Engineering Economics, Cheat Sheet of Mathematics

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2023/2024

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114 ENGINEERING ECONOMICS
ENGINEERING ECONOMICS
Factor Name Converts Symbol Formula
Single Payment
Compound Amount to F given P (F/P,i%,n) (1 + i)n
Single Payment
Present Worth to P given F (P/F,i%,n) (1 + i)n
Uniform Series
Sinking Fund to A given F (A/F,i%,n)
()
11−+n
i
i
Capital Recovery to A given P (A/P,i%,n)
()
()
11
1
+
+
n
n
i
ii
Uniform Series
Compound Amount to F given A (F/A,i%,n)
()
i
in11−+
Uniform Series
Present Worth to P given A (P/A, i%,n)
()
()
n
n
ii
i
+
+
1
11
Uniform Gradient
Present Worth to P given G (P/G,i%,n)
()
() ()
nn
n
ii
n
ii
i
+
+
+
11
11
2
Uniform Gradient †
Future Worth to F given G (F/G,i%,n)
()
i
n
i
in
+
2
11
Uniform Gradient
Uniform Series to A given G (A/G,i%,n)
()
11
1
+
n
i
n
i
NOMENCLATURE AND DEFINITIONS
A .......... Uniform amount per interest period
B .......... Q
BV Book value
C .......... Cost
d Combined interest rate per interest period
Dj Depreciation in year j
F .......... Future worth, value, or amount
f 
G Uniform gradient amount per interest period
i 
ie -
m Number of compounding periods per year
n  7#
of an asset
P .......... Present worth, value, or amount
r ........... Nominal annual interest rate
SnExpected salvage value in year n
Subscripts
j at time j
n at time n
† F/G ~F/A n)/i ~F/A) × (A/G)
NON-ANNUAL COMPOUNDING
im
r
22
e
m
=+ -
bl
BREAK-EVEN ANALYSIS
By altering the value of any one of the variables in a situation,
+ 
value for that variable that makes the two alternatives equally
! *%
Q*% 
capacity of operation for a manufacturing plant at which
'#
! *,
 ,

INFLATION
!+ 
f, and then they are shifted
over the time scale using the interest rate per interest period
iq d for
computing present worth values P and Net P
The formula for d is d ~i + f + (i × f)
pf3
pf4
pf5

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ENGINEERING ECONOMICS

Factor Name Converts Symbol Formula

Single Payment

Compound Amount

to F given P ( F/P , i% , n ) (1 + i )

n

Single Payment

Present Worth

to P given F ( P/F , i% , n ) (1 + i )

  • n

Uniform Series

Sinking Fund

to A given F ( A/F , i% , n )

n

i

i

Capital Recovery to A given P ( A/P , i% , n )

n

n

i

i i

Uniform Series

Compound Amount

to F given A ( F/A , i% , n )

i

i

n

Uniform Series

Present Worth

to P given A ( P/A, i% , n )

n

n

i i

i

Uniform Gradient

Present Worth

to P given G ( P/G , i% , n )

n n

n

i i

n

i i

i

2

Uniform Gradient †

Future Worth

to F given G ( F/G , i% , n )

i

n

i

i

n

2

Uniform Gradient

Uniform Series

to A given G ( A/G , i% , n )

− n

i

n

i

NOMENCLATURE AND DEFINITIONS

A .......... Uniform amount per interest period

B .......... Q  

BV  Book value

C .......... Cost

d  Combined interest rate per interest period

D j  Depreciation in year j

F .......... Future worth, value, or amount

f     “        

G  Uniform gradient amount per interest period

i           

i e -      

m  Number of compounding periods per year

n       7  #    

of an asset

P .......... Present worth, value, or amount

r ........... Nominal annual interest rate

S n  Expected salvage value in year n

Subscripts

j  at time j

n  at time n

† F/G ~ F/A – n ) / i ~ F/A ) × ( A/G )

NON-ANNUAL COMPOUNDING

i

m

r

e^2

m

= b^ + l -

BREAK-EVEN ANALYSIS

By altering the value of any one of the variables in a situation,

value for that variable that makes the two alternatives equally

Q *%           

capacity of operation for a manufacturing plant at which

INFLATION

“         f , and then they are shifted

over the time scale using the interest rate per interest period

i q              d for

computing present worth values P and Net P 

The formula for d is d ~ i + f + ( i × f )

DEPRECIATION

Straight Line

D

n

C S

j

n

Accelerated Cost Recovery System (ACRS)

