Verify Unit of Measure in a Multivariate Equation Intermediate Cost Analysis and Management 2.2

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Terminal Learning Objective Action: 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 

Importance of Units of Measure You can’t 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

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 person’s salary $30,000 per year Cleaning person’s 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

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

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

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

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

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

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

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)

LSA #1 Check on Learning Q1. If two components of a cost expression have the same unit of measure, they may be either: A1. ______ or _________ Q2. Which mathematical operation using two different units of measure results in a new unit of measure? A1. Q1. If two components of a cost expression have the same unit of measure, they may be either: A1. ______ or _________ Q2. Which mathematical operation using two different units of measure results in a new unit of measure? A1. Multiplication – results in a new unit of measure that is the product of the two different units of measure.

LSA #1 Summary During this lesson, we discussed the important mathematical computations (adding, subtracting, multiplying, dividing) as it pertains to Units of Measure.

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 + $ Driver’s time

Variables Affecting Cost of Trip All of the following items will affect our trip’s 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) Average speed in miles per hour (represented by d) Driver’s cost in dollars per hour (represented by e)

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

Cost of Gasoline Example Your car gets 20 miles/gallon, gas costs $4/gallon and you drive 100 miles: 100 miles * gallon/20 miles * $4/gallon (100 miles * gallon/20 miles) * $4/gallon (100 * gallon/20) * $4/gallon (100 5* gallon)/20 * $4/gallon (5* gallon)* $4/gallon 5 gallons * $4/gallon 5 * $4 = $20

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

Cost of Driver’s Time Cost of Driver’s 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

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

Proving the Unit of Measure Cost of Gasoline + Cost of Insurance Cost of Driver’s Time x miles * gallon a miles b $ gal c $ mile hour d miles e $

Proving the Unit of Measure Cost of Gasoline + Cost of Insurance Cost of Driver’s Time x miles * gallon a miles b $ gal c $ mile hour d miles e $ x miles * ( Cost of Gasoline /mile + Cost of Insurance Cost of Driver’s Time /mile ) gallon a miles b $ gal c $ mile hour d miles e $

Proving the Unit of Measure x miles * ( Cost of Gasoline /mile + Cost of Insurance Cost of Driver’s Time /mile ) gallon a miles b $ gal c $ mile hour d miles e $ b a $ c e d

Proving the Unit of Measure x miles * ( Cost of Gasoline /mile + Cost of Insurance Cost of Driver’s Time /mile ) b a $ mile c e d x miles * $ mile ( Gasoline + Insurance Driver’s Time ) b a c e d 1 x

Plugging Values into the Equation $ * x ( b a + c e d ) 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 average speed is 80 miles per hour (d) The driver’s cost per hour is $20 (e)

Plugging Values into the Equation $ * x ( b a + c e d ) 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 average speed is 80 miles per hour (d) The driver’s cost per hour is $20 (e)

LSA #2 Check on Learning Q1. What is the procedure when dividing by a fractional unit of measure? A1. Q1. What is the procedure when dividing by a fractional unit of measure? A1. Invert the second fraction and then multiply.

LSA #2 Summary During this lesson, we identified variables and units of measure in a cost expression, and then put it into a practical application via a formula

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

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?

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?

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 Full Tank – Gallons Used + Gallons Added = Full Tank – Gallons Used + Gallons Added = 0 Gallons Used = Gallons Added

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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

Using the Input-Output Equation (Cont.) 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

LSA #3 Check on Learning Q1. What are three useful equations that represent common cost relationships? A1. Q1. What are three useful equations that represent common cost relationships? A1. Revenue – Cost = Profit Total Cost = Fixed Cost + Variable Cost Beginning + Input – Output = Ending

LSA #3 Summary During this lesson, we discussed the use of “key cost equations” and applied this to a formula.

Conduct Practical Exercises

TLO Summary Action: 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