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Variable sales commission split between AE and Solutions Engineer (XLOOKUP)
Allocate a deal commission pool dynamically across Account Executive and Solutions Engineer roles using a tiered deal-size split table.
Read the guide →Financial & AccountingVariable-Rate Compound Growth Schedule (PRODUCT(1 + Rates) & FVSCHEDULE)
Compound an initial principal across a sequence of changing periodic interest or return rates using PRODUCT(1 + rateRange) and compute cumulative growth and equivalent geometric return.
Read the guide →Retail Pricing & Game TheoryVarian Model of Sales Promotional Price Floor Formula
Calculate Hal Varian’s (1980) mixed-strategy promotional discount floor p_min = c + (U / (U + I)) * (r - c), equilibrium store profit, and sale-frequency CDF F(p).
Read the guide →Quantitative Volatility Derivatives, OTC Variance Swaps & Vega ConvexityVariance Swap Vega-to-Variance Notional & Convexity Formula
Calculate Variance Notional N_var = N_vega / (2*K_vol), Variance Swap Settlement P&L, Linear Volatility Swap P&L, and Strictly Non-Negative Volatility Convexity Gain.
Read the guide →Machine Learning & Information TheoryVariation of Information (VI) and Normalized Partition Distance Formula
Compute Meilă’s (2007) true metric distance between clusterings VI(U, V) = 2*H(U, V) - H(U) - H(V) in bits and Kraskov’s Normalized Variation of Information in [0, 1].
Read the guide →Engineering & ScienceVarshni Semiconductor Bandgap Temperature Dependence Formula Eg(T)
Calculate semiconductor bandgap shrinkage with temperature E_g(T) = E_g0 - (alpha * T^2) / (T + beta) (eV) and cutoff absorption wavelength lambda_g (nm) for Silicon, GaAs, and LED/laser diodes.
Read the guide →Credit Risk & Banking RegulationVasicek ASRF Basel Conditional Default & Capital Formula
Compute Vasicek’s (2002) Asymptotic Single-Risk-Factor stressed default rate CPD = Phi((Phi^-1(PD) + SQRT(rho)*z_q) / SQRT(1 - rho)) and Basel unexpected-loss capital K_UL.
Read the guide →Financial ModelingVasicek Short-Rate Term Structure: Zero-Coupon Bond Price P(0,T) & Spot Yield
Price default-free zero-coupon bonds and derive continuously compounded spot yields under Oldrich Vasicek’s (1977) Ornstein-Uhlenbeck mean-reverting interest-rate model.
Read the guide →Venture Capital & Startup Cap TablesVC Option Pool Shuffle & Effective Pre-Money Formula
Calculate the Series A Share Price P_A, New Option Pool Top-Up Shares, Effective True Pre-Money Valuation of existing shares, and Founder Ownership under Investor-Friendly Pre-Money Pool Creation vs Shared Post-Money Pool.
Read the guide →Venture Capital & Startup Cap TablesVC Participating vs Non-Participating Preference Formula
Calculate Venture Capital exit payouts under 1x Non-Participating Preferred versus Capped Participating Preferred MIN(W, MAX(alpha*W, MIN(Cap, Pref + alpha*MAX(0, W - Pref)))) across all four exit zones.
Read the guide →Venture Capital Cap Tables, Down-Round Restructuring & Pay-to-Play ClausesVC Pay-to-Play Pro-Rata & Preferred-to-Common Formula
Calculate Venture Capital Pay-to-Play Down-Round Pro-Rata Check Requirement ($M), Preserved Preferred Shares vs Penalty Shares Forcibly Converted 1:1 into Common Stock, and Surviving Senior Liquidation Preference Stack ($M).
Read the guide →EngineeringVehicle stopping distance formula: reaction distance + v² / (2·μ·g) braking
Calculate thinking distance, kinetic braking distance, and total stopping distance in meters from speed (km/h), reaction time, and road friction μ.
Read the guide →Charts & Visual DataVenn Diagram Overlap Formula: 2- & 3-Set Inclusion-Exclusion Union & Exclusive Counts
Reconcile audience or SKU overlaps across three overlapping sets using the Principle of Inclusion-Exclusion so Venn diagram region labels and unique-reach totals never double-count intersections.