Dj ~ C

Sum of the Years Digits

D

j

n j

C S

j

j

n n

2

=

_ i

BOOK VALUE

BV ~   XΣ D j

TAXATION

Taxable income is total income less depreciation and

CAPITALIZED COSTS

  5 ~ P

i

A

BONDS

Bond Value equals the present worth of the payments the

purchaser (or holder of the bond) receives during the life of

the bond at some interest rate i 

Q  ,        

RATE-OF-RETURN

interest rate that one is willing to accept, or the rate one

BENEFIT-COST ANALYSIS

  %  +    B ' 

exceed the estimated costs C 

B – C ≥ )+ B/C ≥ 2

MODIFIED ACRS FACTORS

Recovery Period (Years)

Year Recovery Rate (Percent)

n P/F P/A P/G F/P F/A A/P A/F A/G

Factor Table - i = 2.00%

n P/F P/A P/G F/P F/A A/P A/F A/G

n P/F P/A P/G F/P F/A A/P A/F A/G

Factor Table - i = 6.00%

n P/F P/A P/G F/P F/A A/P A/F A/G

n P/F P/A P/G F/P F/A A/P A/F A/G

Factor Table - i = 18.00%

n P/F P/A P/G F/P F/A A/P A/F A/G

      1.  - 0. - 0. - 2. - 5. - 9. - 14. - 20. - 27. - 34. - 43. - 52. - 63. - 74. - 86. - 99. 
        • 1,035.
        • 1,448.
        • 3,562. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 3. - 4. - 5. - 6. - 7. - 8. - 9. - 10. - 11. - 12. - 13. - 14. - 15. - 16. - 17. - 18. - 19. - 20. - 22. - 23. - 24. - 25. - 26. - 32. - 44. - 56. - 69. - 129. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 5. - 5. - 6. - 6. - 7. - 7. - 8. - 8. - 9. - 9. - 10. - 10. - 11. - 11. - 14. - 18. - 23. - 28. - 45.
      1.  - 0. - 0. - 2. - 5. - 9. - 14. - 19. - 26. - 33. - 41. - 50. - 60. - 71. - 82. - 94. 
        • 1,192.
        • 2,605. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 1. - 2. - 3. - 4. - 5. - 6. - 7. - 8. - 9. - 10. - 11. - 12. - 13. - 14. - 16. - 17. - 18. - 19. - 20. - 22. - 23. - 24. - 25. - 26. - 28. - 34. - 48. - 64. - 81. - 170. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 5. - 5. - 6. - 6. - 7. - 7. - 8. - 8. - 9. - 9. - 10. - 10. - 11. - 11. - 13. - 18. - 22. - 26. - 41.
      1.  - 0. - 0. - 2. - 5. - 9. - 13. - 19. - 26. - 32. - 40. - 48. - 58. - 67. - 78. - 89. 
        • 1,937. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 4. - 1. - 2. - 3. - 4. - 5. - 6. - 7. - 8. - 9. - 10. - 11. - 13. - 14. - 15. - 16. - 17. - 19. - 20. - 21. - 23. - 24. - 25. - 27. - 28. - 30. - 37. - 54. - 73. - 96. - 228. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 5. - 5. - 6. - 6. - 7. - 7. - 8. - 8. - 9. - 9. - 9. - 10. - 10. - 11. - 13. - 17. - 21. - 25. - 37.
      1.  - 0. - 0. - 2. - 5. - 9. - 13. - 18. - 24. - 31. - 38. - 46. - 55. - 64. - 74. - 85. - 96. 
        • 1,464. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 3. - 7. - 1. - 2. - 3. - 4. - 5. - 6. - 7. - 8. - 9. - 10. - 12. - 13. - 14. - 15. - 17. - 18. - 20. - 21. - 22. - 24. - 25. - 27. - 28. - 30. - 32. - 40. - 60. - 84. - 114. - 312. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 5. - 5. - 6. - 6. - 7. - 7. - 7. - 8. - 8. - 9. - 9. - 10. - 10. - 10. - 13. - 16. - 20. - 23. - 33.
      1.  - 0. - 0. - 2. - 5. - 8. 
