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

2. You can’t…. +
= ??

3. 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 

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

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 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

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

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

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

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

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

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
= $400

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)

13. 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.

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

15. 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

16. 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)

17. 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

18. 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

19. 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

20. 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

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

22. 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 $

23. 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 $

24. 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

25. 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

26. 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)

27. 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)

28. 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.

29. 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

30. 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

31. 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?

32. 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?

33. 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

34. 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

35. 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

36. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

37. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

38. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

39. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

40. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

41. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

42. 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
$ Finance Charge – $325 = $940
$925 + Finance Charge = $940
Finance Charge = $940 – $925
Finance Charge = $15

43. 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

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

45. Conduct Practical Exercises

46. 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