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Every number you need, and the working underneath it.

Finance

Loans, savings, investing and everyday money maths. Every rate, term and price is an input you control, so nothing here is tied to a particular year’s figures.

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Loan payment & amortisationRepayment on a fixed-rate amortised loan, with a year-by-year schedule.Mortgage repaymentFull monthly housing cost: principal, interest, rates, insurance and strata.Compound interestGrowth of a lump sum plus regular contributions.Simple interestInterest charged on the original principal only.Savings goalWhat you must put away each month to hit a target.Present & future valueWhat a future sum is worth today, and vice versa.Annuity / drawdown payoutHow much a pot pays out each month until it runs dry.Retirement projectionBalance at retirement and how long it lasts.Car loanRepayment after deposit, trade-in and on-road costs.Credit card payoffHow long a balance takes to clear at a fixed payment.APR to effective rate (APY)Converts a nominal rate into the true annual yield.Inflation & purchasing powerWhat money is worth after years of price rises.Return on investmentTotal and annualised return on an investment.CAGR (compound annual growth rate)The smooth yearly rate that links a start and end value.NPV & IRRDiscounted value of a project, and the rate that makes it break even.Rule of 72Quick mental estimate of doubling time.Salary & hourly rate converterMoves between hourly, weekly, monthly and annual pay.Tip & bill splitTip amount and what each person owes.Discount & sale pricePrice after one or two discounts, and what you save.Sales tax / GST / VATAdds tax to a net price, or strips it out of a gross one.Margin & markupTurns cost and price into margin, markup and profit.Break-even pointUnits and revenue needed before a business turns a profit.Debt-to-income ratioThe ratio lenders use to size your borrowing.How much can I borrowMaximum loan a given repayment supports.Bond price & yieldPresent value of a bond’s coupons and face value.Average share cost & positionBlended entry price across several buys.Currency conversion with feesWhat you actually receive after spread and fixed fees.Depreciation (straight-line & declining)Annual depreciation expense and remaining book value under two common methods.Payback periodHow many years of cash flows it takes to recover an initial investment, plain and discounted.Liquidity ratios (current, quick & cash)Three views of whether current assets can cover current liabilities.Leverage & solvency ratiosHow much of a company is financed by debt, and how easily it covers interest.Working capital & turnoverThe cash cushion between current assets and current liabilities, and how hard it is working.Cash conversion cycleDays between paying for inventory and collecting cash from its sale.WACC (weighted average cost of capital)The blended return a company must earn to satisfy both lenders and shareholders.CAPM (cost of equity)The return shareholders should require, given the market risk a stock carries.Dividend discount model (Gordon growth)Values a stock as its dividends, discounted and growing forever.Dividend yield & payout ratioWhat a stock pays relative to its price, and how much of earnings that consumes.EBITDA & marginOperating profitability before interest, tax, depreciation and amortisation.Enterprise valueThe theoretical takeover price of a company: equity plus debt, minus cash on hand.Stock valuation multiples (P/E, P/B, PEG)Common price ratios used to judge whether a stock is cheap or expensive.DuPont analysis (ROE breakdown)Splits return on equity into profitability, efficiency and leverage.Sharpe ratioReturn earned above the risk-free rate, per unit of volatility taken on.Perpetuity valuePresent value of a cash flow that continues forever, flat or growing.Balloon loan paymentRegular instalment on a loan amortised over a long schedule, plus the lump sum due early.Sustainable growth rateThe fastest a company can grow using only retained profit, no new borrowing or share issues.Graham numberBenjamin Graham’s quick estimate of a fair share price from earnings and book value.Bond yield to maturity (YTM)The return a bond earns if held to maturity, found by solving price for yield.Bond duration & convexityHow sensitive a bond’s price is to a change in yield, to first and second order.Yield to call (YTC)Approximate return on a callable bond if the issuer redeems it early.Cap rate (capitalisation rate)The unlevered return a rental property generates from its net operating income.Gross rent multiplier (GRM)A quick screen of property price against the rent it collects, before expenses.DSCR (debt service coverage ratio)Whether a property or business generates enough income to cover its debt payments.Asset turnover ratiosHow hard a company’s inventory, receivables and total assets are working to generate revenue.Free cash flow & yieldCash a business generates after the capital spending needed to keep running.Rent or buy comparisonNet cost of renting versus buying a home over the years you plan to stay.Car lease payment (money factor)Monthly lease payment from depreciation and finance charges.Real interest rate (Fisher equation)What an interest rate is worth once expected inflation is stripped out.PVGO (present value of growth opportunities)How much of a stock’s price reflects expected future growth, versus current earnings alone.Growing annuity present valuePresent value of a stream of payments that increases by a fixed rate each period.Degree of operating leverage (DOL)How much operating profit swings for a given swing in sales, given a company’s fixed