Verify Unit of Measure in a Multivariate Equation © 20111.

Verify Unit of Measure in a Verify Unit of Measure in a Multivariate EquationMultivariate Equation

© 2011 1

You can’t….You can’t….

+ = ??

© 2011 2

Terminal Learning ObjectiveTerminal 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

© 2011 3

terri.kelly1

Changes task to Determine versus Verify and added learning step 1 to standard and notes below.

Importance of Units of MeasureImportance 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

© 2011 4

AddingAdding• 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

© 2011 5

AddingAdding

• 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

© 2011 6

SubtractingSubtracting

• 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

© 2011 7

DividingDividing

• “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

© 2011 8

CancellingCancelling

© 2011 9

MultiplicationMultiplication

• 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

© 2011 10

Cross-CancellingCross-Cancelling

© 2011 11

FactoringFactoring

• 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)

© 2011 12

Learning CheckLearning 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?

© 2011 13

Proving a Unit of MeasureProving 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

© 2011 14

Variables Affecting Cost of TripVariables 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) • Driver’s cost in dollars per hour (represented by e)• Average speed in miles per hour (represented by d)

© 2011 15

Cost of GasolineCost 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

© 2011 16

Cost of InsuranceCost of Insurance

© 2011 17

Cost of Driver’s TimeCost 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

© 2011 18

Cost ExpressionCost Expression

gallon/ a miles * x miles * b $/gal+

c $/mile * x miles+

hours/d mile * e $/hour * x miles

© 2011 19

Proving the Unit of MeasureProving 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

© 2011 20

Proving the Unit of MeasureProving 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 )

© 2011 21

Proving the Unit of MeasureProving the Unit of Measure

© 2011 22

Proving the Unit of MeasureProving the Unit of Measure

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 )

© 2011 23

Plugging Values into the EquationPlugging 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)

© 2011 24

Plugging Values into the EquationPlugging 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)

© 2011 25

The Value of EquationsThe 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

© 2011 27

Input-Output EquationInput-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?

© 2011 28

Applications of Input-OutputApplications 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?

© 2011 29

Using the Input-Output EquationUsing 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

© 2011 31


• 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

© 2011 32


• 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

© 2011 33

Verify Unit of Measure in a Multivariate Equation © 20111.

Documents

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