Verify Unit Of Measure In A Multivariate Equation © Dale R. Geiger 20111.

Verify Unit Of Measure In A Multivariate Equation

Principles of Cost Analysis and Management

You can’t….

+ = ??

Terminal Learning Objective

• Task: Verify Unit Of Measure In A Multivariate Equation

• Condition: You are a cost advisor technician with access to all regulations/course handouts, and awareness of Operational Environment (OE)/Contemporary Operational Environment (COE) variables and actors.

• Standard: with at least 80% accuracy:• 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 InnDepreciation on building $60,000 per yearMaintenance person’s salary $30,000 per yearCleaning person’s salary $24,000 per yearReal 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 InnLaundry service $4.00 per person-nightFood $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)

Check on Learning

• 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?

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/gallon5 gallons * $4/gallon 5 gallons * $4/gallon

5 * $4 = $20

Cost of Insurance

• Cost of Insurance = • Insurance $/mile * miles on trip• Insurance $/mile =

Total Insurance $ /yearTotal 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 MeasureCost of Gasoline + Cost of Insurance + Cost of Driver’s Time

x miles * gallona miles

* b $gal

+ c $mile

* x miles + hourd miles

* e $hour

* x miles

Proving the Unit of MeasureCost of Gasoline + Cost of Insurance + Cost of Driver’s Time

x miles * gallona miles

* b $gal

+ c $mile

* x miles + hourd miles

* e $hour

* x miles

x miles * ( Cost of Gasoline/mile + Cost of Insurance

/mile + Cost of Driver’s Time /mile )

x miles * ( gallona miles * b $

gal+ c $

mile + hourd miles * e $

hour )

Proving the Unit of Measurex miles * ( Cost of Gasoline

/mile + Cost of Insurance/mile + Cost of Driver’s

Time /mile )x miles * ( gallon

a miles * b $gal

+ c $mile + hour

d miles * e $hour )

x miles * ( gallona miles * b $

gal + c $mile + hour

d miles * e $hour )

x miles * ( b $a miles + c $

mile + e $d miles )

x miles * ( b a * $

mile + c * $mile + e

d * $mile )

Proving the Unit of Measure

x miles * $mile * ( Gasoline + Insurance + Driver’s

Time )x miles * $

mile * ( b a + c + e

d )x miles

1 * $mile * ( b

a + c + ed )

$ * x * ( b a + c + e

x miles * ( Cost of Gasoline/mile + Cost of Insurance

/mile + Cost of Driver’s Time /mile )

x miles * ( b a * $

mile + c * $mile + e

d * $mile )

Plugging Values into the Equation

$ * x * ( b a + c + e

What is the cost of the trip if:The distance (x) is 300 milesThe car gets 25 miles per gallon (b)The cost of a gallon of gas is $4The insurance cost per mile (c) is $.05The driver’s cost per hour is $20 (e)The average speed is 80 miles per hour (d)

Plugging Values into the Equation

$ * x * ( b a + c + e

d )$ * 300 * ( 4

25 + .05 + 2080 )

What is the cost of the trip if:The distance (x) is 300 milesThe car gets 25 miles per gallon (a)The cost of a gallon of gas is $4 (b)The insurance cost per mile (c) is $.05The driver’s cost per hour is $20 (e)The average speed is 80 miles per hour (d)

Check on Learning

• What is the procedure when dividing by a fractional unit of measure?

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 = EndingOr, chronologically:

Beginning – Output + Input = EndingFull Tank – Gallons Used + Gallons Added = Full TankFull Tank – Gallons Used + Gallons Added = Full Tank

– Gallons Used + Gallons Added = 0Gallons 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

• 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

• 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 = $940Finance Charge = $940 – $925

Finance Charge = $15

Check on Learning

• What are three useful equations that represent common cost relationships?

Practical Exercises

Verify Unit Of Measure In A Multivariate Equation © Dale R. Geiger 20111.

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