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Little's Law (WIP, Throughput, Cycle Time) and Takt Time formulas
Size Kanban WIP caps with Little's Law (WIP = Throughput * CycleTime) and balance production lines using Takt Time and workstation efficiency.
Read the guide →Operations Research & Revenue ManagementLittlewood Rule and EMSR-b Seat Protection Booking Limit Formula
Aggregate high-fare classes via Belobaba’s EMSR-b weighted fare p_pool, pooled SD SQRT(SUM sigma^2), and Littlewood’s critical ratio 1 - p_low / p_pool to set discount booking limits.
Read the guide →Soft-Switching Power Electronics & EV ChargersLLC Resonant Converter Tank Frequency, Ln & Q Formula
Calculate Half-Bridge LLC Series Resonant Frequency f_r = 1000 / (2*PI()*SQRT(L_r_uH * C_r_uF)) (kHz), Characteristic Impedance Z_0 = SQRT(L_r / C_r), Inductance Ratio L_n = L_m / L_r, and Loaded Quality Factor Q_e = Z_0 / R_ac.
Read the guide →MathLN function: calculate continuous growth rates and exact doubling time
Use the natural logarithm LN (inverse of EXP) to solve for unknown compounding time or continuous log returns, and handle zero/negative #NUM! inputs.
Read the guide →Maritime Energy Trading, LNG Cryogenics & Shipping EconomicsLNG Carrier Boil-Off Gas (BOG), Heel & Delivered MMBtu Formula
Calculate LNG Carrier Voyage Boil-Off Gas Loss (m³ LNG), Arrival Tank Volume (m³), Net Discharged Cargo After Cryogenic Heel (m³), and Delivered Regasification Energy (MMBtu).
Read the guide →Quantitative Finance & Time SeriesLo-MacKinlay Variance Ratio VR(q) Random-Walk Test Formula
Test the Random Walk Hypothesis in Excel and Arte Sheets using Lo and MacKinlay’s (1988) variance ratio VR(q) = var_q / (q * var_1) and asymptotic Z(q) statistic.
Read the guide →FinancialLoan amortization schedule: reconcile payment, interest, principal, and ending balance
Combine PMT, IPMT, PPMT, and FV to build a self-balancing loan amortization schedule that amortizes cleanly to zero.
Read the guide →Regional Economics & Urban PlanningLocation Quotient & Economic Base Multiplier Formula
Calculate sectoral Location Quotients LQ_i = (E_ir / E_r) / (E_in / E_n), bifurcate Basic export employment MAX(0, E_ir - E_r*(E_in/E_n)), and compute the Economic Base Multiplier.
Read the guide →Engineering & ScienceLockhart-Martinelli Parameter (X_tt) & Chisholm Two-Phase Multiplier (φ_L²)
Calculate gas-liquid two-phase pipe frictional pressure drop multiplier φ_L^2 = 1 + C / X_tt + 1 / X_tt^2 from vapor quality x, phase densities, and viscosities.
Read the guide →MathLOG and LOG10: calculate custom-base logarithms, binary depth, and decibels
Compute base-10 orders of magnitude and decibels with LOG10, or binary/custom-base logarithms with LOG(number, base).
Read the guide →StatisticalLog-Gamma Scaling and Erlang Multi-Stage Waiting-Time Probability (GAMMALN & GAMMA.DIST)
Evaluate exact log-gamma LN(FACT(k - 1)), gamma density ordinates, and closed-form Erlang k-stage waiting-time cumulative probabilities using EXP, LN, and FACT.
Read the guide →Statistical & Mathematical FormulasLog-Rank (Mantel-Cox) Survival Test: Chi-Square & Hazard Ratio
Compare two censored Kaplan-Meier survival curves across discrete event times via the Mantel-Cox Log-Rank test χ^2 = (Σ O_1j - Σ E_1j)^2 / Σ V_j with exact hypergeometric tie variance.
