1.11: PEMDAS with Positive Real Numbers
Rosa walked into Math class and saw the following question on the board.
\begin{align*}\boxed{6+12 \div 2 \times 3+1}\end{align*}
Her teacher, Ms. Black, directed the class to evaluate the mathematical expression. When the students had completed the task, Ms. Black then asked several students to put their work on the board. Here are the results:
\begin{align*}\boxed{6+12 \div 2 \times 3+1}\end{align*}
Which answer is correct?
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Khan Academy Introduction to Order of Operations
Guidance
\begin{align*}\boxed{6+12 \div 2 \times 3+1}\end{align*}
To avoid confusion in evaluating mathematical expressions like the one shown above, mathematicians have adopted a standard order of operations for arithmetic calculations. This adopted standard of operations consists of the following rules:
- Perform any calculations shown inside parenthesis first.
- Perform any calculations with terms that have exponents.
- Perform all multiplication and division, in the order they occur, working from left to right.
- Perform all addition and subtraction, in the order they occur, working from left to right.
If you look at the letters above that have been highlighted, you will see that they form the word PEMDAS – parenthesis, exponents, multiplication, division, addition, subtraction. The word PEMDAS serves as a method for you to remember the order in which to perform the arithmetic calculations.
Example A
Perform the following calculations, using PEMDAS.
\begin{align*}360 \div (18+6\times 2)-2\end{align*}
\begin{align*}360 \div(18+{\color{blue}12})-2 \quad {\color{blue}(6 \times 2=12)}\end{align*}
When performing the calculations in parenthesis, follow the rules for order of operations.
\begin{align*}360 \div {\color{blue}30} -2 \quad {\color{blue}(18+12=30)}\end{align*}
When the calculations in parenthesis have been completed, the parenthesis is no longer necessary.
\begin{align*}{\color{blue}12}-2 \quad {\color{blue}(360 \div 30 =12)}\end{align*}
There are no exponents in this problem. The next step is to perform the division.
\begin{align*}= {\color{blue}10 \quad (12-2=10)}\end{align*}
The final step is to subtract 2 from 12. The final answer is 10.
Example B
\begin{align*}\left(\frac{3}{4}+\frac{1}{6}\right) \times (5 \times 3^2-5)\end{align*}
\begin{align*}{\color{blue}\frac{11}{12}} \times (5 \times 3^2-5) \quad \left(\left(\frac{3}{3}\right)\frac{3}{4}=\frac{9}{12}\right) \quad \left(\left(\frac{2}{2}\right)\frac{1}{6}=\frac{2}{12}\right) \quad {\color{blue}\left(\frac{9}{12}+\frac{2}{12}=\frac{11}{12}\right)}\end{align*}
There are two sets of parenthesis. Work from left to right in the first set of parenthesis. Calculate the sum of the two fractions.
\begin{align*}\frac{11}{12} \times (5 \times {\color{blue}9} -5) \quad {\color{blue}(3^2=3 \times 3=9)}\end{align*}
In the second set of parenthesis, simplify three that has an exponent of 2. The order of operations must be applied inside of the parenthesis.
\begin{align*}\frac{11}{12} \times ({\color{blue}45}-5) \quad {\color{blue}(5 \times 9=45)}\end{align*}
Perform the multiplication in the parenthesis.
\begin{align*}\frac{11}{12} \times {\color{blue}40} \quad {\color{blue}(45-5=40)}\end{align*}
Complete the calculations in the parenthesis by doing the subtraction.
\begin{align*}={\color{blue}36 \frac{2}{3} \quad \left(\frac{11}{12} \times 40 = \frac{440}{12}=36 \frac{2}{3}\right)}\end{align*}
The final step is to multiply the answer of the first parenthesis by the answer of the second parenthesis.