cost base.Revenue valuation multiples (EV/Sales & P/S)Prices a company against its revenue, useful when earnings are thin or negative.Debt-to-capital ratioWhat share of a company’s permanent funding comes from debt rather than equity.Interest-only mortgageThe interest-only payment, and what it jumps to once principal repayments begin.Biweekly mortgage paymentTime and interest saved by paying half the monthly payment every two weeks instead.Credit utilization ratioHow much of your available credit is in use, per card and overall.Bond equivalent yieldConverts a discount instrument price into an annualised coupon-style yield.Taxable equivalent yieldThe taxable yield needed to match a tax-free bond after tax.Insurance combined ratioWhether premiums collected cover claims and the cost of writing policies.Balance transfer savingsWhether moving a balance to a promo-rate card actually saves money.Blended interest rateThe single weighted rate that several loans are effectively costing.Sortino ratioRisk-adjusted return that only penalises downside volatility.Maximum drawdownThe worst peak-to-trough decline across a series of portfolio values.Modified internal rate of return (MIRR)A more realistic IRR that uses separate finance and reinvestment rates.Debt snowball vs avalancheCompares paying off smallest balances first against paying off highest rates first.Net effective rentThe real average rent after free months, allowances and operating costs.Days cash on handHow many days a business could cover its operating costs from cash reserves.Weighted average shares outstandingTime-weighted share count used as the denominator for EPS.Buyback impact on EPSHow a share repurchase changes earnings per share, all else equal.Annuity due vs ordinary annuityHow payment timing (start vs end of period) changes an annuity’s value.Economic value added (EVA)Profit left over after paying for the capital used to generate it.Altman Z-ScoreBankruptcy-risk score for a company from five balance-sheet and income ratios.Black-Scholes option priceTheoretical price of a European call and put option.Treynor ratioReturn earned above the risk-free rate, per unit of market (beta) risk taken.EV/EBITDA multiplePrices a company against its cash operating profit, independent of capital structure.Mortgage refinance breakevenHow many months of lower payments it takes to recoup the cost of refinancing.Deferred annuity present valueWhat a stream of future payments is worth today if it does not start for several years.Futures fair value (cost of carry)Theoretical futures price implied by the spot price, financing rate and yield.Mortgage points breakevenWhether paying upfront points for a lower rate pays off before you sell or refinance.Additional funds needed (AFN)External financing a growing business needs beyond what retained earnings cover.Accrual ratioEarnings-quality check comparing the change in net operating assets to their average.Cash flow coverage ratiosWhether operating cash flow covers current liabilities and total debt.High-low methodSplits a mixed cost into fixed and variable parts from its highest and lowest activity levels.Economic Order Quantity (EOQ)Order size that minimises combined ordering and holding costs for inventory.Herfindahl-Hirschman Index (HHI)Market concentration measured from the sum of squared market shares of each firm.Cobb-Douglas Production FunctionTotal output from labour and capital inputs given their productivity and output elasticities.Covered Interest Rate ParityNo-arbitrage forward exchange rate implied by the interest rate gap between two currencies.Put-call parityChecks or solves the no-arbitrage relationship between a call, a put and the underlying.Optimal hedge ratioShare of a spot position to cover with futures to minimise portfolio variance.Implied forward rateThe interest rate implied for a future period by two spot rates of different maturities.Credit spread (bond)The extra yield a corporate bond pays over a government bond of the same maturity.Jensen's alphaRisk-adjusted return a portfolio earned above what CAPM predicted.Information ratioExcess return over a benchmark per unit of tracking-error risk taken.Profitability indexValue created per dollar invested, for ranking or screening projects.Sinking fund paymentThe regular deposit needed to reach a target sum by a future date.Holding period returnTotal return from an investment, combining price change and income received.Actual cash value (ACV)What an insurer would pay for an item today, after wear and depreciation.ADR & RevPAR (hotel revenue)Average daily rate and revenue per available room for a lodging property.Bank reconciliationReconciles your cash book balance against the bank statement balance.Customer acquisition cost (CAC)What it costs, on average, to win one new customer, and how fast that cost is repaid.Customer churn & retention rateShare of customers lost over a period, and the flip side: how many you kept.GMROI (gross margin return on investment)How many dollars of gross profit each dollar tied up in inventory returns.Direct material price varianceWhether materials cost more or less than the standard (budgeted) rate.ARM (adjustable-rate) mortgagePayment during the fixed period, and the worst-case payment once the rate can adjust.Bond current yieldThe income return a bond offers at its present market price, ignoring capital gain or loss.Return on capital employed (ROCE)Operating profit generated per dollar of capital tied up in the business.Operating marginWhat share of revenue is left as operating profit, after cost of sales and running costs.SaaS customer lifetime value (LTV)What an average subscriber is worth over their time, weighed against acquisition cost.