Read the guide →Statistical AnalysisLognormal Cumulative Probability, Mean-Median Skew & Quantiles (LOGNORMDIST & LOGNORM.INV)
Model strictly positive right-skewed claim severities and response latencies by mapping log-space mean mu and volatility sigma into lognormal medians, expected means, cumulative probabilities, and VaR quantiles.
Read the guide →Engineering & ScienceLondon Magnetic Penetration Depth & BCS Superconducting Energy Gap Formula
Calculate the two-fluid temperature-dependent London penetration depth lambda_L(T) = lambda_0 / SQRT(1 - (T / Tc)^4) in nm and weak-coupling BCS Cooper-pair energy gap 2*Delta_0 = 3.528 * kB * Tc in meV.
Read the guide →StatisticalLongest winning or daily habit streak formula (current and best streak)
Track consecutive wins, incident-free days, or SLA hits with a resetting streak counter and summarize Best and Current Streak.
Read the guide →Lookup & ReferenceLook Up & Concatenate All Matches Into One Comma-Separated Cell (TEXTJOIN + FILTER)
Return every invoice, serial number, or owner matching a lookup key inside a single comma-delimited summary cell using TEXTJOIN(", ", TRUE, FILTER(..., "No Matches")).
Read the guide →Lookup & ReferenceLook up a table row and transpose it into a vertical summary card (HSTACK + TRANSPOSE)
Build a clean 2-column Field | Value inspector card from a wide horizontal table by pairing XLOOKUP, TRANSPOSE, and HSTACK.
Read the guide →Lookup & ReferenceLookup a value between two numbers (explicit Min and Max columns)
Match an input inside inclusive [Min, Max] bracket rows using FILTER—rejecting out-of-bounds values that approximate VLOOKUP accepts.
Read the guide →Lookup & ReferenceLookup Across Selectable Rate Tables Without Volatile INDIRECT (IFS & VLOOKUP)
Route SKU price lookups dynamically between Standard and Partner price books based on a customer tier cell using non-volatile IFS and VLOOKUP.
Read the guide →Lookup & ReferenceLookup and sum an entire column or row dynamically (SUM + XLOOKUP)
Match a quarter, month, or scenario header horizontally and SUM, AVERAGE, or MAX the entire matched column in one readable formula.
Read the guide →Financial & Business FormulasLookup Hourly Rates & Sum Total Invoice Billing (XLOOKUP & SUMPRODUCT)
Pull role-based hourly billing rates from a master rate card with XLOOKUP and roll up total project invoice value via SUMPRODUCT(Hours, Looked_Up_Rates).
Read the guide →Lookup & ReferenceLookup Latest File Version, Artifact Filename & Author (XLOOKUP Reverse Search)
Retrieve the most recently appended file revision, version integer, and uploader for any document family in a chronological release log using XLOOKUP search_mode = -1 and MAXIFS.
Read the guide →Lookup & ReferenceLookup latest price or revision by date (unsorted history safe)
Retrieve the most recent unit cost or contract rate for an item even when historical log rows are NOT sorted chronologically.
Read the guide →Lookup & ReferenceLookup price with client-specific contract override and standard rate fallback
Check a negotiated client contract rate table first and automatically fall back to your master catalog price when no custom override exists.
Read the guide →Lookup & ReferenceLookup Product Unit Cost & Extended Gross Profit Without #N/A (XLOOKUP)
Retrieve standard unit cost from a master catalog with built-in XLOOKUP if_not_found protection and compute extended gross profit (Unit_Price - Unit_Cost) * Qty safely.
Read the guide →Lookup & ReferenceLookup the date or name of a MAX or MIN value: first and last matching rows
Return the label associated with a maximum or minimum, and choose the first or last matching row when values tie.
Read the guide →EngineeringLorentz Factor (gamma), Relativistic Time Dilation, Length Contraction & Kinetic Energy
Calculate Hendrik Lorentz & Albert Einstein's relativistic factor gamma = 1 / SQRT(1 - beta^2), dilated coordinate time, contracted length, and (gamma - 1)*mc^2.
