I.

ENGINEERING ECONOMICS F, measured today’s pesos of a P:

A. Interests: 𝑃
𝑭=
1. Simple interest (1 + 𝑖𝑓 )𝑛
𝑰 = 𝑃𝑖𝑛
F, when interest being compounded while
Accumulated Amount: inflation is occurring.
𝑭 = 𝑃 + 𝐼 = 𝑃 + 𝑃𝑖𝑛 = 𝑃(1 + 𝑖𝑛) 𝑃(1 + 𝑖)𝑛
𝑭=
(1 + 𝑖𝑓 )𝑛
Ordinary Simple Interest:
1 year = 12 months = 360 days
C. Nominal and Effective Interest Rates:
Exact Simple Interest: 1. Nominal interest rate
1 year = 12 months = 365 or 366 days 𝑖𝑛
𝑖 = = 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑝𝑒𝑟 𝑝𝑒𝑟𝑖𝑜𝑑
(If Year/4 is whole number, use 366 days 𝑚
otherwise 365 days) 𝒊𝒏 = 𝑛𝑜𝑚𝑖𝑛𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒

2. Compound interest 2. Effective interest rate per year

Present Worth: (EIR > nominal interest rate)
𝑷 = 𝐹(1 + 𝑖)−𝑛 𝑖𝑛 𝑚
𝒊𝒆 = (1 + ) − 1
𝑚
Single Payment Present Worth Factor: 𝒎 = 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑝𝑒𝑟 𝑦𝑒𝑎𝑟
𝑷 semi-annually = 2 Annually = 1
( , 𝒊%, 𝒏) = (1 + 𝑖)−𝑛 quarterly = 4 Bi-monthly = 6
𝑭
Monthly = 12 Daily = 360
Future Worth:
𝑭 = 𝑃(1 + 𝑖)𝑛 3. Continuously compounding interest rate
𝑭 = 𝑃𝑒 𝑖𝑛 𝑛
Single Payment Compound Amount Factor:
𝑭
( , 𝒊%, 𝒏) = (1 + 𝑖)𝑛
𝑷 D. Annuities:
1. Ordinary annuity
For Present Worth:
B. Cash Flow Diagram, Discount & Inflation Rate: 1 − (1 + 𝑖)−𝑛
1. Discount 𝑷 = 𝐴[ ]
𝑖
𝑫 = 𝐹 − 𝑃 = 𝑆𝑃 − 𝑁𝑃
Uniform Series Present Worth Factor:
Discount Rate: 𝑷 1 − (1 + 𝑖)−𝑛
𝐹 − 𝑃 𝑆𝑃 − 𝑁𝑃 1 ( , 𝒊%, 𝒏) = [ ]
𝒅= = =1− 𝑨 𝑖
𝐹 𝑆𝑃 (1 + 𝑖)
A when P is given:
2. Inflation
𝑖
𝑭𝑪 = 𝑃𝐶(1 + 𝑖𝑓 )𝑛 𝑨 = 𝑃[ ]
1 − (1 + 𝑖)−𝑛
if = annual inflation rate
PC = present cost of a commodity Capital Recovery Factor:
FC = future cost of the same commodity 𝑨 𝑖
( , 𝒊%, 𝒏) = [ ] = 𝑖 + 𝑆𝐹
𝑷 1 − (1 + 𝑖)−𝑛
 For Future Worth: 𝐺 (1 + 𝑖)𝑛 − 1
𝑪= [ − 𝑛]
(1 + 𝑖)𝑛 − 1 𝑖 𝑖
𝑭 = 𝐴[ ] G = uniform gradient amount
𝑖