Health

Body composition, energy needs, training load and cycle tracking, using the standard published clinical and sports-science formulas.

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Body mass index (BMI)Weight relative to height, with the WHO category.BMR: basal metabolic rateEnergy burned at complete rest (Mifflin-St Jeor).TDEE: daily calorie needsBMR scaled by how active you are.Weight change timelineHow long a given calorie gap takes to shift a target.Body fat percentage (US Navy)Tape-measure estimate of body composition.Ideal body weightFour classic clinical formulas side by side.Macronutrient splitGrams of protein, carbs and fat for a calorie target.Daily water intakeBaseline fluid target adjusted for exercise and heat.Calories burned (MET)Energy cost of an activity from its metabolic equivalent.Heart rate training zonesKarvonen zones from resting and maximum heart rate.VO₂ max (Cooper test)Aerobic fitness from a 12-minute run.One-rep maxStrength ceiling estimated from a submaximal set.Running pace & finish timeConverts between pace, speed and race time.Pregnancy due dateEstimated delivery date from the last period (Naegele’s rule).Body surface areaUsed for drug dosing and cardiac index.Waist-to-hip & waist-to-heightFat distribution measures that track health risk.Ovulation & fertile windowLikely fertile days across a cycle.Sleep cycle timingWake times that fall between 90-minute sleep cycles.Anion gapUnmeasured ions in the blood, a first step in narrowing the cause of a metabolic acidosis.Corrected calcium for albuminAdjusts a total calcium reading for a low albumin level, using the Payne formula.Blood alcohol content (Widmark formula)Rough estimate of blood alcohol content from drinks consumed, body weight and time elapsed.Estimated average glucose from A1CConverts an HbA1c percentage into an average blood glucose reading (ADAG study formula).Corrected QT interval (QTc)Heart-rate-adjusted QT interval from an ECG reading, using the Bazett and Fridericia formulas.Creatinine clearance (Cockcroft-Gault)Estimated kidney filtration rate from age, weight and serum creatinine.Lean body mass (Boer formula)Fat-free body mass estimated from height and weight.Mean arterial pressure (MAP)Average pressure driving blood flow through the body over one heartbeat cycle.Total body water (Watson formula)Estimated volume of water in the body from age, height and weight.Fat-free mass index (FFMI)Muscularity measure that adjusts fat-free mass for height, comparable across body sizes.Smoking pack-yearsCumulative smoking exposure used in clinical risk screening.LDL cholesterol (Friedewald equation)Estimated LDL cholesterol from a standard lipid panel.Total cholesterol to HDL ratioA single number summarizing cardiovascular risk from a lipid panel.Daily protein intakeProtein target range based on body weight and activity level.Daily fiber intake targetRecommended dietary fiber based on total calorie intake.A Body Shape Index (ABSI)Waist circumference adjusted for height and BMI, an alternative to the waist-to-height ratio.Relative fat mass (RFM)Body fat percentage estimated from height and waist circumference (Woolcott-Bernstein).Shock indexHeart rate divided by systolic blood pressure, a quick marker of circulatory stress.Rate pressure productHeart rate times systolic blood pressure, an index of how hard the heart is working.HOMA-IR (insulin resistance)Estimated insulin resistance from fasting glucose and fasting insulin levels.Calculated serum osmolalityEstimates blood osmolality from a basic metabolic panel, plus the osmolal gap.Corrected sodium for hyperglycemiaAdjusts a measured sodium reading for high blood glucose, using the Katz and Hillier formulas.Delta ratio (delta gap)Compares the rise in anion gap to the fall in bicarbonate, to spot a mixed acid-base disorder.BUN-to-creatinine ratioCompares blood urea nitrogen to creatinine, used to help distinguish causes of kidney injury.AST-to-ALT ratio (De Ritis ratio)Compares two liver enzymes, sometimes used to point toward a pattern of liver injury.Albumin-to-globulin ratioSplits total blood protein into albumin and globulin and compares the two.Transferrin saturation (TSAT)The share of iron-carrying protein that is actually loaded with iron.Mentzer indexScreens a routine blood count for thalassemia trait versus iron-deficiency anemia.FEV1/FVC ratioShare of lung capacity exhaled in the first second of a forced breath, from spirometry.Minute ventilationThe total volume of air moved in and out of the lungs in one minute.Estimated height from knee heightStanding height for a patient who cannot be measured directly (Chumlea equations).Adjusted body weightBlends actual and ideal body weight, a middle-ground figure used for larger patients.Ponderal indexA leanness measure like BMI, but using height cubed for consistency at any height.Body roundness index (BRI)Estimates body shape and visceral fat from waist and height, an alternative to BMI.Osteoporosis risk score (OST index)A quick screening score from age and weight, flagging who might need bone density testing.Pregnancy weight gain guidelineRecommended weight gain for a singleton pregnancy, by pre-pregnancy BMI (IOM guidelines).Gestational age from crown-rump lengthDates an early pregnancy from a first-trimester ultrasound measurement.Fundal height gestational age (McDonald’s rule)Between 20 and 36 weeks, fundal height in centimetres roughly tracks gestational age.Glycemic loadCombines a food’s glycemic index with the actual carbohydrate in a serving.Net