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Numerical Expression Evaluation with Grouping Symbols

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Order of Operations with Positive Real Numbers
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Rosa walked into Math class and saw the following question on the board.

\boxed{6+12 \div 2 \times 3+1}

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:

Which answer is correct?

Watch This

Khan Academy Introduction to Order of Operations

Guidance

\boxed{6+12 \div 2 \times 3+1}

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:

  1. Perform any calculations shown inside p arentheses first.
  2. Perform any calculations with terms that have e xponents.
  3. Perform all m ultiplication and d ivision, in the order they occur, working from left to right.
  4. Perform all a ddition and s ubtraction, 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 – parentheses, 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.

360 \div (18+6\times 2)-2

Solution: When performing the calculations in parentheses, follow the rules for order of operations.

360 \div(18+{\color{blue}12})-2

When the calculations in parentheses have been completed, the parentheses are no longer necessary.

360 \div {\color{blue}30} -2

There are no exponents in this problem. The next step is to perform the division.

{\color{blue}12}-2

The final step is to subtract 2 from 12. The final answer is 10.

= 10

Example B

\left(\frac{3}{4}+\frac{1}{6}\right) \times (5 \times 3^2-5)

Solution: There are two sets of parentheses. Work from left to right in the first set of parentheses.

{\color{blue}\frac{11}{12}} \times (5 \times 3^2-5)

\frac{11}{12} \times (5 \times {\color{blue}9} -5)

\frac{11}{12} \times ({\color{blue}45}-5)

\frac{11}{12} \times {\color{blue}40}

=36 \frac{2}{3}

Example C

(1+6)^2 - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)

Solution: To start, add the numbers in the first parentheses.

({\color{blue}7})^2 - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)

{\color{blue}49} - \frac{2+4 \times 12}{18-4 \times 2}+(72 \div 8)

49- \frac{2+4 \times 12}{18-4 \times 2}+ {\color{blue}9}

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.

49 - \frac{2+{\color{blue}48}}{18-{\color{blue}8}}+9

49 - \frac{{\color{blue}50}}{{\color{blue}10}}+9

49-{\color{blue}5}+9

={\color{blue}53}

Example D

6.12+8.6\times0.9-(10.26\div3.8)

Solution:

& =6.12+8.6\times 0.9- {\color{blue}2.7} \\& =6.12+{\color{blue}7.74}-2.7 \\& ={\color{blue}13.86}-2.7 \\& ={\color{blue}=11.16}

Example E

If m=2 and n=3 , evaluate m^2+3n-7 .

Solution: The first step is to substitute the values into the given statement.

m^2+3n-7

=(2)^2+3 \times 3-7

&= {\color{blue}4}+3\times 3-7 \\& =4+{\color{blue}9}-7\\& ={\color{blue}13}-7 \\& = {\color{blue}6}

Concept Problem Revisited

The final solution is correct.

Ms. Black could have minimized the confusion by writing the statement with parentheses.

\boxed{6+(12\div2\times 3)+1}

Vocabulary

Parentheses
Parentheses , ( ), are symbols that are used to group numbers in mathematics.
PEMDAS
The letters PEMDAS represent the standard order of operations for calculating mathematical statements.

P - Parentheses E - Exponents M - Multiplication D - Division A - Addition S - Subtraction

Guided Practice

1. Perform the following operations using PEMDAS: 8\times 9+19 \div(30-11)-6

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 \frac{12+6}{6+3}+\frac{36}{4}-(12\div 12) by using the rules for the standard order of operations.

Answers:

1.

& 8 \times 9 + 19 \div(30-11)-6\\& 8 \times 9 +19 \div {\color{blue}19}-6 \\& {\color{blue}72}+19\div 19-6 \\& 72+{\color{blue}1}-6 \\& {\color{blue}73}-6 \\& {\color{blue}= 67}

2. The first option is $3102 more than the second option.

& (132 \times \$ 66)-(78 \times \$ 56 + 54 \times \$ 23)\\ & {\color{blue} \$8712}-(78 \times \$ 56+54 \times \$ 23) \\& \$8712 - ({\color{blue}\$4368+\$1242}) \\& \$ 8712 - {\color{blue}\$5610} \\& = \$3102

3.

& \frac{12+6}{6+3} + \frac{36}{4} - (12 \div 12)\\& \frac{12+6}{6+3} + \frac{36}{4} - {\color{blue}1} \\& \frac{12+6}{6+3}+{\color{blue}9}-1 \\& {\color{blue}\frac{18}{9}}+9-1 \\& {\color{blue}2}+9-1 \\& {\color{blue}11}-1 \\& ={\color{blue}10}

Problem Set

Perform the indicated calculations, using PEMDAS to determine the answer.

  1. \frac{4^2(8+7)}{6}
  2. \frac{2 \times 6}{4}(5-2)
  3. \frac{15 \times 3}{5}+4(7 \times 1)-2 \times 3
  4. 4+27 \div 3\times 2-6
  5. 7^2-3 \times 2^3-5

For each of the following problems write a single mathematical statement to represent the problem. Then use the statement to determine the answer.

  1. 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.
  2. 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. What is the estimated cost of the job?
  3. 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?
  4. 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?
  5. 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 a=2, b=3 and c=5 , evaluate, using PEMDAS to determine the answer.

  1. 6a-3b+4c
  2. 2a^2-3a+b^2
  3. 3ac-2ab+bc
  4. a^2+b^2+c^2
  5. 3a^2(4c-3b)

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