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COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Boot camp: Order of Operations Peer Tutor Slides Instructor: Mr. Ethan W. Cooper, Lead Tutor © 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE EMAIL MR. COOPER AT THE FOLLWING: [email protected] Order of Operations Many equations only contain one mathematical equation. However, most of the equations in JUST 3900 contain two or more operations. Example: 2 + 2 Example: 22 + 4 ∗ 32 /2 2 In such equations it is important to perform the operations in the proper order. Simply working from left to right will not yield the correct answer. LEFT TO RIGHT: WRONG!!! ORDER OF OPERATIONS: CORRECT!!! Order of Operations Operations should be performed in the proper order and can be remembered by the following acronym: Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Please Excuse My Dear Silly Aunt Sally (P) (E) (M) (D) (S) (A) (S) Parentheses Exponents Multiplication Division Summation Addition Subtraction This operation was not taught in high school. Order of Operations: Then and Now High School Please () Excuse X2 My Dear (*) (÷) Aunt Sally (+) (-) JUST 3900 Please () Excuse X2 My Dear (*) (÷) Silly Ʃ Aunt Sally (+) (-) Order of Operations Left to Right 22 + 4 ∗ 32 /2 4 + 4 ∗ 32 /22 8 ∗ 32 /22 242 /22 576/22 2882 82944 2 Order of Operations 22 + 4 ∗ 32 /2 2 22 + 4 ∗ 9/2 2 22 + 36/2 2 22 + 18 2 4 + 324 328 Order of Operations: Parentheses Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Any operations set off by parentheses should be done first. 2∗ 3+4 2∗7 14 Sometimes there are multiple operations within a set off parentheses. In these cases, we still follow the proper Exponents order. 2 ∗ 32 + 4 ∗ 5 2∗ 9+4∗5 2 ∗ 9 + 20 2 ∗ 29 58 Multiplication Addition Order of Operations: Parentheses Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Often times, you’ll come across equations written like this: 10+2 5+1 In such cases, we must solve the numerator, first and the denominator, second. 10+2 5+1 12 5+1 12 6 2 It’s as if it were written: 10 + 2 / 5 + 1 We do what’s in parentheses, first, working from left to right. Order of Operations: Exponents Please Excuse My Dear Silly Aunt Sally (PEMDSAS) After performing all operations written in parentheses, we move on to exponents: 22 + 4 4+4 8 Order of Operations: Exponents Please Excuse My Dear Silly Aunt Sally (PEMDSAS) When there are multiple exponents within an equation, complete them in order from left to right before moving on to the next operation: 22 + 43 + 54 4 + 43 + 54 4 + 64 + 54 4 + 64 + 625 68 + 625 693 Order of Operations: Exponents Please Excuse My Dear Silly Aunt Sally (PEMDSAS) There are times when a set of parentheses are raised to the nth power. In these cases, perform the operations in parentheses first, working from left to right: 22 + 43 − 5 + 3 2 4 + 43 − 5 + 3 2 4 + 43 − 5 + 3 2 4 + 64 − 5 + 3 2 68 − 5 + 3 2 68 − 8 2 68 − 64 4 Order of Operations: Multiplication and Division Please Excuse My Dear Silly Aunt Sally (PEMDSAS) After parentheses and exponents, multiplication and division are performed, in order, from left to right: 2 ∗ 2 + 3/3 4 + 3/3 4+1 5 Here, we multiply, first. Order of Operations: Multiplication and Division Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Another example: This time we divide, first. 20 + 5 4+6∗3+ 4 + 18 + 4 + 18 + 3 − 10 22 + 3 − 10 25 − 10 15 21 − 10 7 21 − 10 7 6∗3+ 21 7 − 10 Remember two things: 1) Multiplication and division should be done after parentheses and exponents. 2) Multiplication and division should be done, in order, from left to right. Order of Operations: Multiplication and Division Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Put it all together: 25 18 + 5 3 25 18 + 5 3 18 5+ 3 2 5∗3 152 − 225 − 225 − 5 + 6 225 − 11 214 − Order of Operations: Summation Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Summation is an operation not usually covered in high school algebra classes, but it will be important to know for JUST 3900. Summation involves summing a set of numbers. Because we’re using data sets in statistics, summation will become an important operation. Values for X: 1, 2, 3, 4, 5 =1+2+3+4+5 =3+3+4+5 =6+4+5 = 10 + 5 = 15 Order of Operations: Summation Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Using the same set of numbers: 2 2 2 2 2 2 = 12 + 22 + 32 + 42 + 52 = 1 + 4 + 9 + 16 + 25 = 5 + 9 + 16 + 25 = 14 + 16 + 25 = 30 + 25 = 55 Order of Operations: Summation Please Excuse My Dear Silly Aunt Sally (PEMDSAS) And again…. 2 2 2 2 2 2 = 1+2+3+4+5 = 3+3+4+5 2 = 6+4+5 2 = 10 + 5 2 = 15 2 = 225 2 Order of Operations: Addition and Subtraction Please Excuse My Dear Silly Aunt Sally (PEMDSAS) Just like multiplication and division, addition and subtraction are performed in order from left to right. 6+5−1 11 − 1 10 Or 6−5+1 1+1 2 Order of Operations Let’s put it all together and solve the following equation: 3 + 5 − 10 + 2 8 − 10 + 2 8 − 12 2 2 6 3 −42 ∗ 5 + 2 −42 ∗ 7 16 ∗ 7 112 ∗ 5+ ∗ 5+ ∗ 5+ −42 ∗ 5 + 2 6 3 6 3 6 3