carbsCarbohydrate grams likely to affect blood sugar, after subtracting fibre and sugar alcohols.Mid-parental height predictionPredicted adult height for a child from both parents heights (Tanner method).Drug elimination by half-lifeFraction of a substance remaining or cleared after a given time, from its half-life.Predicted vital capacityExpected lung vital capacity for a healthy adult, from height, age and sex.Free water deficitEstimated water needed to correct a high serum sodium (hypernatraemia).Urine anion gapUrinary electrolyte gap that helps distinguish causes of a normal anion-gap metabolic acidosis.Fractional excretion of sodium (FeNa)Compares urine and plasma sodium and creatinine to help classify acute kidney injury.Urine protein-to-creatinine ratioEstimates 24-hour urinary protein loss from a single spot urine sample.MCH & MCHC (red cell indices)Average haemoglobin content and concentration per red blood cell, from a full blood count.Absolute neutrophil count (ANC)Total mature and immature neutrophils in circulation, used to gauge infection risk.PSA densityPSA level adjusted for prostate size, used alongside other tests to gauge cancer risk.RMR from lean mass (Katch-McArdle)Resting metabolic rate estimated from lean body mass rather than total weight.PaO₂/FiO₂ ratio (Horowitz index)Oxygenation index used to gauge the severity of acute respiratory failure.Alveolar-arterial (A-a) gradientGap between calculated and measured arterial oxygen, a clue to the cause of hypoxaemia.Bolton tooth-size ratioCompares upper and lower tooth widths to flag a size mismatch before orthodontic treatment.Amplitude of accommodation (Hofstetter)Expected eye-focusing range for a given age, and the approximate near point for reading.Number needed to treat (NNT)How many patients must be treated to prevent one additional bad outcome, from two event rates.Vaccine efficacyPercentage reduction in disease risk among a vaccinated group versus an unvaccinated group.Ankle-brachial index (ABI)Peripheral artery disease screening ratio of ankle to arm blood pressure.Left ventricular ejection fractionPercentage of blood pumped out of the left ventricle with each heartbeat.Cardiac output & cardiac indexBlood pumped by the heart each minute, scaled to body size.Cerebral perfusion pressure (CPP)Net pressure driving blood flow to the brain, from blood pressure and intracranial pressure.Lung-protective tidal volume (ARDSnet)Low tidal volume ventilation target from predicted body weight and height.Physiologic dead space (Bohr equation)Share of each breath that does not take part in gas exchange.Alcohol units (UK)UK standard units of pure alcohol in a drink, from volume and strength.Reticulocyte production index (RPI)Corrects a reticulocyte percentage for anaemia severity and cell maturation time.Albumin-to-creatinine ratio (ACR)Urine albumin relative to creatinine, a standard kidney-damage screening test.FIB-4 index (liver fibrosis)Screens for advanced liver fibrosis risk from age plus a routine liver panel and platelet count.Transtubular potassium gradient (TTKG)Estimates how well the kidney is handling potassium from paired urine and serum readings.Urea reduction ratio (URR)Percentage drop in blood urea over one haemodialysis session, a dialysis-adequacy check.Winters’ formula (expected pCO₂)Checks whether respiratory compensation for a metabolic acidosis looks appropriate.Neutrophil-to-lymphocyte ratio (NLR)A simple inflammation marker calculated from a routine blood differential.Caffeine remaining in the bodyEstimates caffeine left in your system over time using first-order elimination kinetics.Karvonen target heart rateTarget heart rate training zone from heart rate reserve, using the Karvonen formula.Body frame sizeClassifies frame size as small, medium or large from the height-to-wrist-circumference ratio.IQ percentileWhere an IQ score ranks against the population, from the normal distribution.Blood glucose unit converterConverts blood glucose readings between mmol/L and mg/dL.Adjusted age for premature babiesCorrects a premature baby’s age for weeks born early, for milestone tracking.DIY hand sanitizer recipe (WHO formula)Scales the WHO alcohol-based handrub formulations to any batch size.Safe noise exposure timeMaximum daily exposure time allowed at a given noise level, NIOSH or OSHA rules.Added sugar daily limitCompares a day’s added sugar against the AHA recommended maximum.Sleep debt trackerCumulative sleep shortfall over several nights against a target amount of sleep.Next blood donation eligibility dateEarliest date you can donate again, from your last donation and the donation type.Estimated conception dateBack-calculates a likely conception date from a due date or the start of the last period.Child blood type probabilityPossible ABO and Rh blood types of a child from both parents, classic Mendelian genetics.Baby formula daily intakeApproximate daily formula volume for a bottle-fed baby, from weight and age.US standard alcoholic drinksNumber of US standard drinks (14 g pure alcohol each) in a beverage, from volume and ABV.Baby eye color probabilityLikely eye colors for a baby from both parents’ eye colors, using a simplified genetic model.Breastfeeding extra calorie needsExtra daily calories needed while exclusively breastfeeding, on top of your baseline TDEE.Vitamin mcg to IU converterConverts vitamin A or vitamin D between micrograms (mcg) and International Units (IU).Cholesterol unit converterConverts total cholesterol, HDL, LDL or triglycerides between mg/dL and mmol/L.

