1. Verify Unit of Measure in a Multivariate Equation © 20111

2. You cant…. += ?? © 20112

3. Terminal Learning Objective Task: Determine Unit of Measure in a Multivariate Equation Condition: You are training to become an ACE with access to ICAM course handouts, readings, and spreadsheet tools and awareness of Operational Environment (OE)/Contemporary Operational Environment (COE) variables and actors Standard: with at least 80% accuracy: Describe mathematical operations using units of measure Solve unit of measure equations Describe key cost equations © 20113

4. Importance of Units of Measure You cant add apples and oranges but you can add fruit Define the Unit of Measure for a cost expression Use algebraic rules to apply mathematical operations to various Units of Measure © 20114

5. Adding If two components of the cost expression have the same unit of measure, they may be added together Example: Smoky Mountain Inn Depreciation on building$60,000 per year Maintenance persons salary $30,000 per year Cleaning persons salary $24,000 per year Real estate taxes $10,000 per year Depreciation, maintenance, cleaning, and taxes are all stated in $ per year, so they may be added to equal $124,000 per year © 20115

6. Adding If two components of the cost expression have the same unit of measure, they may be added together Example: Smoky Mountain Inn Laundry service $4.00 per person-night Food $6.00 per person-night Laundry and food are both stated in $ per person- night, so they may be added to equal $10 per person-night © 20116

7. Subtracting If two components of the cost expression have the same unit of measure, they may be subtracted Example: Selling price is $10 per widget Unit cost is $3.75 per widget Since both Selling price and Unit cost are stated in $ per widget, they may be subtracted to yield Gross Profit of $6.25 per widget © 20117

8. Dividing Per represents a division relationship and should be expressed as such Example: Cost per unit = Total $ Cost / # Units Total Cost = $10,000 # Units = 500 $10,000/500 units = $20/unit © 20118

9. Cancelling If the same Unit of Measure appears in both the numerator and denominator of a division relationship, it will cancel Example: $25 thousand 10 thousand units = $2.50/unit © 20119

10. Multiplication When multiplying different units of measure, they become a new unit of measure that is the product of the two factors Example: 10 employees * 40 hrs = 400 employee-hrs 2x * 3y = 6xy © 201110

11. Cross-Cancelling When multiplying two division expressions, common Units of Measure on the diagonal will cancel Example: Variable Cost Variable cost $4/unit * 100 units = $4 * 100 units Unit 1 = $400 © 201111

12. Factoring If the same unit of measure appears as a factor in all elements in a sum, it can be factored out Example: $4/hour + $6/hour = $/hr *(4 + 6) © 201112

13. Learning Check If two components of a cost expression have the same unit of measure, they may be either or. Which mathematical operation using two different units of measure results in a new unit of measure? © 201113

14. Proving a Unit of Measure What is the cost expression for a driving trip? The cost will be the sum of the following components: $ Gasoline + $ Insurance + $ Drivers time © 201114

15. Variables Affecting Cost of Trip All of the following items will affect our trips cost: Distance in miles (represented by x) Gas usage in miles per gallon (represented by a) Cost per gallon of gas in dollars (represented by b) Insurance cost in dollars per mile (represented by c) Drivers cost in dollars per hour (represented by e) Average speed in miles per hour (represented by d) © 201115

16. Cost of Gasoline Cost of gasoline = # gallons * $/gallon # gallons = x miles ÷ a miles/gallon When dividing fractions, invert the second fraction and multiply Cost of gasoline = x miles * gallon /a miles * b $/gallon © 201116

17. Cost of Insurance Cost of Insurance = Insurance $/mile * miles on trip Insurance $/mile = Total Insurance $ /year Total miles /year So: c $/mile * x miles © 201117

18. Cost of Drivers Time Cost of Drivers Time = # hours * $/hour # hours = x miles ÷ d miles/hour Or: x miles * hour/d miles Hours/mile * $/hour * miles on trip So: hours/d mile * e $/hour * x miles © 201118

19. Cost Expression gallon/ a miles * x miles * b $/gal + c $/mile * x miles + hours/d mile * e $/hour * x miles © 201119

20. Proving the Unit of Measure Cost of Gasoline+Cost of Insurance+Cost of Drivers Time x miles*gallon a miles *b $ gal +c $ mile *x miles+hour d miles *e $ hour *x miles © 201120

