CELE Engineering Mathematics — Engineering EconomyCheat Sheet

Formula

F = P(1 + i)^n

Meaning

F = future worth; P = present worth; i = interest rate per period; n = number of periods

Watch Out

CRITICAL: i and n MUST use the SAME TIME PERIOD. If i is monthly, n must be in months. Mixing annual rate with monthly periods = automatic failure.

When To Use

Single lump-sum payment with compound interest; any time you need to find future value from a present amount.

Formula

P = F(1 + i)^{-n} = F / (1 + i)^n

Meaning

Discounting: bring future cash back to present value; (1 + i)^{-n} is the discount factor.

Watch Out

The exponent is NEGATIVE or use division. Students often write (1 + i)^n in denominator and forget the negative sign.

When To Use

When you have a future amount and need its present worth; used in NPV and cost-benefit analysis.

Formula

i_{eff} = (1 + r/m)^m - 1

Meaning

i_eff = effective annual interest rate; r = nominal annual rate; m = number of compounding periods per year

Watch Out

NEVER use nominal rate directly for calculations if compounding ≠ annual. Always convert to effective rate first or adjust i and n to match compounding frequency.

When To Use

When compounding frequency ≠ annual (monthly, quarterly, semi-annual). REQUIRED to compare rates with different compounding periods.

Formula

F = P(1 + i \cdot n)

Meaning

Simple interest: F = future amount; i = interest rate; n = number of periods (rarely used in engineering economy)

Watch Out

DO NOT use simple interest unless explicitly told. Board exams assume compound interest.

When To Use

Only when problem explicitly states 'simple interest'; almost never in PRC exams. Default is COMPOUND interest.

Value

Bank savings 2–4%, bonds 4–6%, stocks/investments 8–15%, credit cards 12–24%

Symbol

i or r

Quantity

Typical interest rates (annual)

Value

2.71828

Symbol

e

Quantity

Euler's number

Term

Present Worth (PW)

Example

₱10,000 five years from now at 10% interest has PW = ₱10,000 / (1.10)^5 = ₱6,209.

Definition

The equivalent value of all future cash flows expressed in today's pesos (discounted to time zero).

Term

Future Worth (FW)

Example

₱10,000 invested today at 10% compounded annually for 5 years grows to FW = ₱16,105.

Definition

The equivalent value of all present (or past) cash flows expressed at a specified future time.

Term

Interest Rate per Period (i)

Example

12% annual compounded monthly → i = 0.12 / 12 = 0.01 (1% per month).

Definition

The fractional return on money per compounding period (decimal form: 5% = 0.05).

Term

Nominal Rate (r)

Example

Bank advertises '12% nominal compounded monthly'; actual effective rate is (1 + 0.12/12)^12 − 1 = 12.68%.

Definition

The stated annual interest rate before adjusting for compounding frequency; NOT the rate to use in calculations unless compounding is annual.

Term

Effective Annual Rate (i_eff)

Example

12% nominal compounded monthly → i_eff = 12.68%; use this to compare against other investment options.

Definition

The equivalent annual interest rate that accounts for the effect of compounding within the year.

Formula

F = A \left[ \frac{(1+i)^n - 1}{i} \right]

Meaning

F = future worth of annuity; A = uniform payment per period; [(1+i)^n − 1]/i = future worth factor (FWF) or 'F/A factor'

Watch Out

This assumes payments at the END of each period. If payments are at the START (annuity due), multiply result by (1+i). Misidentifying timing = wrong answer.

When To Use

Finding total future value of equal periodic payments (ordinary annuity: payments at END of each period).

Formula

P = A \left[ \frac{(1+i)^n - 1}{i(1+i)^n} \right]

Meaning

P = present worth of annuity; [(1+i)^n − 1] / [i(1+i)^n] = present worth factor (PWF) or 'P/A factor'

Watch Out

Denominator has i(1+i)^n, NOT just i. Students often drop the (1+i)^n in denominator. Also, this is ordinary annuity (end-of-period); adjust if payments are at period start.

When To Use

Finding lump-sum value TODAY equivalent to equal periodic payments (loan repayment, pension present value).

Formula

P = \frac{A}{i}

Meaning

P = present worth of perpetuity (infinite annuity); A = constant periodic payment forever

Watch Out

Only works if n → ∞. If n is finite (even very large like 100 years), use the standard P/A formula, NOT perpetuity. Perpetuity is strictly infinite.