Maths

Percentages, fractions, algebra, number theory and trigonometry: the working shown alongside every answer.

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PercentageThe three everyday percentage questions in one place.Percentage change & differenceIncrease, decrease and the symmetric difference.Fraction arithmeticAdd, subtract, multiply or divide two fractions and simplify.Decimal ↔ fractionClosest fraction to a decimal, at a chosen precision.GCD & LCMGreatest common divisor and lowest common multiple.Quadratic equationRoots, vertex and discriminant of ax² + bx + c.Ratio & proportionSolves a : b = c : d for the missing term.Exponents & rootsPowers, nth roots and scientific notation of the result.LogarithmLog in any base, plus the natural and common logs.Scientific notationConverts between plain numbers and a × 10ⁿ.Prime check & factorisationTests primality and lists the prime factors.Permutations & combinationsOrdered and unordered selections from a set.Number base converterBinary, octal, decimal, hex and any base to 36.Roman numeral converterArabic to Roman and back, 1 to 3999.Rounding & significant figuresRound to decimals, significant figures or a nearest step.Trigonometry: solve a triangleSine and cosine rules from two sides and the included angle.Trigonometric functionssin, cos, tan and their inverses for one angle.Modulo & integer divisionQuotient, remainder and the true mathematical modulo.Random number generatorDraws numbers from a range: press recalculate for a new set.Matrix determinant (2×2 or 3×3)Determinant of a 2×2 or 3×3 matrix from its entries.2×2 matrix inverseInverse of a 2×2 matrix using the adjugate formula.2×2 matrix operationsAdd, subtract or multiply two 2×2 matrices, and scale one by a constant.Vector dot productScalar dot product of two 3D vectors, and the angle between them.Vector cross productThe perpendicular vector from two 3D vectors, and the area of the parallelogram they span.Vector magnitude & unit vectorLength of a 3D vector and the unit vector pointing the same way.Arithmetic sequenceThe nth term and the sum of the first n terms of an arithmetic sequence.Geometric sequenceThe nth term and the sum of a geometric sequence, plus the infinite sum where it converges.Complex number arithmeticAdd, subtract, multiply or divide two complex numbers, plus modulus and argument.Synthetic divisionDivides a polynomial by (x − r) and returns the quotient and remainder.Equation of a lineSlope-intercept, point-slope and standard form of the line through two points.Distance & midpoint between two points (3D)Straight-line distance and midpoint between two points in 3D space (set z = 0 for 2D).Law of sines (solve a triangle)Finds the remaining sides and angles from two angles and one side.Arithmetic in standard formAdd, subtract, multiply or divide two numbers written as a × 10ⁿ.Binomial expansionExpands (x + y)ⁿ term by term using the binomial theorem.Distance from a point to a linePerpendicular distance from a point to a line given in general form.Geometric & harmonic meanThe geometric mean and the harmonic mean of a list of positive numbers.Golden ratioSplits a length into golden-ratio proportions, or scales a rectangle side by φ.Partial fraction decomposition (2 factors)Splits (ax+b)/((x−p)(x−q)) into simple fractions, including the repeated-root case.System of two linear equationsSolves a₁x+b₁y=c₁ and a₂x+b₂y=c₂ for x and y.Intersection of a circle and a lineWhere a line y = mx + c crosses a circle centred at (h, k).Matrix trace & transpose (2×2 or 3×3)Sum of the diagonal, and the matrix flipped over its diagonal.Hyperbolic functionssinh, cosh and tanh of a value, and their inverses.Circle equation: general to standard formConverts x²+y²+Dx+Ey+F=0 into centre-radius form.Absolute value equationSolves |ax + b| = c for x.Consecutive integers from a sumFinds n evenly-spaced integers (consecutive, even or odd) that add to a target sum.Divisibility testChecks which numbers from 2 to 12 divide a whole number evenly.Digital rootRepeatedly sums the digits of a number until one digit remains.Cubic equationReal roots of ax³ + bx² + cx + d = 0 via Cardano’s method.Eigenvalues of a 2×2 matrixThe two eigenvalues (real or complex) of [[a,b],[c,d]] from the characteristic polynomial.Solve a·bˣ = c for xUses logarithms to isolate x in an exponential equation.Rationalize a denominatorRemoves a square root