2.3: Order of Operations
Introduction
Try It
1) You have just won [latex]\$50,000[/latex] in a contest. Congratulations! But before you can claim it, you are required to answer a mathematical skill-testing question, and no calculators are permitted. After you hand over your winning ticket to the redemption agent, she hands you your time-limited skill-testing question: [latex]2\times5+30\div5[/latex]. What does question this equal?
Solution
If you figured out the solution is [latex]16[/latex], you are on the right path. If you thought it was something else, this is a great time to review order of operations.
The Symbols
While some mathematical operations such as addition use a singular symbol (+), there are other operations, like multiplication, for which multiple representations are acceptable. With the advent of computers, even more new symbols have crept into mathematical symbology. The table below lists the various mathematical operations and the corresponding mathematical symbols you can use for them.
| Mathematical Operation | Symbol or Appearance | Comments |
|---|---|---|
| Brackets | [latex]()[/latex] or [latex][][/latex] or [latex]\{\}[/latex] | In order, these are known as round, square, and curly brackets. |
| Exponents | [latex]2^3[/latex] or [latex]2\hat{}3[/latex] |
|
| Multiplication | [latex]\times[/latex] or [latex]\ast[/latex] or [latex]\cdot[/latex] or [latex]2(2)[/latex] or [latex](2)(2)[/latex] | In order, these are known as times, star, or bullet. The last two involve implied multiplication, which is when two terms are written next to each other joined only by brackets. |
| Division | [latex]/[/latex] or [latex]\div[/latex] or [latex]\frac42[/latex] | In order, these are known as the slash, divisor, and divisor line. Note in the last example that the division is represented by the horizontal line. |
| Addition | [latex]+[/latex] | There are no other symbols. |
| Subtraction | [latex]−[/latex] | There are no other symbols. |
You may wonder if the different types of brackets mean different things. Although mathematical fields like calculus use specialized interpretations for the different brackets, business math uses all the brackets to help the reader visually pair up the brackets. Consider the following two examples:
Example 1: [latex]3 × (4 / (6 - (2 + 2)) + 2)[/latex]
Example 2: [latex]3 × [4 / \{6 - (2 + 2)\} + 2][/latex]
Notice that in the second example you can pair up the brackets much more easily, but changing the shape of the brackets did not change the mathematical expression. This is important to understand when using a calculator, which usually has only round brackets. Since the shape of the bracket has no mathematical impact, solving example 1 or example 2 would involve the repeated usage of the round brackets.
BEDMAS
In the section opener, your skill-testing question was [latex]2\times5+30\div5[/latex]. Do you just solve this expression from left to right, or should you start somewhere else? To prevent any confusion over how to resolve these mathematical operations, there is an agreed-upon sequence of mathematical steps commonly referred to as BEDMAS.
HOW TO
Use BEDMAS
BEDMAS is an acronym for Brackets, Exponents, Division, Multiplication, Addition, and Subtraction.
Step 1: Brackets must be resolved first. As brackets can be nested inside of each other, you must resolve the innermost set of brackets first before proceeding outwards to the next set of brackets. When resolving a set of brackets, you must perform the mathematical operations within the brackets by following the remaining steps in this model (EDMAS). If there is more than one set of brackets but the sets are not nested, work from left to right and top to bottom.
Step 2: If the expression has any exponents, you must resolve these next. Remember that an exponent indicates how many times you need to multiply the base against itself. For example, [latex]2^3=2\times2\times2[/latex]. More review of exponents is found in Section 2.5.
Step 3: The order of appearance for multiplication and division does not matter. However, you must resolve these operations in order from left to right and top to bottom as they appear in the expression.
Step 4: The last operations to be completed are addition and subtraction. The order of appearance doesn’t matter; however, you must complete the operations working left to right through the expression.
Key Takeaway
Before proceeding with the Texas Instruments BAII Plus calculator, you must change some of the factory defaults, as explained in the table below. To change the defaults, open the Format window on your calculator. If for any reason your calculator is reset (either by removing the battery or pressing the reset button), you must perform this sequence again.