Read the guide →Venture Capital & Structured CreditVenture Debt Warrant Coverage & All-In Cost Formula
Calculate Net Funded Venture Debt Proceeds, Cash-Only Effective Annual Cost (with upfront commitment and end-of-term final payment fees), Warrant Intrinsic Payoff, and All-In Equity-Adjusted Cost.
Read the guide →Engineering & ScientificVenturi Meter Differential-Pressure Flow Rate & Beta^4 Velocity-of-Approach
Convert differential pressure dP across a converging Venturi tube (ISO 5167) into volumetric and mass flow rate using throat beta ratio d_2/d_1 and approach factor 1 / SQRT(1 - beta^4).
Read the guide →Financial & AccountingVerdoorn’s Law (Kaldor’s Second Growth Law) Formula: Dynamic Productivity & Employment Elasticity
Forecast manufacturing labor-productivity growth g_p = a + b * g_Y and induced employment growth g_E = (1 - b) * g_Y - a from output expansion g_Y using Petrus J. Verdoorn (1949) and Nicholas Kaldor’s (1966) increasing-returns law.
Read the guide →MathVerhulst Logistic S-Curve Growth & Carrying Capacity formula (EXP & LN)
Model bounded market adoption and population saturation via P(t) = K / (1 + ((K - P0)/P0) * EXP(-r * t)) and solve for the K/2 peak-growth inflection time.
Read the guide →Data Validation & AuditVerify Value Does Not Exist in a Blocklist (COUNTIF = 0 Anti-Join)
Enforce "must not exist in list" blocklist rules on vendor IDs, reserved usernames, or sanctioned codes using COUNTIF($E$2:$E$6, A2) = 0 and audit batch blocklist hits.
Read the guide →International Trade & Global Value ChainsVertical Specialization (VS) & VAX Export Ratio Formula
Calculate Hummels, Ishii, and Yi’s (2001) Vertical Specialization Foreign Value Added in Exports VS = (M_int / Y_gross) * X_gross, Domestic Value Added (DVA), and Johnson-Noguera VAX ratio.
Read the guide →Aggregate Sizing & Mineral Processing, VSMA Vibrating Screen G-Force & Bed StratificationVibrating Screen G-Force, Bed Depth & VSMA Ratio Formula
Calculate Peak-to-Peak Circular Stroke (mm), Centrifugal Vibrator Acceleration Kv (G), Discharge-End Oversize Bed Depth (mm), and VSMA Bed-to-Aperture Ratio (<= 4.0).
Read the guide →EngineeringVibration Damping Ratio (zeta), Damped Frequency & Logarithmic Decrement
Classify mass-spring-damper dynamics into underdamped, critically damped, and overdamped regimes via zeta = c / (2 * SQRT(k * m)) and compute omega_d and logarithmic decrement.
Read the guide →Vibratory Bowl & Pan Feeders, Packaging Automation & Granular Micro-Hopping MechanicsVibratory Feeder Throw Factor Gamma & Conveying Formula
Calculate Peak Trough Acceleration in Gs, Dimensionless Vertical Throw Factor Gamma = (a*omega²*SIN(beta))/g, Ideal Horizontal Peak Velocity (mm/s), and Effective Transport Speed (mm/s).
Read the guide →Metallurgy, Mechanical Testing & Materials ScienceVickers Microhardness (HV) & Tabor Yield-Strength Formula
Calculate ASTM E384 / ISO 6507 Equivalent Test Load (kgf), Vickers Diamond-Pyramid Hardness (HV), True Pyramidal Contact Pressure (MPa), and Tabor Yield-Strength Estimate (MPa).
Read the guide →Urban Economics & Congestion PricingVickrey Bottleneck Congestion & Dynamic Toll Formula
Calculate Vickrey’s (1969) equilibrium rush-hour commute cost delta*(N/s), peak queue length, and pure queuing deadweight loss eliminated by a dynamic congestion toll.