from the denominator of a/(b + c√n) using the conjugate.Pythagorean triple generatorBuilds a right-triangle integer triple from two generator numbers (Euclid’s formula).Clock hand angleThe angle between the hour and minute hands at a given time.Reference & coterminal angleReduces any angle to its 0–360° coterminal angle, quadrant and reference angle.Round to the nearest fractionRounds a decimal to the nearest 1/2, 1/4, 1/8, 1/16 or any denominator you choose.Percent errorHow far a measured or estimated value is from the accepted true value.Set operationsUnion, intersection and differences of two sets of numbers.Polygon area from vertices (shoelace formula)Area, perimeter and centroid of a simple polygon from its ordered vertices.Triangle incenter & inradiusIncenter coordinates and inscribed-circle radius from a triangle’s three vertices.Perfect, abundant & deficient number checkClassifies a whole number by comparing it to the sum of its own proper divisors.Amicable numbers checkTests whether two numbers form an amicable pair: each is the sum of the other’s divisors.Continued fraction expansionExpands a fraction p/q into continued-fraction terms and lists its convergents.Linear Diophantine equation (ax + by = c)Finds all integer solutions to ax + by = c using the extended Euclidean algorithm.System of three linear equations (Cramer’s rule)Solves a 3×3 system of linear equations using determinants.Vector projectionProjects vector a onto vector b in 3D (set z = 0 for 2D vectors).Distance from a point to a planePerpendicular distance from a point to a plane given as ax + by + cz + d = 0.Chinese remainder theorem (two congruences)Finds x satisfying x≡a₁ (mod n₁) and x≡a₂ (mod n₂), including non-coprime moduli.Modular multiplicative inverseFinds x such that a·x ≡ 1 (mod n), using the extended Euclidean algorithm.nth roots of a complex numberAll n complex roots of a+bi using De Moivre’s theorem, in polar and rectangular form.Fibonacci sequenceThe nth Fibonacci number, the sequence around it and the sum so far.Egyptian fraction decompositionSplits a fraction into a sum of distinct unit fractions using the greedy algorithm.Linear interpolationEstimates a missing y (or x) on the straight line through two known points.Modular exponentiationComputes baseᵉˣᵖ mod m efficiently using square-and-multiply.Luhn algorithm (checksum)Validates a number sequence or computes the check digit that makes it valid.Manhattan (taxicab) distanceGrid-style distance between two points, moving only along axes.Collatz conjecture (3n+1)Steps to reach 1 by halving even numbers and applying 3n + 1 to odd ones.Cosine similarityAngular similarity between two vectors, from −1 (opposite) to 1 (same direction).Descartes' rule of signsMaximum possible positive and negative real roots of a polynomial, from its sign changes.Sum of digitsAdds up the individual digits of a whole number in one pass (not repeated to a single digit).Order of magnitudeThe power of ten closest to a number, used to compare sizes at a glance.Unit rateSimplifies a ratio of two quantities to a rate per single unit of the second quantity.Golden rectangleFinds the sides, diagonal and area of a rectangle whose side ratio equals the golden ratio.Magic squareFinds the magic constant of an n×n magic square and builds it for odd n via the Siamese method.Discrete convolutionConvolves two finite number sequences to produce their discrete convolution sequence.Two's complementConverts a signed decimal integer to its binary two’s-complement form at a chosen bit width.Harmonic numberSums the reciprocals of the first n positive integers to give the nth harmonic number.Gamma functionEvaluates the gamma function Γ(x) for any real x, extending the factorial to non-integers.Error functionComputes the Gauss error function erf(x) and its complement erfc(x) for a given x.Involute functionFinds the involute of a pressure angle, used in gear-tooth geometry: φ = tan(α) − α.Absolute value inequalitiesSolves |ax + b| compared to c (<, ≤, >, ≥) and shows the solution in interval notation.Euler's formula for polyhedraFinds the missing vertex, edge or face count of a convex polyhedron from V − E + F = 2.Cyclomatic complexityComplexity of a control-flow graph from its edge, node and connected-component counts.Bilinear interpolationEstimates a value at a point inside a rectangular grid from the four surrounding known values.