| Buttons Pushed | Calculator Display | What It Means |
|---|---|---|
| 2nd Format | DEC=2.00 | You have opened the Format window to its first setting. DEC tells your calculator how to round the calculations. In business math, it is important to be accurate. Therefore, we will set the calculator to what is called a floating display, which means your calculator will carry all of the decimals and display as many as possible on the screen. |
| 9 Enter | DEC=9. | The floating decimal setting is now in place. Let us proceed. |
| ↓ | DEG | This setting has nothing to do with business math and is just left alone. If it does not read DEG, press 2nd Set to toggle it. |
| ↓ | US 12-31-1990 | Dates can be entered into the calculator. North Americans and Europeans use slightly different formats for dates. Your display is indicating North American format and is acceptable for our purposes. If it does not read US, press 2nd Set to toggle it. |
| ↓ | US 1,000 | In North America it is common to separate numbers into blocks of 3 using a comma. Europeans do it slightly differently. This setting is acceptable for our purposes. If your display does not read US, press 2nd Set to toggle it. |
| ↓ | Chn | There are two ways that calculators can solve equations. This is known as the Chain method, which means that your calculator will simply resolve equations as you punch it in without regard for the rules of BEDMAS. This is not acceptable and needs to be changed. |
| 2nd Set | AOS | AOS stands for Algebraic Operating System. This means that the calculator is now programmed to use BEDMAS in solving equations. |
| 2nd Quit | 0. | Back to regular calculator usage. |
Also note that on the BAII Plus calculator you have two ways to key in an exponent:
- If the exponent is squaring the base (e.g., [latex]3^2[/latex]), press [latex]3[/latex] [latex]x^2[/latex]. It calculates the solution of [latex]9[/latex].
- If the exponent is anything other than a [latex]2[/latex], you must use the [latex]y^x[/latex] button. For [latex]2^3[/latex], you press [latex]2[/latex] [latex]y^x[/latex] [latex]3 =[/latex]. It calculates the solution of [latex]8[/latex].
Things to Watch For
Negative Signs
Remember that mathematics use both positive numbers (such as [latex]+3[/latex]) and negative numbers (such as [latex]−3[/latex]). Positive numbers do not need to have the [latex]+[/latex] sign placed in front of them since it is implied. Thus [latex]+3[/latex] is written as just [latex]3[/latex]. Negative numbers, though, must have the negative sign placed in front of them. Be careful not to confuse the terminology of a negative number with a subtraction or minus sign. For example, [latex]4 + (−3)[/latex] is read as “four plus negative three” and not “four plus minus three.” To key a negative number on a calculator, enter the number first followed by the [latex]±[/latex] button, which switches the sign of the number.
Horizontal Divisor Line
One of the areas in which people make the most mistakes involves the “hidden brackets.” This problem almost always occurs when the horizontal line is used to represent division. Consider the following mathematical expression:
[latex](4 + 6)\div(2 + 3)[/latex]
If you rewrite this expression using the horizontal line to represent the divisor, it looks like this:
[latex]\begin{align*}\frac{4+6}{2+3}\end{align*}[/latex]
Notice that the brackets disappear from the expression when you write it with the horizontal divisor line because they are implied by the manner in which the expression appears. Your best approach when working with a horizontal divisor line is to reinsert the brackets around the terms on both the top and bottom. Thus, the expression looks like this:
[latex]\begin{align*}\frac{(4+6)}{(2+3)}\end{align*}[/latex]
Employing this technique will ensure that you arrive at the correct solution, especially when using calculators.
Paths to Success
Hidden and Implied Symbols
If there are hidden or implied symbols in the expressions, your first step is to reinsert those hidden symbols in their correct locations. In the example below, note how the hidden multiplication and brackets are reinserted into the expression: [latex]4\left[\frac{3+2^2\times3}{(2+8)\div2}\right][/latex] transforms into [latex]4\times\left[\frac{\{3+2^2\times3\}}{\{(2+8)\div2\}}\right][/latex].
Once you have reinserted the symbols, you are ready to follow the BEDMAS model.
Calculators are not programmed to be capable of recognizing implied symbols. If you key in “[latex]3(4 + 2)[/latex]” on your calculator, failing to input the multiplication sign between the “[latex]3[/latex]” and the “[latex](4 + 2)[/latex],“ you get a solution of [latex]6[/latex]. Some calculators ignore the “[latex]3[/latex]” since they don’t know what mathematical operation to perform on it. To have your calculator solve the expression correctly, you must punch the equation through as “[latex]3 \times (4 + 2) =[/latex]”. This produces the correct answer of [latex]18[/latex].
Simplifying Negatives
If your question involves positive and negative numbers, it is sometimes confusing to know what symbol to put when simplifying or solving. Remember these two rules:
Rule #1: A pair of the same symbols is always positive.
Thus “[latex]4+(+3)[/latex]” and “[latex]4−(−3)[/latex]” both become “[latex]4+3[/latex].”
Rule #2: A pair of the opposite symbols is always negative.
Thus “[latex]4+(−3)[/latex]” and “[latex]4−(+3)[/latex]” both become “[latex]4–3[/latex].”
A simple way to remember these rules is to count the total sticks involved, where a “[latex]+[/latex]” sign has two sticks and a “[latex]-[/latex]” sign has one stick. If you have an odd number of total sticks, the outcome is a negative sign. If you have an even number of total sticks, the outcome is a positive sign.