Example C
\begin{align*}(1+6)^2 - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)\end{align*}
\begin{align*}({\color{blue}7})^2 - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)\end{align*}
Add the numbers in the first parenthesis.
\begin{align*}{\color{blue}49} - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)\end{align*}
The calculation with the exponent must be completed by multiplying seven times itself.
\begin{align*}49- \frac{2+4 \times 12}{18-4 \times 2}+ {\color{blue}9}\end{align*}
Perform the division in the second parenthesis.
\begin{align*}49 - \frac{2+{\color{blue}48}}{18-{\color{blue}8}}+9\end{align*}
Remember that the line of a fraction means divide. Before the division can be completed, you must obtain an answer for the calculations in the numerator and in the denominator. PEMDAS must be applied when doing the calculations.
\begin{align*}49 - \frac{{\color{blue}50}}{{\color{blue}10}}+9\end{align*}
Add the numbers in the numerator. Subtract the numbers in the denominator.
\begin{align*}49-{\color{blue}5}+9\end{align*}
Complete the division.
\begin{align*}={\color{blue}53}\end{align*}
Working from left to right, in the order they occur, perform the addition and subtraction.
Example D
Although the numbers in the following statement are decimals, the standard order of operations does not change. PEMDAS is applied to calculate the answer.
\begin{align*}& 6.12+8.6\times0.9-(10.26\div3.8)\\ & 6.12+8.6\times 0.9- {\color{blue}2.7} \quad {\color{blue}(10.26\div3.8=2.7)} \ \text{Parenthesis}\\ & 6.12+{\color{blue}7.74}-2.7 \quad {\color{blue}(8.6\times0.9=7.74)} \ \text{Multiplication}\\ & {\color{blue}13.86}-2.7 \quad {\color{blue}(6.12+7.74=13.86)} \ \text{Addition}\\ & {\color{blue}=11.86 \quad (13.86-2.7=11.6)} \ \text{Subtraction}\end{align*}
Example E
The order of operations is often done by writing the statement with variables and then substituting the given values of each variable into the statement. The calculations are then performed when all of the value substitutions have been made.
If \begin{align*}m=2\end{align*} and \begin{align*}n=3\end{align*}, evaluate
\begin{align*}m^2+3n-7\end{align*}
The first step is to substitute the values into the given statement.
\begin{align*}m^2+3n-7\end{align*}
\begin{align*}(2)^2+3 \times 3-7\end{align*}
Follow the steps for PEMDAS and evaluate.
\begin{align*}& (2)^2+3 \times 3-7\\ & {\color{blue}4}+3\times 3-7 \ \text{Exponents}\\ & 4+{\color{blue}9}-7 \ \text{Multiplication}\\ & {\color{blue}13}-7 \ \text{Addition}\\ & = {\color{blue}6} \ \text{Subtraction}\end{align*}
Concept Problem Revisited
The final solution is correct.
In all fairness to the students, Ms. Black could have minimized the confusion by writing the statement with parenthesis.
\begin{align*}\boxed{6+(12\div2\times 3)+1}\end{align*}
Vocabulary
- Parenthesis
- A parenthesis ( ), is a symbol that is used to group numbers in mathematics. The parenthesis shows where an expression starts and stops.
- PEMDAS
- The letters PEMDAS represent the standard order of operations for calculating mathematical statements.
P - Parenthesis E - Exponents M - Multiplication D - Division A - Addition S - Subtraction
Guided Practice
1. Perform the following operations using PEMDAS: \begin{align*}8\times 9+19 \div(30-11)-6\end{align*}
2. A remodeling job requires 132 square feet of countertops. Two options are being considered. The more expensive option is to use all Corian at $66 per sq ft. The less expensive option is to use 78 sq ft of granite at $56 per sq ft and 54 sq ft of laminate at $23 per sq ft. Write a mathematical statement to calculate the difference in cost between the more expensive option and the less expensive option. What is the cost difference?
3. Determine the answer to \begin{align*}\frac{12+6}{6+3}+\frac{36}{4}-(12\div 12)\end{align*} by using the rules for the standard order of operations.