Uniform Series Compound Worth Factor: 2. Geometric uniform gradient

𝑭 (1 + 𝑖)𝑛 − 1 Present Worth:
( , 𝒊%, 𝒏) = [ ] Case 1: z and r are not equal to 1
𝑨 𝑖
𝐺 1 − 𝑧𝑛
𝑷= [ ]
1+𝑖 1−𝑧
A when F is Given: 1+𝑟
𝑖 𝒛= ( )
𝑨 = 𝐹[ ] 1+𝑖
(1 + 𝑖)𝑛 − 1 r = common ratio = rate of increase
= decrease in disbursement
Sinking Fund Factor:
𝑨 𝑖 Case 2: r = 1, then z =1
( , 𝒊%, 𝒏) = [ ]
𝑭 (1 + 𝑖)𝑛 − 1 𝐺𝑛
𝑷=
1+𝑖
2. Annuity due
Present Worth: Future Worth:
1 − (1 + 𝑖)−(𝑛−1) 𝑭 = 𝑃(1 + 𝑖)𝑛
𝑷 = 𝐴 [1 + ]
𝑖

Future Worth: F. Amortization:

(1 + 𝑖)(𝑛+1) − 1 1. Amortization
𝑭 = 𝐴[ − 1] 𝑃1
𝑖
𝑨𝑴 = = 𝑃𝑎𝑦𝑚𝑒𝑛𝑡
1 − (1 + 𝑖)−𝑛
[ 𝑖 ]
3. Deferred annuity
1 − (1 + 𝑖)−𝑛
𝑷 = 𝐴[ ] (1 + 𝑖)−𝑚 2. Amortization scheduling:
𝑖
• First Accumulate Worth
𝑭𝟏 = 𝑃1 (1 + 𝑖)1
4. Perpetuity
𝐴
𝑷= • First Interest
𝑖
𝑰𝟏 = 𝐹1 − 𝑃1

E. Uniform Gradient: • Second Outstanding Principal

1. Arithmetic uniform gradient 𝑷𝟐 = 𝐹1 − 𝐴𝑀
Present Worth:
1 − (1 + 𝑖)−𝑛 • Second Accumulate Worth
𝑷 = 𝐴[ ]+𝐵 𝑭𝟐 = 𝑃2 (1 + 𝑖)1
𝑖

𝐺 1 − (1 + 𝑖)−𝑛 • Second Interest

𝑩= [ − 𝑛(1 + 𝑖)−𝑛 ] 𝑰𝟐 = 𝐹2 − 𝑃2
𝑖 𝑖

Future Worth: • First Principal Repaid

(1 + 𝑖)𝑛 − 1 𝑷𝑹𝟏 = 𝐴𝑚 − 𝐼1
𝑭 = 𝐴[ ]+𝐶
𝑖
II. DEPRECIATION 5. DOUBLE DECLINING BALANCE METHOD
A. Methods of Computing Depreciation: Depreciation Charge (first year):
1. STRAIGHT LINE METHOD 2
𝑫𝟐𝑫𝑩 = (𝐹𝐶)
Annual Depreciation: 𝑛
𝐹𝐶 − 𝑆𝑉
𝑫𝑺𝑳 = Depreciation Charge to Date:
𝑛
n = useful life 2
𝑫𝟐𝑫𝑩𝑻𝑫 = (𝐹𝐶 − 𝐷𝑇𝐷 )
𝑛
Book Value of m years: DTD = depreciation to date
𝑩𝑽𝑺𝑳 = 𝐹𝐶 − 𝑚𝐷𝑆𝐿
6. SERVICE OUTPUT or PRODUCTION UNITS’
2. SINKING FUND METHOD METHOD
Annual Depreciation: 𝐹𝐶 − 𝑆𝑉
𝑫𝑺𝑶 =
𝐹𝐶 − 𝑆𝑉 𝑁𝑜. 𝑜𝑓 𝑈𝑛𝑖𝑡𝑠 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
𝑫𝑺𝑭 =
(1 + 𝑖)𝑛 − 1
𝑖 7. WORKING HOURS or MACHINE HOURS
𝐹𝐶 − 𝑆𝑉
Book Value after m years: 𝑫𝑺𝑶 =
𝑁𝑜. 𝑜𝑓 𝐻𝑜𝑢𝑟𝑠 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦
(1 + 𝑖)𝑚 − 1
𝑩𝑽𝑺𝑭 = 𝐹𝐶 − ( ) 𝐷𝑆𝑓
𝑖 8. DEPRECIATION RATE
𝐷 1 𝐹𝐶 − 𝑆𝑉
𝒅= = ( ) ; 𝑥 100%
3. SUM-OF-THE-YEARS-DIGIT METHOD 𝐹𝐶 𝐹𝐶 𝑛
Sum-Of-The-Years-Digit: D = annual depreciation
𝑛(𝑛 + 1) FC = First Cost
𝑺𝒀𝑫 = SV = Salvage Value
2