Note the following examples:
- [latex]4+(−3) = 3[/latex] total sticks is odd and therefore simplifies to negative [latex]4−3 = 1[/latex]
- [latex](−2) \times (−2)=2[/latex] total sticks is even and therefore simplifies to positive [latex](−2) \times (−2)=+4[/latex]
Example 2.3.1
Evaluate each of the following expressions:
- [latex]2\times5+30\div5[/latex]
- [latex]{(6+3)}^2+18\div2[/latex]
- [latex]\begin{align*}4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{(2+8)\div2\}}\right]\end{align*}[/latex]
Solution
a.
Step 1: B – Are there brackets? No.
Step 2: E – Are there exponents? No.
Step 3: D – Resolve the division.
[latex]{\color{black}{30}}{\color{black}{\div}}{\color{black}{5}}{\color{black}{=}}{\color{red}{6}}[/latex]
The expression now looks like this:
[latex]2\times5+{\color{red}{6}}[/latex]
Step 4: M – Resolve the multiplication.
[latex]2\times5={\color{blue}{10}}[/latex]
The expression becomes:
[latex]{\color{blue}{10}}+{\color{red}{6}}[/latex]
Step 5: A – Perform the remaining addition.
[latex]{\color{blue}{10}}+{\color{red}{6}}{\;}{\color{black}{=}}{\;}{\color{black}{16}}[/latex]
Step 6: S – There is no subtraction, so we state the final solution.
[latex]16[/latex]
b.
Step 1: B – Are there brackets? Yes, start with the brackets.
[latex]{{(6+3)}}^{2}{\;}{=}{\;}{{({\color{red}{9}})}}^{2}[/latex]
The expression now looks like this:
[latex]{{({\color{red}{9}})}}^{2}+18\div2[/latex]
Step 2: E – Are there exponents? Yes, resolve exponents next.
[latex]{{({\color{red}{9}})}}^{\color{blue}{2}}{=}{\;}{\color{blue}{81}}[/latex]
The expression now looks like this:
[latex]{\color{blue}{81}}{+}{18}{\div}{2}[/latex]
Step 3: D – Perform the division.
[latex]{\color{blue}{81}}{+}{\color{purple}{18}}{\color{purple}{\div}}{\color{purple}{2}}{\;}{=}{\;}{\color{purple}{9}}[/latex]
The expression now looks like this:
[latex]{\color{blue}{81}}{+}{\;}{\color{purple}{9}}[/latex]
Step 4: M – Is there multiplication? No.
Step 5: A – Perform the remaining addition.
[latex]{\color{blue}{81}}{+}{\;}{\color{purple}{9}}{\;}{=}{\;}{90}[/latex]
Step 6: S – There is no subtraction, so we state the final solution.
[latex]90[/latex]
c.
Step 1: B – Are there brackets?
Yes, start with the innermost set of brackets and perform EDMAS.
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{{\color{red}{(}}{\color{red}{2}}{\color{red}{+}}{\color{red}{8}}{\color{red}{)}}\div2\}}\right]\;=\;4\;\times\;\left[\frac{\{3+2^2\times3\}}{\{{\color{red}{10}}\div2\}}\right]\end{align*}[/latex]
Now solve the curly brackets, starting with the top. Perform EDMAS.
Step 2: E – Are there exponents? Yes, resolve exponents next.
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{blue}{2}}^{\color{blue}{2}}{\color{blue}{=}}{\color{blue}{4}}\times3\}}{\{{\color{red}{10}}\div2\}}\right]\end{align*}[/latex]
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{blue}{4}}\times3\}}{\{{\color{red}{10}}\div2\}}\right]\end{align*}[/latex]
Step 3: D – Perform the division.
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{blue}{4}}\times3\}}{\{{\color{purple}{10}}{\color{purple}{\div}}{\color{purple}{2}}{=}{\color{purple}{5}}\}}\right]\end{align*}[/latex]
We no longer need the curly brackets on the bottom, so they are dropped.
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{blue}{4}}\times3\}}{\color{purple}{5}}\right]\end{align*}[/latex]
Step 4: M – Perform multiplication on the top.
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{green}{4}}{\color{green}{\times}}{\color{green}{3}}{=}{\color{green}{12}}\}}{\color{purple}{5}}\right]\end{align*}[/latex]
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{\{3+{\color{green}{12}}\}}{\color{purple}{5}}\right]\end{align*}[/latex]
Step 5: A – Perform the remaining addition.
We no longer need the curly brackets on the top, so they are dropped.
[latex]\begin{align*}4\;\times\;\left[\frac{\{{\color{chocolate}{3}}{\color{chocolate}{+}}{\color{chocolate}{12}}{=}{\color{chocolate}{15}}\}}{\color{purple}5}\right]\end{align*}[/latex]
The expression now looks like this:
[latex]\begin{align*}4\;\times\;\left[\frac{\color{chocolate}{15}}{\color{purple}{5}}\right]\end{align*}[/latex]
Step 6: Repeating the division, we solve the fraction.