21. Proving the Unit of Measure Cost of Gasoline+Cost of Insurance+Cost of Drivers Time x miles*gallon a miles *b $ gal +c $ mile *x miles+hour d miles *e $ hour *x miles * ( Cost of Gasoline /mile + Cost of Insurance /mile + Cost of Drivers Time /mile ) x miles* ( gallon a miles * b $ gal +c $ mile + hour d miles * e $ hour ) © 201121

22. Proving the Unit of Measure x miles* ( Cost of Gasoline /mile + Cost of Insurance /mile + Cost of Drivers Time /mile ) x miles * ( gallon a miles * b $ gal +c $ mile + hour d miles * e $ hour ) x miles* ( gallon a miles * b $ gal + c $ mile + hour d miles * e $ hour ) x miles* ( b $ a miles + c $ mile + e $ d miles ) x miles* ( baba * $ mile + c * $ mile + eded * $ mile ) © 201122

23. Proving the Unit of Measure x miles* $ mile * ( Gasoline+Insurance+ Drivers Time ) x miles* $ mile * ( baba + c + eded ) x miles 1 * $ mile * ( baba + c + eded ) $ * x * ( baba + c + eded ) x miles* ( Cost of Gasoline /mile + Cost of Insurance /mile + Cost of Drivers Time /mile ) x miles* ( baba * $ mile + c * $ mile + eded * $ mile ) © 201123

24. Plugging Values into the Equation $ * x * ( baba + c + eded ) What is the cost of the trip if: The distance (x) is 300 miles The car gets 25 miles per gallon (b) The cost of a gallon of gas is $4 The insurance cost per mile (c) is $.05 The drivers cost per hour is $20 (e) The average speed is 80 miles per hour (d) © 201124

25. Plugging Values into the Equation $ * x * ( baba + c + eded ) $ * 300 * ( 4 25 +.05 + 20 80 ) What is the cost of the trip if: The distance (x) is 300 miles The car gets 25 miles per gallon (a) The cost of a gallon of gas is $4 (b) The insurance cost per mile (c) is $.05 The drivers cost per hour is $20 (e) The average speed is 80 miles per hour (d) © 201125

26. Learning Check What is the procedure when dividing by a fractional unit of measure? © 201126

27. The Value of Equations Equations represent cost relationships that are common to many organizations Examples: Revenue – Cost = Profit Total Cost = Fixed Cost + Variable Cost Beginning + Input – Output = Ending © 201127

28. Input-Output Equation Input-Output Equation Beginning + Input – Output = End If you take more water out of the bucket than you put in, what happens to the level in the bucket? © 201128

29. Applications of Input-Output Account Balances What are the inputs to the account in question? Raw materials? Work In process? Finished goods? Your checking account? What are the outputs from the account? © 201129

30. Applications of Input-Output Gas Mileage: Miles/Gallon = Miles Driven/Gallons Used Calculate Miles Driven using the odometer How do you know Gallons Used? If you always fill the tank completely, then: Beginning + Input – Output = Ending Or, chronologically: Beginning – Output + Input = Ending Full Tank – Gallons Used + Gallons Added = Full Tank – Gallons Used + Gallons Added = 0 Gallons Used = Gallons Added © 201130

31. Using the Input-Output Equation If any three of the four variables is known, it is possible to solve for the unknown The beginning balance on your credit card is $950. During the month you charge $300 and make a payment of $325. At the end of the month your balance is $940. What was the finance charge? What are the inputs? Charges and finance charge What are the outputs? Payments © 201131

32. Using the Input-Output Equation If any three of the four variables is known, it is possible to solve for the unknown The beginning balance on your credit card is $950. During the month you charge $300 and make a payment of $325. At the end of the month your balance is $940. What was the finance charge? What are the inputs? Charges and finance charge What are the outputs? Payments © 201132

33. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201133

34. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201134

35. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201135

36. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201136

37. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201137

38. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201138

39. Using the Input-Output Equation Set up the equation: Beginning + Inputs – Outputs = Ending Beg + Charges + Finance Charges – Payments = End $950 + $300 + Finance Charge – $325 = $940 $1250 + Finance Charge – $325 = $940 $925 + Finance Charge = $940 Finance Charge = $940 – $925 Finance Charge = $15 © 201139

40. Learning Check What are three useful equations that represent common cost relationships? © 201140

41. Practical Exercises © 201141