Statistics

Averages, spread, distributions, confidence and correlation for data sets you paste straight in.

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Descriptive statisticsMean, median, mode, spread and quartiles for a data set.Weighted averageAverage where each value carries its own weight.Z-score & percentileHow far a value sits from the mean, in standard deviations.Normal distribution probabilityArea under the bell curve between two points.Confidence interval for a meanMargin of error around a sample average.Required sample sizeHow many responses a survey needs.Correlation & linear regressionPearson r and the line of best fit for paired data.Binomial probabilityChance of exactly k successes in n independent trials.Probability of eventsCombines two event probabilities: and, or, neither.Odds ↔ probabilityConverts between fractional odds, decimal odds and percent.GPA / weighted gradeGrade point average across units with credit weights.Grade needed on the finalWhat the last assessment must score to hit a target.Coefficient of variationRelative variability of a data set: standard deviation as a percentage of the mean.SkewnessSample skewness: whether the data leans left or right of the mean.Poisson probabilityChance of a given number of independent events in a fixed interval, given the average rate.Geometric probabilityChance the first success happens on a given trial, for repeated independent trials.Hypergeometric probabilityChance of drawing an exact number of successes from a finite population, without replacement.Chi-square goodness of fitTests whether observed category counts match expected counts.Cohen's d (effect size)Standardised difference between two group means, in pooled standard deviations.Margin of error (proportion)Margin of error for a survey proportion at a chosen confidence level.Relative riskCompares event risk between an exposed and an unexposed (or treated and control) group.T-statistic (one or two sample)How many standard errors a sample mean is from the reference value, or from another mean.Z-test for a proportionTests a sample proportion against a hypothesised population proportion.Expected value of a discrete distributionMean and standard deviation of a discrete random variable from its outcomes and probabilities.Outlier fences (Tukey)The IQR-based cutoffs beyond which a data point counts as an outlier.CovarianceHow two variables move together, in sample or population form.Standard error of the meanHow much a sample mean is expected to vary from the true population mean.F-test for equal variancesCompares the spread of two samples using the ratio of their variances.Negative binomial probabilityChance that the rth success happens on exactly the kth independent trial.Percentile rankWhat percentage of a data set falls at or below a given value.Value at a percentileFinds the data value at any percentile using linear interpolation.Mean absolute deviationThe average distance of each data point from the mean (or median).Bayes’ theoremUpdates the probability of an event given new evidence.Class width for a histogramHow wide each bin should be, from either a target class count or Sturges’ rule.Spearman’s rank correlationMeasures a monotonic relationship between two variables using their ranks, ties included.Chebyshev’s theoremMinimum share of any data set within k standard deviations of the mean, whatever its shape.Two-proportion z-testTests whether two independent samples have significantly different proportions.Pooled standard deviationCombines two sample standard deviations into one pooled estimate for equal-variance tests.Z-test for a single meanTests a sample mean against a hypothesised mean when the population standard deviation is knownQuartile deviationHalf the interquartile range: a robust spread measure, less sensitive to outliers than the SD.Range rule of thumb for standard deviationA quick estimate of standard deviation from just the minimum and maximum of a data set.Mean & standard deviation from grouped dataEstimates the mean and standard deviation of a frequency table using class midpoints.Empirical rule (68-95-99.7)The share of normal data expected within 1, 2 and 3 standard deviations of the mean.KurtosisSample excess kurtosis: how heavy-tailed a data set is versus a normal distribution.Diagnostic test accuracy metricsSensitivity, specificity, predictive values and accuracy from a 2×2 test result table.Birthday paradox probabilityChance that at least two people in a group share the same birthday.Bonferroni correctionAdjusts the significance threshold when running many statistical tests at once.Benford's law leading-digit checkCompares a data set's leading digits against the distribution natural data follows.One-way ANOVA (F-statistic)Tests whether three or more group means differ, from the raw data in each group.McNemar's testTests whether paired binary outcomes changed, using the two discordant-pair counts.Mann-Whitney U test (Wilcoxon rank-sum)Compares two independent samples for a shift in distribution, using ranks instead of means.MidrangeThe midpoint between the smallest and largest value in a data set, a quick measure of centre.Sum of squaresTotal squared deviation of each data point from the mean, the building block of variance.Confusion matrixTurns the four counts of a binary confusion matrix into accuracy, precision, recall and F1.Youden's indexCombines a test’s sensitivity and specificity into one score of diagnostic performance.Weibull distributionComputes the PDF, CDF, mean and variance of a Weibull distribution from its shape and scale.Rayleigh distributionComputes the PDF, CDF, mean and variance of a Rayleigh distribution from its scale parameter.Fisher's exact testRuns a two-tailed Fisher exact test on a 2×2 contingency table of counts.Process capability indexCalculates Cp and Cpk from specification limits, the process mean and standard deviation.Degrees of freedomWorks out degrees of freedom for a one-sample, two-sample or chi-square test.Index of qualitative variationMeasures diversity among categorical frequencies on a 0 to 1 scale using the IQV formula.Beta distributionPDF, mean and variance of a beta distribution from its two shape parameters.Monty Hall problemWin probability from staying versus switching after the host reveals losing doors.Statistical power analysisAchieved power for a given sample size, and the sample size needed for a target power.Error propagationPropagates the uncertainty of two measurements through a sum, difference, product or quotient.