[latex]\begin{eqnarray*}&\;&4\;\times\;\left[{\color{purple}{\frac{15}5}}\right]\\[1ex]&=&4\;\times\;{\color{purple}{3}}\\\end{eqnarray*}[/latex]
We no longer need the square brackets so they are dropped here.
Step 7: The last step is to perform multiplication.
[latex]4\;\times\;{\color{purple}{3}}{\;}{=}{\;}{12}[/latex]
Step 6: S – There is no subtraction, so we state the final solution.
[latex]12[/latex]
Section 2.3 Exercises
Solve the following.
Hint: If a question involves money, round the answers to the nearest cent.
Mechanics
- [latex]81\div 27+3\times 4[/latex]
- [latex]100\div (5\times 4–5\times 2)[/latex]
- [latex]3^3–9+(1+7\times 3)[/latex]
- [latex](6+3)^2–17\times 3+70[/latex]
- [latex]100–(4^2+3)–(3+9\times 3–4)[/latex]
Solutions
- [latex]15[/latex]
- [latex]10[/latex]
- [latex]40[/latex]
- [latex]100[/latex]
- [latex]55[/latex]
Applications
- [latex][(7^2−\{−41\})−5\times 2]\div (80\div 10)[/latex]
- [latex]\begin{align*}\$1,000\left(1+0.09\times\frac{88}{365}\right)\end{align*}[/latex]
- [latex]3[\$2,000(1 + 0.003)^8]+\$1,500[/latex]
- [latex]\begin{align*}\frac{\$20,000}{1+0.07\times\frac{7}{12}}\end{align*}[/latex]
- [latex]4\times [(5^2+15)^2\div (13^2–9)]^2[/latex]
- [latex]\begin{align*}\$500\left[\frac{\left(1+0.00875\right)^{43}-1}{0.00875}\right]\end{align*}[/latex]
- [latex]\begin{align*}\$1,000\left(1+\frac{0.12}{6}\right)^{15}\end{align*}[/latex]
Solutions
- [latex]10[/latex]
- [latex]\$1,021.70[/latex]
- [latex]\$7,645.52[/latex]
- [latex]\$19,215.37[/latex]
- [latex]400[/latex]
- [latex]\$25,967.38[/latex]
- [latex]\$1,345.87[/latex]
Challenge, Critical Thinking, & Other Applications
- [latex]\begin{align*}\left(\frac{\$2,500}{1+0.10}\right)+\left(\frac{\$7,500}{\left(1+0.10\right)^2}\right)+\left(\frac{-\$1,500}{\left(1+0.10\right)^3}\right)+\left(\frac{-\$2,000}{\left(1+0.10\right)^4}\right)\end{align*}[/latex]
- [latex]\begin{align*}\$175,000\left(1+0.07\right)^{15}+\$14,000\left[\frac{\left(1+0.07\right)^{20}-1}{0.07}\right]\end{align*}[/latex]
- [latex]\begin{align*}\$5,000\left[\frac{\left\{1+\left[\left(1+0.08\right)^{0.5}-1\right]\right\}^{75}-1}{\left(1+0.08\right)^{0.5}-1}\right]\end{align*}[/latex]
- [latex]\begin{align*}\$800\left[\frac{\left(1+0.07\right)^{20}-\left(1+0.03\right)^{20}}{0.07-0.03}\right]\end{align*}[/latex]
- [latex]\begin{align*}\$60,000\left(1+0.0058\right)^{80}−\$450\left[\frac{\left(1+0.0058\right)^{25}-1}{0.0058}\right]\end{align*}[/latex]
- [latex]\begin{align*}\left(\frac{0.08}{2}\right)\$1,000\left[\frac{1-\frac{1}{\left\{1+\left(\frac{0.08}{2}\right)\right\}^{16}}}{\left(\frac{0.08}{2}\right)}\right]+\$1,000\left[1+\left(\frac{0.08}{2}\right)\right]^{16}\end{align*}[/latex]
- [latex]\begin{align*}\$1,475\left[\frac{\left(1+\frac{0.06}{4}\right)^{16}-1}{\frac{0.06}{4}}\right]\end{align*}[/latex]
- [latex]\begin{align*}\$6,250\left(1+0.0525\right)^{10}+\$325\left[\frac{\left(1+0.0525\right)^{10}-1}{0.0525}\right]\end{align*}[/latex]
Solutions
- [latex]\$5,978.08[/latex]
- [latex]\$1,056,767.41[/latex]
- [latex]\$2,156,764.06[/latex]
- [latex]\$41,271.46[/latex]
- [latex]\$83,228.60[/latex]
- [latex]\$2,339.07[/latex]
- [latex]\$26,450.25[/latex]
- [latex]\$14,561.43[/latex]
Attribution
“2.1: Order of Operations” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.
