}\\ \amp = 4 + 21 = 25 \amp \amp \blert{\text{Add.}} \end{align*}
  2. \begin{align*} \amp \dfrac{6 + \sqrt{64}}{9-\sqrt{4}} \amp \amp \blert{\text{Compute the square roots.}}\\ \amp = \dfrac{6+8}{9-2} \amp \amp \blert{\text{Simplify above and below the fraction bar.}}\\ \amp = \dfrac{14}{7} = 2 \end{align*}

Checkpoint 4.5.16.

Simplify \(\sqrt{9+16}\)
Answer.
\(5\)

Note 4.5.17.

Be careful! In Checkpoint 4.5.16, did you add 9 + 16 to get 25 before you took the square root? Look at Step 1 of the order of operations: Perform any operations ... or inside a radical. In other words,
\begin{equation*} \sqrt{9+16} \hphantom{0000} \text{is not equal to} \hphantom{0000} \sqrt{9} + \sqrt{16} \end{equation*}
You can also use your calculator to find approximations for expressions that involve radicals. Let’s agree to round all approximate values to three decimal places, unless otherwise specified.

Example 4.5.18.

Simplify \(~5 + 2\sqrt{7}\)

Solution.

The order of operations tells us how to perform the calculations in this order:
  1. Compute \(\sqrt{7}\)
  2. Multiply \(\sqrt{7}\) times 2
  3. Add 5 to the result
If we enter the calculations in exactly this order, we will be sure to get the correct result. Enter:
\begin{equation*} 7 \hphantom{0} \boxed{\sqrt{\hphantom{0}}} \hphantom{0} \boxed{\times} \hphantom{0} 2 \hphantom{0} \boxed{+} \hphantom{0} 5 \hphantom{0} \boxed{=} \end{equation*}
The calculator displays the result, 10.29150262. Rounding to three places gives 10.292. However, most scientific calculators know the order of operations and will automatically perform the operations in the correct order. Test your calculator by entering
\begin{equation*} 5 \hphantom{0} \boxed{+} \hphantom{0} 2 \hphantom{0} \boxed{\times} \hphantom{0} 7 \hphantom{0} \boxed{\sqrt{\hphantom{0}}} \hphantom{0} \boxed{=} \end{equation*}
If your calculator knows the order of operations, you should get the same result as before.

Checkpoint 4.5.19.

Approximate \(6 - \sqrt{5}\) to three decimal places.
Answer.
\(3.764\)

Subsection 4.5.7 Vocabulary

  • fraction bar
  • order of operations

Exercises 4.5.8 Practice 4.5

Exercise Group.

In Problems 1-12, simplify. Use the order of operations.
1.
\(2(4) - 3\)
2.
\(4(3) - 5\)
3.
\(2 + 4 \cdot 3\)
4.
\(9 - 3 \cdot 2\)
5.
\(25 - 3(6.4)\)
6.
\(7 + 2(18.9)\)
7.
\(24 \div 6 + 2\)
8.
\(40 \div 10 - 2\)
9.
\(80 - 56 \div 8\)
10.
\(70 - 35 \div 7\)
11.
\(18 + \dfrac{18}{3}\)
12.
\(12 - \dfrac{36}{4}\)

Exercise Group.

In Problems 13-22, simplify. Notice the difference between (a) and (b) in each pair of expressions.
13.
  1. \(\displaystyle 8 + 2 \cdot 5\)
  2. \(\displaystyle (8 + 2) \cdot 5\)
14.
  1. \(\displaystyle 10 - 6 \div 2\)
  2. \(\displaystyle (10 - 6) \div 2\)
15.
  1. \(\displaystyle \dfrac{24}{2 + 6}\)
  2. \(\displaystyle \dfrac{24}{2} + 6\)
16.
  1. \(\displaystyle \dfrac{8 + 4}{2}\)
  2. \(\displaystyle 8 + \dfrac{4}{2}\)
17.
  1. \(\displaystyle (9 - 4) - 3\)
  2. \(\displaystyle 9 - (4 - 3)\)
18.
  1. \(\displaystyle 27 \div (9 \div 3)\)
  2. \(\displaystyle (27 \div 9) \div 3\)
19.
  1. \(\displaystyle 6 \cdot 8 - 6\)
  2. \(\displaystyle 6(8 - 6)\)
20.
  1. \(\displaystyle 24 \div 3 \cdot 2\)
  2. \(\displaystyle 24 \div (3 \cdot 2)\)
21.
  1. \(\displaystyle \dfrac{36}{6(3)}\)
  2. \(\displaystyle \dfrac{36}{6}(3)\)
22.
  1. \(\displaystyle 20 - 8 + 2\)
  2. \(\displaystyle 20 - (8 + 2)\)

Exercise Group.