When To Use

Perpetual payments (e.g., endowments, government bonds, infrastructure maintenance funds lasting forever).

Formula

A = P \left[ \frac{i(1+i)^n}{(1+i)^n - 1} \right]

Meaning

A = uniform periodic payment to repay a loan of amount P; this is the capital recovery factor (CRF) or 'A/P factor'.

Watch Out

This is the INVERSE of the P/A formula. Denominator is (1+i)^n − 1 (without the i multiplication in numerator). Used for loan amortization.

When To Use

Finding monthly/annual payment for a loan, mortgage, or to accumulate a future amount via regular savings.

Formula

A = F \left[ \frac{i}{(1+i)^n - 1} \right]

Meaning

A = periodic payment to accumulate a future amount F; i / [(1+i)^n − 1] = sinking fund factor (SFF) or 'A/F factor'.

Watch Out

Similar to A/P but numerator is JUST i (not i(1+i)^n). This is the inverse of F/A formula. Easy to confuse A/F with A/P.

When To Use

Finding deposit amount needed periodically to reach a target future goal (e.g., saving for equipment replacement).

Value

Annual (n in years) or monthly (n in months)

Symbol

1 year = 12 months

Quantity

Standard annuity period (engineering practice)

Term

Ordinary Annuity

Example

Monthly loan payments of ₱5,000 due on the last day of each month.

Definition

Equal periodic payments occurring at the END of each period (most common in engineering practice and PRC exams).

Term

Annuity Due

Example

Apartment rent of ₱10,000 due on the 1st of each month (beginning of period).

Definition

Equal periodic payments occurring at the BEGINNING of each period (rent, lease).

Term

Perpetuity

Example

Endowment fund yielding ₱500,000 annually in perpetuity for scholarship grants.

Definition

An annuity with infinite duration; payments continue forever at a constant amount.

Term

Deferred Annuity

Example

Loan where payments start 6 months after disbursement (12-month deferment before first ₱5,000 payment).

Definition

An annuity whose first payment occurs more than one period away from the present.

Formula

d = \frac{C - S}{n}

Meaning

d = annual depreciation; C = initial cost; S = salvage value (book value at end of life); n = useful life in years

Watch Out

MUST subtract salvage value S before dividing by n. Common error: forgetting to subtract S, or using S = 0 when it shouldn't be. Also, n and d timing must align (annual depreciation if n is years).

When To Use

Straight-line (SL) depreciation — the simplest and most commonly used method in PRC exams. Assumes constant depreciation each year.

Formula

BV_t = C - d \cdot t

Meaning

BV_t = book value after t years; C = original cost; d = annual depreciation; t = year of interest

Watch Out

This is a LINEAR equation. BV decreases by the SAME amount d each year. If you calculate d wrong, all book values will be wrong.

When To Use

Finding the carrying value (remaining undepreciated value) of an asset at any year t during its life.

Formula

d_t = \frac{(n - t + 1)}{\sum k=1^{n} k} (C - S) = \frac{(n - t + 1)}{n(n+1)/2} (C - S)

Meaning

d_t = depreciation in year t (SYD method); sum of years' digits denominator = n(n+1)/2; numerator = remaining useful life fraction

Watch Out

Denominator is sum of digits 1+2+...+n = n(n+1)/2, NOT just n. Year 1 depreciation is the LARGEST; depreciates in declining amounts. Easy to invert numerator/denominator.

When To Use

Sum-of-years-digits (SYD): accelerated depreciation method favoring larger deductions early. Used for assets that lose value faster initially (vehicles, equipment).

Formula

BV_t = C \cdot (1 - k)^t

Meaning

BV_t = book value in year t (declining balance method); k = depreciation rate (fixed percentage, typically 2/n for double-declining); C = original cost

Watch Out

This is EXPONENTIAL (multiplicative), not linear. If k = 2/n (double-declining), depreciation is aggressive early. Rate k does NOT guarantee the asset reaches salvage value S; it may undershoot or overshoot.

When To Use

Declining balance (DB) or double-declining balance (DDB): accelerated method with exponential decay; larger early deductions.

Formula

d_{1} = \frac{2}{n} \cdot C \quad (\text{double-declining balance, year 1})

Meaning

First-year depreciation in DDB method: use rate 2/n (double the straight-line rate of 1/n) applied to original cost.

