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© Dale R. Geiger 2011 1
Verify Unit Of Measure In A Multivariate Equation
Principles of Cost Analysis and Management
© Dale R. Geiger 2011 2
You can’t….
+ = ??
© Dale R. Geiger 2011 3
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
© Dale R. Geiger 2011 4
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
© Dale R. Geiger 2011 5
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
© Dale R. Geiger 2011 6
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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 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 = • $/hr *(4 + 6)
© Dale R. Geiger 2011 13
Check on Learning
• If two components of a cost expression have the same unit of measure, they may be either
or .• Which mathematical operation using two
different units of measure results in a new unit of measure?
© Dale R. Geiger 2011 14
Proving a Unit of Measure
• What is the cost expression for a driving trip?• The cost will be the sum of the following
components:$ Gasoline + $ Insurance + $ Driver’s time
© Dale R. Geiger 2011 15
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)
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 17
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
© Dale R. Geiger 2011 18
Cost of Insurance
• Cost of Insurance = • Insurance $/mile * miles on trip• Insurance $/mile =
Total Insurance $ /yearTotal miles /year
So:c $/mile * x miles
© Dale R. Geiger 2011 19
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
© Dale R. Geiger 2011 20
Cost Expression
gallon/ a miles * x miles * b $/gal+
c $/mile * x miles+
hours/d mile * e $/hour * x miles
© Dale R. Geiger 2011 21
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
© Dale R. Geiger 2011 22
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 )
© Dale R. Geiger 2011 23
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 )
© Dale R. Geiger 2011 24
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
d )
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 )
© Dale R. Geiger 2011 25
Plugging Values into the Equation
$ * x * ( b a + c + e
d )
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)
© Dale R. Geiger 2011 26
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)
© Dale R. Geiger 2011 27
Check on Learning
• What is the procedure when dividing by a fractional unit of measure?
© Dale R. Geiger 2011 28
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
© Dale R. Geiger 2011 29
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?
© Dale R. Geiger 2011 30
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?
© Dale R. Geiger 2011 31
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
© Dale R. Geiger 2011 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
© Dale R. Geiger 2011 33
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
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 40
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 = $940Finance Charge = $940 – $925
Finance Charge = $15
© Dale R. Geiger 2011 41
Check on Learning
• What are three useful equations that represent common cost relationships?
© Dale R. Geiger 2011 42
Practical Exercises