For Problems 23-26, write each phrase as a mathematical expression.
23.
  1. Three times the sum of 5 and 8.
  2. The sum of 3 times 5 and 8.
24.
  1. Find the sum of 18 and 30, then divide by 6.
  2. Add to 18 the quotient of 30 divided by 6.
25.
  1. Subtract the difference of 12 and 8 from 25.
  2. Subtract 8 from the difference of 25 and 12.
26.
  1. Find the difference of 34 and 7, then multiply by 3.
  2. Multiply 7 times 3, then subtract the product from 34.

Exercise Group.

For Problems 27-48, simplify according to the order of operations.
27.
\(45 - 24 \div 4(3)\)
28.
\(100 - 75 \div 25(3)\)
29.
\(84 - 2(5)(6)\)
30.
\(5 + 3(4)(10)\)
31.
\(18 \cdot 5 - 3 \cdot 12\)
32.
\(24 \cdot 5 + 5 \cdot 13\)
33.
\(3(8.2) - 6(2.1)\)
34.
\(5(1.3) + 4(2.2)\)
35.
\(2 + 3\cdot 8 - 6 + 3\)
36.
\(4 \cdot 9 - 3 \cdot 2 + 4\)
37.
\(24 \div 6 + 2 \cdot 8 \div 4\)
38.
\(48 \div 3 \cdot 4 - 20 - 8\)
39.
\(3 + 2(6 - 1)\)
40.
\(5 + 3(2 + 3)\)
41.
\(12 - 2(1 + 3)\)
42.
\(20 - 3(6 - 4)\)
43.
\(3(9 - 5) + 5(12 - 4)\)
44.
\(7(5 + 6) - 4(9 - 5)\)
45.
\(\dfrac{16}{20 - 12}\)
46.
\(\dfrac{25}{15 - 10}\)
47.
\(\dfrac{30 - 9}{12 - 9}\)
48.
\(\dfrac{6 + 18}{6 + 6}\)

Exercise Group.

For Problems 49-58, simplify according to the order of operations.
49.
\(7\sqrt{16}\)
50.
\(8\sqrt{25}\)
51.
\(3 + 5\sqrt{81}\)
52.
\(4 + 6\sqrt{64}\)
53.
\(20 - \sqrt{225}\)
54.
\(16 - \sqrt{100}\)
55.
\(18 - 2\sqrt{9}\)
56.
\(32 - 5\sqrt{36}\)
57.
\(\dfrac{6 + \sqrt{144}}{3}\)
58.
\(\dfrac{20 + \sqrt{225}}{5}\)

Exercise Group.

For Problems 59-68, simplify the pair of expressions. Are they the same?
59.
  1. \(\displaystyle \sqrt{9 + 16}\)
  2. \(\displaystyle \sqrt{9} + \sqrt{16}\)
60.
  1. \(\displaystyle 4\sqrt{25}\)
  2. \(\displaystyle \sqrt{4(25)}\)
61.
  1. \(\displaystyle \sqrt{9}\sqrt{36}\)
  2. \(\displaystyle \sqrt{9(36)}\)
62.
  1. \(\displaystyle 5 + 2\sqrt{64}\)
  2. \(\displaystyle (5 + 2)\sqrt{64}\)
63.
  1. \(\displaystyle \sqrt{\dfrac{16}{49}}\)
  2. \(\displaystyle \dfrac{\sqrt{15}}{\sqrt{49}}\)
64.
  1. \(\displaystyle \sqrt{100} - \sqrt{64}\)
  2. \(\displaystyle \sqrt{100 - 64}\)
65.
  1. \(\displaystyle \dfrac{\sqrt{81}}{4}\)
  2. \(\displaystyle \sqrt{\dfrac{81}{4}}\)
66.
  1. \(\displaystyle 10 - 3\sqrt{144}\)
  2. \(\displaystyle (10 - 3)\sqrt{144}\)
67.
  1. \(\displaystyle \sqrt{4\sqrt{81}}\)
  2. \(\displaystyle \sqrt{4}\sqrt{81}\)
68.
  1. \(\displaystyle \sqrt{4 + \sqrt{25}}\)
  2. \(\displaystyle \sqrt{4} + \sqrt{25}\)

Exercise Group.

For Problems 69-74, use a calculator to simplify the expression. Round your answers to two decimal places.
69.
\(\dfrac{12.8 + 24.6}{3.5}\)
70.
\(\dfrac{45.2 - 16.3}{2.4}\)
71.
\(\dfrac{38}{79-24}\)
72.
\(\dfrac{56}{21 + 51}\)
73.
\(\dfrac{156 - 36.7}{2.8(7.4)}\)
74.
\(\dfrac{64.5 + 59.1}{0.2(43)}\)
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