Depreciation at Any Year (y): 9. BOOK VALUE AT ANY TIME

𝐹𝐶 − 𝑆𝑉 𝑩𝑽𝒎 = 𝐹𝐶 − 𝐷𝑚
𝑫𝑺𝒀𝑫 =
𝑆𝑌𝐷 Dm = total depreciation for m years
𝑛−𝑦+1

Book Value after m years: B. Capital Recovery: (Factors of Annual Cost)

𝑩𝑽𝑺𝒀𝑫 = 𝐹𝐶 − ∑ 𝐷𝑆𝑌𝐷(1 𝑡𝑜 𝑚) 1. CAPITAL RECOVERY (USING SINKING FUND
METHOD):
Annual Capital Recovery Rate:
4. DECLINING BALANCE METHOD
Constant Ratio: CRSL = DSL + IFC
𝐹𝐶 − 𝑆𝑉
= + 𝑖(𝐹𝐶)
𝑛 𝑆𝑉 (1 + 𝑖)𝑛 − 1
𝒌= 1− √ 𝑖
𝐹𝐶

2. CAPITAL RECOVERY (USING STRAIGHT LINE

Depreciation at Any Year (y):
METHOD):
𝑫𝑫𝑩 = 𝑘(𝐹𝐶)(1 − 𝑘)𝑦−1
Annual Capital Recovery Rate:
CRSL = DSL + IAVE + ISV
Book Value after m years: 𝐹𝐶 − 𝑆𝑉 𝐹𝐶 − 𝑆𝑉
𝑩𝑽𝑫𝑩 = 𝐹𝐶(1 − 𝑘)𝑚 𝑪𝑹𝑺𝑳 = + + 𝑖(𝑆𝑉)
𝑛 2𝑛
𝑖(𝑛 + 1)
 C. Capitalized Cost: B. Selection of Alternatives:
1. CAPITALIZED COST FOR PERPETUAL LIFE • RATE OF RETURN
𝑂𝑀 𝑁𝑒𝑡 𝑃𝑟𝑜𝑓𝑖𝑡
𝑪𝑪𝑷𝑳 = 𝐹𝐶 + 𝒓𝒐𝒓 =
𝑖 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡

2. CAPITALIZED COST FOR LIFE n: • PAYOUT PERIOD

𝑂𝑀 𝐹𝐶 − 𝑆𝑉 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 − 𝑆𝑎𝑙𝑣𝑎𝑔𝑒 𝑉𝑎𝑙𝑢𝑒
𝑪𝑪𝑳 = 𝐹𝐶 + + 𝒑𝒑 =
𝑖 (1 + 𝑖)𝑛 − 1 𝑁𝑒𝑡 𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤
OM = Operation & Maintenance annually
• ANNUAL COST
𝑨𝒏𝒏𝒖𝒂𝒍 𝑪𝒐𝒔𝒕 = 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 +
D. Break-Even Analysis: 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑜𝑛 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 +
1. TO BREAK-EVEN 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 +
𝐼𝑁𝐶𝑂𝑀𝐸 = 𝐸𝑋𝑃𝐸𝑁𝑆𝐸𝑆 𝑂𝑢𝑡 𝑜𝑓 𝑃𝑜𝑐𝑘𝑒𝑐𝑡 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑠
𝑷(𝒙) = 𝑀(𝑥) + 𝐿(𝑥) + 𝑉(𝑥) + 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡
x = no. of units produced and sold
P = selling price per unit C. Replacement Studies
M = material cost per unit 1. REPLACEMENT STUDIES
L = labor cost per unit • RATE OF RETURN
V = variable cost per unit 𝑆𝑎𝑣𝑖𝑛𝑔𝑠 𝐼𝑛𝑐𝑢𝑟𝑟𝑒𝑑 𝑏𝑦 𝑅𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝒓𝒐𝒓 =
𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑎𝑙 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑

• ANNUAL COST
III. BUSINESS STUDIES 𝑨𝒏𝒏𝒖𝒂𝒍 𝑪𝒐𝒔𝒕 = 𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 +
1. BOND VALUE n PERIODS BEFORE MATURITY 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑜𝑛 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 +
1 − (1 + 𝑖)−𝑛 𝑅 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑎𝑛𝑑 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 +
𝑩𝑶𝑵𝑫 = 𝐹𝑟 [ ]+
𝑖 (1 + 𝑖)𝑛 𝑂𝑢𝑡 𝑜𝑓 𝑃𝑜𝑐𝑘𝑒𝑐𝑡 𝐸𝑥𝑝𝑒𝑛𝑠𝑒𝑠
F = face or par value of the bond = R
Fr = periodic dividend
R = redeemable value (usually equal to F) D. Benefit to Cost Ratio in Public Project:
n = no. of periods 1. BENEFIT-TO-COST RATIO
i = investment rate 𝑩 𝐵 − 𝑂𝑀
=
𝑪 𝐶
B = annual benefits, that is, the annual
worth of benefits incurred because of the
IV. BASIC INVESTMENT STUDIES existence of the project.
A. Basic Investment Studies:
• RATE OF RETURN
C = annual equivalent of the cost
𝑁𝑒𝑡 𝑃𝑟𝑜𝑓𝑖𝑡 = FC/ (P/A, i%, n) – SV/ (F/A, i%, n)
𝒓𝒐𝒓 = 𝐹𝐶 𝑆𝑉
𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑪= −
1 − (1 + 𝑖)−𝑛 (1 + 𝑖)𝑛 − 1
𝑖 𝑖
• PAYOUT PERIOD
𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 − 𝑆𝑎𝑙𝑣𝑎𝑔𝑒 𝑉𝑎𝑙𝑢𝑒
𝒑𝒑 =
𝑁𝑒𝑡 𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤
 E. Economic Order Quantity:
1. ECONOMIC ORDER QUANTITY (EOQ)

2𝑎𝑘
𝑬𝑶𝑸 = √
ℎ
a = the constant depletion rate (items per
unit time)
k = the fixed cost per order, Pesos
h = the inventory storage cost (Pesos per
item per unit time)

V. PRINCIPLE OF ACCOUNTING
A. Balancing System
1. BALANCING SYSTEM
𝑨𝒔𝒔𝒆𝒕𝒔 = 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 + 𝑂𝑤𝑛𝑒𝑟 ′ 𝑠 𝐸𝑞𝑢𝑖𝑡𝑦

B. Terms Used in The Financial Statements:

1. CURRENT RATIO
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠
𝑪𝑹 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠

2. ACID TEST RATIO

𝑄𝑢𝑖𝑐𝑘 𝐴𝑠𝑠𝑒𝑡𝑠
𝑨𝑻𝑹 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠

3. RECEIVABLE TURNOVER
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠 𝑜𝑛 𝐶𝑟𝑒𝑑𝑖𝑡
𝑹𝑻 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑒𝑐𝑒𝑖𝑣𝑎𝑏𝑙𝑒𝑠

4. GROSS MARGIN
𝐺𝑟𝑜𝑠𝑠 𝑃𝑟𝑜𝑓𝑖𝑡
𝑮𝑴 =
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠

5. PROFIT MARGIN RATIO

𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒 𝐵𝑒𝑓𝑜𝑟𝑒 𝑇𝑎𝑥
𝑷𝑴𝑹 =
𝑆𝑎𝑙𝑒𝑠

6. RETURN ON INVESTMENT RATIO

𝑄𝑢𝑖𝑐𝑘 𝐴𝑠𝑠𝑒𝑡𝑠
𝑹𝑶𝑰𝑹 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