Watch Out

Year 1 ONLY uses original cost C; years 2+ use the NEW book value. Students often apply it to C every year (which is wrong).

When To Use

Double-declining balance (DDB) year 1 only; subsequent years apply the rate to declining book value, not original cost.

Value

5, 10, or 20 years depending on industry and asset type

Symbol

n

Quantity

Common useful life (machinery/equipment)

Value

2/n (e.g., for 5-year asset, rate = 40% per year on declining book value)

Symbol

k = 2/n

Quantity

Double-declining balance rate (DDB)

Term

Depreciation

Example

A ₱100,000 machine with ₱10,000 salvage over 5 years depreciates by ₱18,000/year (straight-line).

Definition

The systematic allocation of an asset's cost minus salvage value over its useful life for accounting and tax purposes.

Term

Book Value (BV)

Example

After 3 years of ₱18,000/year depreciation, BV = ₱100,000 − 3(₱18,000) = ₱46,000.

Definition

The undepreciated remaining value of an asset on the balance sheet (original cost minus accumulated depreciation).

Term

Salvage Value (S)

Example

A vehicle with salvage value ₱50,000 after 5 years can be sold for that amount at end of life.

Definition

The estimated value of an asset at the end of its useful life (scrap, resale, or trade-in value).

Term

Useful Life (n)

Example

Heavy equipment with useful life of 10 years depreciates over that 10-year period.

Definition

The expected period (years, months, hours, units) during which an asset will be used profitably.

Term

Accumulated Depreciation

Example

After 3 years of ₱18,000/year depreciation, accumulated depreciation = ₱54,000.

Definition

The sum of all depreciation charges from acquisition to the current date; deducted from cost to get book value.

Formula

NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+i)^t}

Meaning

NPV = net present value; CF_t = cash flow (inflow positive, outflow negative) in year t; i = discount rate (MARR); n = project life

Watch Out

ALL cash flows must be discounted using the SAME discount rate (MARR). Sign convention: inflows are +, outflows are −. Year 0 (present) is NOT discounted.

When To Use

Comparing alternatives on a present-worth basis. Choose the alternative with the HIGHEST (least negative) NPV.

Formula

AW = \frac{NPV}{P/A \text{ factor}} \quad \text{or} \quad AW = NPV \cdot \frac{i(1+i)^n}{(1+i)^n - 1}

Meaning

AW = annual worth (equivalent annual cost or EAC); converts NPV into a uniform annual series over n years.

Watch Out

AW 'spreads' the NPV evenly across n periods. If comparing assets of different lives, AW handles the discrepancy better than NPV. Must use same discount rate i.

When To Use

Comparing alternatives with DIFFERENT life spans or timescales. Choose the alternative with the HIGHEST AW (or LOWEST annual cost if all are costs).

Formula

BC = \frac{PW_{\text{benefits}}}{PW_{\text{costs}}}

Meaning

B/C = benefit-cost ratio; PW_benefits = present worth of all benefits; PW_costs = present worth of all costs (including initial investment)

Watch Out

B/C ≥ 1 means the project is justified (benefits ≥ costs). B/C < 1 rejects the project. Different B/C formulations (e.g., net B/C) exist; stick to the definition given in the problem.

When To Use

Public projects (infrastructure, dams, highways) mandated to justify with B/C ≥ 1. Highest B/C is preferred among alternatives.

Formula

ROR: \text{NPV} = 0 \Rightarrow \sum_{t=0}^{n} \frac{CF_t}{(1+ROR)^t} = 0

Meaning

ROR (or IRR) = rate of return that makes NPV = 0; solve for ROR by trial-and-error or interpolation.

Watch Out

Setting NPV = 0 and solving for i = ROR is non-linear; use trial-and-error or interpolation (linear approximation between two interest rates). Multiple roots possible with non-conventional cash flows (multiple sign changes).

When To Use

Finding the intrinsic return rate of an investment. Compare ROR to MARR: if ROR > MARR, accept; if ROR < MARR, reject.

Formula

\text{Break-even}: C_A(i, n) = C_B(i, n)

Meaning

Set the costs (or present worths) of two alternatives equal to find the interest rate or quantity at which they have equal worth.

Watch Out

Break-even may occur at a specific i, n, or production volume. The 'break-even quantity' is different from 'break-even interest rate'; problem context determines which.