          1.  - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 2. - 2. - 2. - 2. - 2. - 2. - 3. - 4. - 7. - 10. - 50. - 1. - 2. - 3. - 4. - 5. - 6. - 7. - 9. - 10. - 12. - 13. - 15. - 16. - 18. - 20. - 21. - 23. - 25. - 27. - 29. - 31. - 34. - 36. - 39. - 41. - 56. - 95. - 152. - 237. - 1,237. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 5. - 5. - 5. - 6. - 6. - 7. - 7. - 7. - 8. - 8. - 8. - 9. - 9. - 9. - 11. - 14. - 16. - 18. - 22. 
      1.  - 0. - 0. - 2. - 4. - 7. 
          1.  - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 2. - 2. - 2. - 2. - 2. - 3. - 3. - 3. - 3. - 3. - 4. - 4. - 5. - 10. - 18. - 32. 
              1.  - 1. - 2. - 3. - 4. - 5. - 6. - 8. - 9. - 11. - 13. - 14. - 16. - 18. - 21. - 23. - 25. - 28. - 30. - 33. - 36. - 39. - 43. - 46. - 50. - 54. - 79. - 154. - 290. - 533. 
                • 5,638. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 4. - 4. - 5. - 5. - 5. - 6. - 6. - 6. - 7. - 7. - 7. - 8. - 8. - 8. - 9. - 10. - 12. - 13. - 14. - 16.
      1.  - 0. - 0. - 2. - 4. - 7. 
          1.  - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 2. - 2. - 2. - 3. - 3. - 3. - 3. - 4. - 4. - 5. - 5. - 5. - 6. - 6. - 10. - 21. - 46. - 101. - 2,199. - 1. - 2. - 3. - 4. - 5. - 7. - 8. - 10. - 12. - 14. - 16. - 18. - 21. - 24. - 27. - 30. - 33. - 37. - 41. - 45. - 50. - 55. - 60. - 66. - 73. - 113. - 259. - 573. - 1,253. - 27,484. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 3. - 4. - 4. - 4. - 5. - 5. - 5. - 6. - 6. - 6. - 7. - 7. - 7. - 7. - 8. - 8. - 9. - 10. - 11. - 11. - 12. 
        1. - 0. - 0. - 2. - 4. - 6. - 9. 
            1.  - 1. - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 2. - 2. - 3. - 3. - 3. - 4. - 4. - 5. - 5. - 6. - 6. - 7. - 8. - 8. - 9. - 10. - 17. - 45. - 117. - 304. 
              • 13,780. - 1. - 2. - 3. - 4. - 6. - 7. - 9. - 11. - 13. - 15. - 18. - 21. - 24. - 27. - 31. - 35. - 40. - 45. - 51. - 57. - 64. - 71. - 79. - 88. - 98. - 164. - 442. - 1,163. - 3,034. - 137,796. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 3. - 3. - 3. - 4. - 4. - 4. - 4. - 5. - 5. - 5. - 6. - 6. - 6. - 6. - 6. - 7. - 7. - 7. - 8. - 9. - 9. - 9. - 9.
        1.  - 1. - 1. - 1. - 1. - 1. - 1. - 2. - 2. - 2. - 3. - 3. - 3. - 4. - 4. - 5. - 6. - 6. - 7. - 8. - 9. - 10. - 12. - 13. - 15. - 17. - 29. - 93. - 289. - 897. 
          • 83,522. - 1. - 2. - 3. - 4. - 6. - 8. - 10. - 12. - 14. - 17. - 20. - 24. - 28. - 32. - 37. - 42. - 48. - 55. - 63. - 72. - 81. - 92. - 104. - 118. - 133. - 241. - 767. - 2,400. - 7,471. - 696,010. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 2. - 3. - 3. - 3. - 4. - 4. - 4. - 4. - 5. - 5. - 5. - 5. - 6. - 6. - 6. - 6. - 6. - 6. - 7. - 7. - 8. - 8. - 8.
        1.  - 1. - 1. - 1. - 1. - 2. - 2. - 3. - 3. - 4. - 5. - 6. - 7. - 8. - 10. - 11. - 14. - 16. - 19. - 23. - 27. - 32. - 38. - 45. - 53. - 62. - 143. - 750. - 3,927. 
          • 20,555.
          • 15,424,131. - 1. - 2. - 3. - 5. - 7. - 9. - 12. - 15. - 19. - 23. - 28. - 34. - 42. - 50. - 60. - 72. - 87. - 103. - 123. - 146. - 174. - 206. - 244. - 289. - 342. - 790. - 4,163. - 21,813. - 114,189. - 85,689,616. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 0. - 1. - 1. - 2. - 2. - 2. - 2. - 3. - 3. - 3. - 3. - 4. - 4. - 4. - 4. - 4. - 4. - 4. - 4. - 4. - 5. - 5. - 5. - 5. - 5. - 5. - 5. - 5.