Read the guide →Mechanism Design & Public EconomicsVickrey-Clarke-Groves (VCG) Pivotal Externality Fee Formula
Make truthful valuation reporting a dominant strategy in shared public-project decisions by computing each participant’s Clarke pivotal externality payment.
Read the guide →Marketing Science & Dynamic Media Mix ModelingVidale-Wolfe Advertising Saturation & Decay Formula
Calculate Vidale & Wolfe’s Untapped Market Share (M - S)/M, Instantaneous Sales Drift dS/dt = r*A*(M - S)/M - lambda*S, Zero-Drift Maintenance Ad Spend, and Steady-State Equilibrium Sales.
Read the guide →Healthcare & Clinical CalculatorsVienna CATS Score for Cancer-Associated Venous Thromboembolism (0–8)
Augment the Khorana chemotherapy VTE score (0–6) with D-dimer (>= 1.44 µg/mL, +1) and soluble P-selectin (>= 53.1 ng/mL, +1) to stratify 6-month cancer VTE incidence.
Read the guide →Digital Media & Programmatic Ad-TechViewable vCPM, Publisher eCPM & Fill Rate Formula
Calculate programmatic Ad-Request Fill Rate, MRC Advertiser Viewable CPM (vCPM), Publisher Net Served eCPM, and Ad-Request Yield (rCPM) to optimize price floors without killing total revenue.
Read the guide →Engineering & ScienceVinet (Rose-Smith-Ferrante) Universal Equation of State Formula (GPa)
Compute ultra-high-compression crystal pressure P(V) = 3 * K_0 * ((1 - x) / x^2) * EXP(1.5 * (K′_0 - 1) * (1 - x)) where linear lattice ratio x = (V / V_0)^(1/3).
Read the guide →EngineeringVis-Viva Orbital Velocity Equation: perigee, apogee & escape speed
Calculate ideal two-body elliptical and circular orbital speeds from center-based radii, plus local parabolic escape speed.
Read the guide →Clinical Medicine & PhysiologyVisceral Adiposity Index (VAI) Calculator: Amato Sex-Specific Waist-BMI-Lipid Formula
Quantify visceral adipose dysfunction and cardiometabolic risk using Marco C. Amato et al.’s (2010, Diabetes Care) sex-specific Visceral Adiposity Index combining Waist Circumference, BMI, Triglycerides (mmol/L), and HDL-C (mmol/L).
Read the guide →Structural & Civil EngineeringVlasov Open I-Beam Warping Constant C_w and Torsion Length Formula
Calculate open I-beam Saint-Venant torsion J_sv, Vlasov warping constant C_w = t_f*b_f^3*h_0^2 / 24, and characteristic warping length a_warp = SQRT(E*C_w / (G*J_sv)).
Read the guide →LookupsVLOOKUP approximate match: assign parcel rates from sorted weight thresholds
Approximate VLOOKUP (TRUE) is dangerous for product IDs, yet ideal for numeric tier schedules such as postage bands or commission rates—provided the threshold column starts at the lowest valid value and stays sorted ascending.
Read the guide →LookupsVLOOKUP exact match: find a price by product code
Look up a product code in a small price list, return its price, and make missing codes visible instead of accepting the wrong match.
Read the guide →Lookup & ReferenceVLOOKUP If Blank Return Blank Without Wiping Out Legitimate Zero Balances
Test LEN(VLOOKUP(...)) = 0 instead of VLOOKUP(...) = 0 so empty source cells return a clean blank string "" while genuine numeric 0.00 balances remain intact.
Read the guide →Lookup & ReferenceVLOOKUP with Concatenated Composite Key: Delimiter-Safe Multi-Criteria Lookup
Perform fast O(1) VLOOKUP multi-column lookups in legacy and modern workbooks by building a delimiter-protected leftmost composite key A2 = B2 & "|" & C2.
Read the guide →Lookup & ReferenceVLOOKUP with MATCH: make the column index dynamic when headers move
Replace a hard-coded VLOOKUP column number with MATCH so formulas adapt to selected period headers and inserted columns without breaking.