Answers:
1.
\begin{align*}& 8 \times 9 + 19 \div(30-11)-6\\ & 8 \times 9 +19 \div {\color{blue}19}-6 \ \text{Do the calculations in parenthesis.}\\ & {\color{blue}72}+19\div 19-6 \ \text{Do the multiplication.}\\ & 72+{\color{blue}1}-6 \ \text{Do the division.}\\ & {\color{blue}73}-6 \ \text{Do the addition.}\\ & {\color{blue}= 67} \ \text{Do the subtraction.}\end{align*}
2.
\begin{align*}& (132 \times \$ 66)-(78 \times \$ 56 + 54 \times \$ 23)\\ & {\color{blue} \$8712}-(78 \times \$ 56+54 \times \$ 23) \ \text{Do the multiplication in the first parenthesis.}\\ & \$8712 - ({\color{blue}\$4368+\$1242}) \ \text{Do the multiplication in the second parenthesis.}\\ & \$ 8712 - {\color{blue}\$5610} \ \text{Do the addition in the second parenthesis.}\\ & = \$3102 \ \text{Do the subtraction.}\end{align*}
The first option is $3102 more than the second option.
3.
\begin{align*}& \frac{12+6}{6+3} + \frac{36}{4} - (12 \div 12)\\ & \frac{12+6}{6+3} + \frac{36}{4} - {\color{blue}1} \ \text{Do the division in the parenthesis.}\\ & \frac{12+6}{6+3}+{\color{blue}9}-1 \ \text{Do the division.}\\ & {\color{blue}\frac{18}{9}}+9-1 \ \text{Do the addition in the numerator and denominator of the fraction.}\\ & {\color{blue}2}+9-1 \ \text{Do the division.}\\ & {\color{blue}11}-1 \ \text{Do the addition}\\ & ={\color{blue}10} \ \text{Do the subtraction}\end{align*}
Problem Set
Perform the indicated calculations, using PEMDAS to determine the answer.
- \begin{align*}\frac{4^2(8+7)}{6}\end{align*}
- \begin{align*}\frac{2 \times 6}{4}(5-2)\end{align*}
- \begin{align*}\frac{15 \times 3}{5}+4(7 \times 1)-2 \times 3\end{align*}
- \begin{align*}4+27 \div 3\times 2-6\end{align*}
- \begin{align*}7^2-3 \times 2^3-5\end{align*}
For each of the following problems write a single mathematical statement to represent the problem. Then use the statement to determine the answer.
- At the beginning of the day on Monday, the cafeteria has 520 tortilla wraps. The supervisor estimates that she will need 68 wraps each day. A new shipment of 300 wraps will arrive on Thursday. Calculate the number of wraps she will have at the end of the day on Friday.
- The students enrolled in the masonry course are estimating the cost for building a stone wall and gate. They estimate that the job will require 40 hours to complete. They will need the services of two laborers and they will be paid $12 per hour. They will also need three masons who will be paid $16 per hour. The cost of the materials is $2140. The students want to make $500 profit for doing the job. What is the estimated cost of the job and the actual profit?
- Mrs. Forsythe purchased 15 scientific calculators at $19 each and received $8 credit for each of the seven regular calculators that she returned. How much money did she spend to buy the scientific calculators?
- A landscaper charged a customer $472 for labor and $85 each for eight flats of Bedding plants. What was the total cost of the job?
- A painter had a one hundred dollar bill when he went to the hardware store to purchase supplies for a job. He bought 2 quarts of white latex paint for $8 a quart and 4 gallons of white enamel paint for $19 a gallon. How much change did he receive?
If \begin{align*}a=2, b=3\end{align*} and \begin{align*}c=5\end{align*}, evaluate, using PEMDAS to determine the answer.
- \begin{align*}6a-3b+4c\end{align*}
- \begin{align*}2a^2-3a+b^2\end{align*}
- \begin{align*}3ac-2ab+bc\end{align*}
- \begin{align*}a^2+b^2+c^2\end{align*}
- \begin{align*}3a^2(4c-3b)\end{align*}
Image Attributions
Here you will learn the standard order of operations for arithmetic calculations. This standard order of operations is applied when the basic arithmetic operations (addition, subtraction, multiplication and division) are combined in a single mathematical statement. You will see how the order in which calculations are performed affect the answer.
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