Read the guide →Engineering & ScienceLorentz-Lorenz Molar Refractivity (RM) & Clausius-Mossotti Polarizability Formula
Calculate phase-invariant molar refractivity R_M = ((n^2 - 1) / (n^2 + 2)) * (MW / rho) in cm^3/mol and electronic molecular polarizability alpha_0 (in Angstrom^3) from optical refractive index n and density rho.
Read the guide →ScienceLotka-Volterra Equilibrium and Small-Amplitude Period in Excel
Explore predator-prey equilibria, local rates of change and a constant per-capita removal scenario, with a clear distinction between a linearized period and a full nonlinear trajectory.
Read the guide →Private Equity Secondaries, Portfolio Pricing & SPA MechanicsLP Secondary NAV Roll-Forward, Effective Discount & MOIC
Calculate LP Secondary Sale-and-Purchase Agreement (SPA) Closing Price ($M), Effective Closing Discount (%) After Interim Cash Flows, Day-1 Upfront Wire ($M), and Buyer Gross MOIC (x).
Read the guide →Metal Additive Manufacturing, Laser Powder Bed Fusion (LPBF/SLM) & Melt-Pool QALPBF Metal 3D Printing Volumetric Energy Density Formula
Calculate Linear Energy Density P/v (J/mm), Volumetric Energy Density VED = P/(v*h*t) (J/mm³), Theoretical Laser Melt Build Rate (cm³/hr), and Lack-of-Fusion vs Keyhole Porosity Window.
Read the guide →Engineering & ScienceLucas Corresponding-States Low-Pressure Gas Viscosity Formula (µPa·s)
Predict non-polar gas dynamic viscosity η (µPa·s) solely from critical temperature T_c, critical pressure P_c (bar), and molar mass M without Lennard-Jones collision integrals!
Read the guide →Engineering & ScienceLucas-Washburn Capillary Wicking Length & Assay Flow Time
Calculate horizontal capillary penetration distance L(t) = SQRT((γ * r * cos(θ) / (2 * μ)) * t) (mm) and lateral-flow diagnostic transit time t_flow (s) in porous paper strips and heat-pipe wicks.
Read the guide →Engineering & ScienceLuedeking-Piret Bioprocess Kinetics: Growth vs. Non-Growth Product Rate
Calculate specific cellular product formation rate q_p = alpha * mu + beta (g product / g DCW / h) and volumetric bioreactor productivity r_p = (alpha * mu + beta) * X (g/(L*h)) across exponential and stationary fermentation phases.
Read the guide →Math & TrigLuhn Algorithm (Mod 10) formula: generate and validate account check digits
Generate and check a Luhn checksum for an illustrative eight-digit payload. A passing checksum does not validate an account or identity.
Read the guide →EngineeringLumber Board Feet (BF) & MBF pricing formula: (Thick" × Width" × Length') / 12
Convert nominal hardwood and framing tally lists into Board Feet, Linear Feet, and $/MBF (per 1,000 board feet) lumber orders.
Read the guide →Engineering & ScientificLumped-Capacitance Biot Gate (Bi <= 0.1) & Newtonian Thermal Quench Time
Verify lumped-system validity via Characteristic Length L_c = V / A_s and Biot number Bi = h * L_c / k, then calculate thermal time constant tau and exact cooling/heating time.
Read the guide →Clinical Medicine & PhysiologyLund-Browder Pediatric vs Wallace Rule of Nines TBSA% Formula: Age-Adjusted Head & Legs
Calculate age-adjusted Total Body Surface Area %TBSA in infants, children, and adults using Charles C. Lund and Newton C. Browder’s (1944) developmental anatomy chart (where infant Head = 19% vs adult 7%) alongside Alexander Wallace’s (1951) Adult Rule of Nines.
Read the guide →Biopharmaceutical Fill-Finish, Freeze-Drying (Lyophilization) & BiologicsLyophilization Primary Drying Vial Sublimation & Time Formula
Calculate Single-Vial Shelf Heat Transfer Input Q_v (W), Ice Sublimation Rate dm/dt (g/hr), Nominal Primary Drying Duration (hr), and Cake Collapse Safety Margin (K).