Read the guide →StatisticalVO2 Max & Aerobic METs formulas: Cooper 12-Minute Run & Uth-Sorensen HR Ratio
Practice arithmetic for two indirect VO2max estimates using fictional data, and convert an estimate into conventional MET units.
Read the guide →Petroleum Production & Nodal AnalysisVogel Two-Phase Oil Inflow Performance (IPR) Formula
Calculate Absolute Open Flow Potential q_max and Target Bottomhole Drawdown Rate q_o = q_max * (1 - 0.2*(P_wf/P_r) - 0.8*(P_wf/P_r)^2) for saturated solution-gas-drive oil wells.
Read the guide →Glass Science, Polymer Rheology & Optical Fiber DrawingVogel-Fulcher-Tammann Glass Viscosity & Fragility Formula
Calculate Supercooled Glass Melt Log10 Viscosity A + B/(T - T0), Glass Transition Temperature Tg at 10^12 Pa*s, Angell Kinetic Fragility Index m = B*Tg / (Tg - T0)^2, and Dynamic Melt Viscosity (Pa*s).
Read the guide →EngineeringVoltage Divider (Vout) & LED current-limiting series resistor formulas
Calculate resistive divider output voltage Vout = Vin * R2/(R1+R2), solve R2 for ADC scaling, and size LED series resistors with Wattage checks.
Read the guide →Math & TrigVolume and surface area formulas: sphere, cone, and rectangular box
Compute packaging and tank volumes plus total surface areas for spheres, right circular cones, and rectangular prisms using PI and SQRT.
Read the guide →Statistical & Mathematical FormulasVon Bertalanffy Growth Function (VBGF): Asymptotic Length & Age
Predict organism length-at-age L(t) = L_∞ * (1 - EXP(-K * (t - t_0))) and invert catch length into biological age t(L) = t_0 - LN(1 - L / L_∞) / K for fisheries stock assessment and wildlife ecology.
Read the guide →Spatial Economics & Agricultural Land UseVon Thünen Land Rent & Crop-Ring Boundary Formula
Calculate Johann Heinrich von Thünen’s (1826) agricultural bid-rent R_i(d) = Y_i*(p_i - c_i - f_i*d), concentric crop-switch boundary d*_12, and zero-rent extensive frontier.
Read the guide →Dynamic arraysVSTACK function: stack separate regional tables into one consolidated list
Append multiple ranges top-to-bottom into a single dynamic table for unified totals, sorting, or filtering with VSTACK.
Read the guide →FinancialWACC formula: Weighted Average Cost of Capital with interest tax shield
Calculate corporate WACC = (E/V)*Re + (D/V)*Rd*(1 - TaxRate) to discount unlevered Free Cash Flows in DCF models.
Read the guide →Operations & Supply ChainWagner-Whitin Dynamic Lot-Sizing Forward DP Formula vs Silver-Meal
Implement Wagner and Whitin’s (1958) zero-inventory regeneration dynamic programming recurrence F_t in Excel and Arte Sheets and expose where myopic Silver-Meal overpays.
Read the guide →Mechanical Design & Machine ElementsWahl Helical Spring Shear Stress (K_W) and Spring Rate Formula
Calculate spring index C = D / d, Wahl curvature-shear factor K_W = (4C - 1)/(4C - 4) + 0.615/C, inner-coil shear stress tau_max, and axial spring rate k = d^4*G / (8*D^3*N_a).
Read the guide →Engineering & ScienceWakao-Kaguei Packed-Bed Sherwood & Nusselt Correlation (Catalyst & Adsorption)
Calculate external particle-to-fluid mass and heat transfer in packed catalytic and adsorption beds via Wakao & Kaguei’s correlation Sh = 2 + 1.1 * Re_p^0.6 * Sc^(1/3) and Nu = 2 + 1.1 * Re_p^0.6 * Pr^(1/3).
Read the guide →Econometrics & Causal InferenceWald Instrumental Variables Estimator and Complier LATE Formula
Correct imperfect randomized encouragement non-compliance using the Angrist-Imbens-Rubin Wald IV ratio LATE = (E[Y|Z=1] - E[Y|Z=0]) / (E[D|Z=1] - E[D|Z=0]).
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