When To Use

Finding the decision point: at what i, n, or volume do two alternatives break even?

Value

8–15% annually depending on risk and industry

Symbol

i or MARR

Quantity

Typical MARR (private sector, Philippines)

Value

4–8% annually (lower due to social benefit)

Symbol

i

Quantity

Typical MARR (public sector, infrastructure)

Term

Net Present Value (NPV)

Example

Project costs ₱100,000 today, generates ₱30,000/year for 5 years at 10% discount. NPV = ₱113,723 − ₱100,000 = ₱13,723 (positive = good).

Definition

The difference between present worth of benefits and present worth of costs; measures net gain/loss in today's pesos.

Term

Annual Worth (AW)

Example

NPV of ₱13,723 over 5 years at 10% = AW of ₱3,623/year (using capital recovery factor).

Definition

The equivalent uniform annual series value of all cash flows over the project life; also called equivalent annual cost (EAC) if only costs.

Term

Minimum Attractive Rate of Return (MARR)

Example

If MARR = 12%, any project with ROR < 12% is rejected regardless of positive NPV at other rates.

Definition

The discount rate (hurdle rate) used by the organization; investments must exceed MARR to be acceptable.

Term

Internal Rate of Return (IRR) or Rate of Return (ROR)

Example

A ₱100,000 investment returning ₱30,000/year for 5 years has IRR ≈ 15.24% (the rate at which NPV becomes zero).

Definition

The discount rate that makes NPV = 0; the intrinsic return rate of an investment.

Term

Benefit-Cost Ratio (B/C)

Example

Dam project with PW of benefits ₱500M and PW of costs ₱300M has B/C = 500/300 = 1.67 (justified; B/C > 1).

Definition

The ratio of present worth of benefits to present worth of costs; used for public project justification.

Formula

i_{eff} = (1 + r/m)^m - 1

Meaning

i_eff = effective annual rate; r = nominal annual rate; m = number of compounding periods per year

Watch Out

Nominal rate is NOT the rate to use in calculations unless m = 1 (annual compounding). Must convert to effective rate or adjust i and n to match period.

When To Use

Converting between nominal and effective rates. ALWAYS convert before comparing rates with different compounding frequencies.

Formula

i_p = r / m

Meaning

i_p = interest rate per compounding period; r = nominal annual rate; m = compounding periods per year (1 year = 12 months = 4 quarters = 2 semi-annual)

Watch Out

If nominal rate is 12% compounded monthly, i_p = 0.12 / 12 = 0.01 (1% per month); then use this with n in months, NOT years.

When To Use

Use i_p in F = P(1 + i_p)^{n_p} formulas; i_p must match the period of n_p.

Formula

i_{eff} = e^r - 1 \quad (\text{continuous compounding})

Meaning

For continuous compounding at nominal rate r, effective annual rate is e^r − 1; rare but sometimes tested.

Watch Out

e ≈ 2.71828; continuous rate is slightly higher than any discrete m. Almost never used in practice but appears in some advanced problems.

When To Use

Only if problem explicitly states 'continuous compounding'; not common in PRC exams but good to know.

Value

m = 1 (annual), m = 2 (semi-annual), m = 4 (quarterly), m = 12 (monthly), m = 365 (daily)

Symbol

m

Quantity

Common compounding frequencies

Term

Nominal Rate (r)

Example

Bank advertises '12% nominal compounded monthly'; r = 0.12.

Definition

The stated annual interest rate without adjusting for compounding frequency within the year.

Term

Effective Annual Rate (i_eff or APY)

Example

12% nominal compounded monthly has i_eff = (1 + 0.12/12)^12 − 1 = 0.1268 or 12.68%.

Definition

The true annual interest rate accounting for within-year compounding; always ≥ nominal for m > 1.

Term

Compounding Period

Example

Monthly compounding means interest is calculated and added 12 times per year (m = 12).

Definition

The time interval at which interest is calculated and added to the principal (annual, semi-annual, quarterly, monthly, daily, continuous).