Read the guide →StatisticalM/M/1 Queueing formula: server utilization, queue length (Lq) & wait time (Wq)
Model call-center, checkout, and API worker queues using M/M/1 arrival rate lambda, service rate mu, utilization rho = lambda/mu, Lq, and Wq.
Read the guide →Operations & LogisticsM/M/1/K Finite-Buffer Queue Blocking Probability, Drop Rate & Mean Occupancy
Model finite-capacity single-server queues (API connection pools, router packet buffers, call trunks) with exact M/M/1/K blocking probability P_K, admitted throughput lambda_eff, mean occupancy L, and rho = 1 singularity handling.
Read the guide →Investment Banking & M&A StructuringM&A Contingent Earnout Sliding-Scale vs Cliff Formula
Calculate Linear Sliding-Scale Earnout Payout E_max * MIN(1, MAX(0, (R_act - R_min) / (R_max - R_min))), Binary All-or-Nothing Cliff Payout, Present-Value Total Consideration, and Marginal Revenue Slope.
Read the guide →Investment Banking & Corporate M&AM&A EPS Accretion / Dilution & Breakeven Synergy Formula
Calculate Pro-Forma M&A Earnings Per Share under mixed stock-and-debt acquisition financing, EPS Accretion/Dilution percentage, and Pre-Tax Breakeven Synergies.
Read the guide →Mergers & Acquisitions, Transaction Advisory & Deal Closing MechanicsM&A EV-to-Equity Bridge, Working Capital Peg & Wire Formula
Calculate M&A Completion-Accounts Closing Net Debt (Debt - Cash), Dollar-for-Dollar Net Working Capital (NWC) Peg Adjustment (Actual_NWC - Target_Peg), Adjusted Equity Purchase Price, and Day-1 Seller Wire Proceeds Net of Escrow.
Read the guide →Investment Banking & M&A StructuringM&A Floating-Value Stock Collar Exchange Ratio Formula
Calculate the floating stock exchange ratio ER = V_offer / MIN(P_cap, MAX(P_floor, P_acq)), implied target per-share payout, and pro-forma target ownership across M&A collar bands.
Read the guide →FinancialMacaulay Duration and Modified Duration Bond Formulas (DURATION & MDURATION)
Compute present-value-weighted cash-flow maturity (Macaulay Duration) and yield sensitivity (Modified Duration) directly from coupon schedules without opaque Analysis ToolPak black boxes.
Read the guide →EngineeringMach Number, Speed of Sound in Air & Aerodynamic Dynamic Pressure (q) formulas
Calculate local speed of sound a = SQRT(1.4 * 287.05 * (T_C + 273.15)), flight Mach Number V / a, compressible regime, and dynamic pressure 0.5 * rho * V^2.
Read the guide →Engineering & ScienceMach-Zehnder Interferometer (MZI): Phase Transmittance & Extinction Ratio
Calculate integrated silicon-photonics Mach-Zehnder bar-port transmittance T = COS(π * Δn_eff * L / λ)^2 and finite coupler Extinction Ratio ER = -20 * LOG10(ABS(1 - 2 * κ)) (dB).
Read the guide →Personal Finance & PlanningMacro-Nutrient Meal & Grocery Budget Planner (SUMPRODUCT Atwater 4-4-9 & Protein Cost)
Scale serving multiples across daily meals with SUMPRODUCT to track total protein, carbs, fats, Atwater 4-4-9 calories, protein calorie share %, and daily grocery budget limits.
Read the guide →Corporate Tax Accounting, CapEx Modeling & Capital BudgetingMACRS 200% DB & IRC §168(k) Bonus Depreciation Formula
Calculate IRC Section 168(k) First-Year Bonus Depreciation ($M), Combined Year-1 MACRS Half-Year 200% Declining-Balance Deduction ($M), Cash Tax Shield ($M), and Adjusted End-of-Year-1 Tax Basis ($M).
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