Values

• F = P(1+i)^n
• Time → moves forward (single sum) or F = A[(1+i)^n − 1]/i (annuity)
• From present TO future
• Finding amount available at end of project or savings goal

Property

Future Worth (FW)

Values

• P = F(1+i)^{−n}
• Time ← moves backward (single sum) or P = A[(1+i)^n − 1]/[i(1+i)^n] (annuity)
• From future TO present
• Comparing alternatives, NPV analysis, loan valuation (most common in engineering economy)

Property

Present Worth (PW)

Values

• d = (C − S) / n (constant)
• Linear (uniform each year)
• Default; simple accounting; conservative
• Book value reaches EXACT salvage S at year n

Property

Straight-Line (SL)

Values

• d_t = [(n − t + 1) / (n(n+1)/2)] × (C − S)
• Declining (larger early, smaller late)
• Accelerated; for faster asset loss; vehicles, equipment
• Year 1 depreciation is LARGEST; reaches salvage S at year n

Property

Sum-of-Years-Digits (SYD)

Values

• d_t = k × BV_{t−1} (rate k fixed, e.g., k = 1/n or 2/n)
• Exponential decay (curved, flattening)
• Accelerated; aggressive early deductions; tech, machinery
• Book value may NOT reach salvage S; depends on rate k

Property

Declining Balance (DB)

Values

• d_1 = (2/n) × C; d_t = (2/n) × BV_{t−1} for t > 1
• Exponential decay (even steeper than DB)
• Accelerated; maximum early tax deductions
• Year 1 uses ORIGINAL cost C; year 2+ use declining BV; may under/overshoot salvage

Property

Double-Declining Balance (DDB)

Values

• F/A
• [(1+i)^n − 1] / i
• Uniform payment series A → Future amount F
• Savings accounts, accumulation funds, retirement planning

Property

Future Worth of Annuity

Values

• P/A
• [(1+i)^n − 1] / [i(1+i)^n]
• Uniform payment series A → Present amount P
• Loan valuation, pension present value, bond pricing

Property

Present Worth of Annuity

Values

• A/P (CRF)
• [i(1+i)^n] / [(1+i)^n − 1]
• Present amount P → Uniform payment series A
• Loan payment calculation, mortgage, equipment financing (MOST COMMON in practice)

Property

Capital Recovery Factor

Values

• A/F (SFF)
• i / [(1+i)^n − 1]
• Future amount F → Uniform payment series A
• Savings plan to reach future goal, equipment replacement fund

Property

Sinking Fund Factor

Values

• NPV = Σ CF_t / (1+i)^t
• NPV > 0 → accept; NPV < 0 → reject; highest NPV wins
• Private projects, single-criterion comparison
• Confusing sign convention (inflows +, outflows −); discount rate must match

Property

Net Present Value (NPV)

Values

• AW = NPV × [i(1+i)^n / ((1+i)^n − 1)]
• AW > 0 → accept; higher AW wins; AW < 0 → reject (cost problems)
• Comparing alternatives with DIFFERENT life spans; most practical
• Forgetting to convert NPV to AW; using different discount rates for alternatives

Property

Annual Worth (AW)

Values

• B/C = PW(benefits) / PW(costs)
• B/C ≥ 1 → justified; B/C > 1 → more benefit than cost; highest B/C wins
• Public infrastructure projects (mandated by law); water, roads, dams
• Mixing up numerator/denominator; forgetting that B/C < 1 rejects the project

Property

Benefit-Cost Ratio (B/C)

Values

• Solve NPV = 0 for i = ROR (trial-and-error or interpolation)
• ROR > MARR → accept; ROR < MARR → reject; highest ROR wins
• Comparing investments; checking internal return; intuitive 'percent return'
• Non-linear equation; multiple roots possible with non-conventional cash flows; interpolation error

Property

Rate of Return (ROR or IRR)

Values

• Payments at END of each period
• F = A[(1+i)^n − 1] / i (standard formula)
• P = A[(1+i)^n − 1] / [i(1+i)^n] (standard formula)
• Loan repayment (payment on last day of month); most common

Property

Ordinary Annuity

Values

• Payments at BEGINNING of each period
• F = A[(1+i)^n − 1] / i × (1+i) = (standard) × (1+i)
• P = A[(1+i)^n − 1] / [i(1+i)^n] × (1+i) = (standard) × (1+i)
• Rent/lease paid on 1st of month (beginning of period)

Property

Annuity Due

Values

• Payments forever (n → ∞)
• F → ∞ (infinite accumulation)
• P = A / i (perpetual present worth, FINITE)
• Endowment funds, government bonds, 'forever' maintenance (rare in practice)

Property

Perpetuity