Geometry

Area, perimeter, volume and surface area for flat shapes and solids, plus coordinate geometry.

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CircleArea, circumference and diameter from any one measure.Circle sector & arcA wedge of a circle: arc length, area and chord.Triangle from three sidesArea, angles and centres via Heron’s formula.Right triangle / PythagorasThird side, angles and area of a right-angled triangle.Rectangle & squareArea, perimeter and diagonal.TrapezoidArea and perimeter of a four-sided figure with parallel sides.Parallelogram & rhombusArea from base and height, or from sides and angle.EllipseArea, perimeter (Ramanujan) and eccentricity.Regular polygonArea, angles and radii of an n-sided regular shape.SphereVolume and surface area from the radius.CylinderVolume, surface area and capacity of a tube or tank.ConeVolume, slant height and surface area.Rectangular prism & cubeVolume, surface area and space diagonal of a box.PyramidVolume and surface area of a rectangular-based pyramid.Torus (ring)Volume and surface of a doughnut shape.Distance, midpoint & slopeEverything about the line between two points.Area & volume unit scalingHow area and volume change when a shape is scaled.

Conversion

Unit converters that show the full ladder of equivalents at once, not just the one pair you asked for.

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Length & distanceMetric and imperial length, from millimetres to nautical miles.Weight & massGrams, kilograms, pounds, ounces, stone and tonnes.Volume & capacityLitres, millilitres, gallons, pints, cups and cubic units.AreaSquare metres, hectares, acres and square feet.SpeedKilometres per hour, miles per hour, metres per second, knots.TimeSeconds through to centuries.Digital storageBytes, kilobytes, mebibytes and the binary versus decimal split.PressurePascals, bar, psi, atmospheres and millimetres of mercury.EnergyJoules, calories, kilowatt-hours and BTU.PowerWatts, kilowatts, horsepower and BTU per hour.AngleDegrees, radians, gradians and turns.Cooking measuresCups, spoons, millilitres and fluid ounces for the kitchen.TemperatureCelsius, Fahrenheit, Kelvin and Rankine.Fuel economyLitres per 100 km, miles per gallon and km per litre.Height & weight (imperial ↔ metric)Feet-and-inches and stone-and-pounds, both ways.TorqueNewton-metres, foot-pounds, inch-pounds and kilogram-force metres.Paper quantitySheets, quires and reams, the units the printing and stationery trade uses.Concentration (ppm, ppb, %)Parts per million, parts per billion and percentage, as the same ratio expressed differently.12-hour ↔ 24-hour (military) timeClock time converted to 24-hour military format, and back.Astronomical distance (AU, light-year, parsec)Distances across the solar system and beyond: astronomical units, light-years and parsecs.Degrees, minutes, seconds ↔ decimal degreesConverts a geographic coordinate between decimal degrees and degrees-minutes-seconds format.

Date & time

Ages, durations, working days, deadlines and timesheets, calculated on real calendar arithmetic.

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Age calculatorExact age in years, months and days, plus the next birthday.Days between datesElapsed time between two dates, with and without weekends.Add or subtract from a dateThe date a set number of days, weeks or months away.Working daysCounts weekdays between two dates, less any holidays.Time duration arithmeticAdds and subtracts hours, minutes and seconds.Timesheet & payHours worked in a shift and what they are worth.Day of the week & week numberWhich weekday a date falls on, plus ISO week and day of year.Countdown to a dateTime remaining until a deadline or event.Unix timestamp converterEpoch seconds to a readable date and back.

Physics

Mechanics, electricity, energy and waves: first-principles formulas with every constant stated.

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Ohm’s lawVoltage, current, resistance and power from any two.Resistors in series & parallelCombined resistance of up to four resistors.Voltage dividerOutput voltage across the lower resistor of a pair.Electricity running costWhat an appliance costs to run over time.Force, mass & accelerationNewton’s second law, with weight under gravity.Kinetic & potential energyEnergy of a moving or raised mass.Speed, distance & timeJourney arithmetic in metric and imperial.Acceleration & stopping distanceUniform acceleration, braking distance and reaction gap.Projectile motionRange, peak height and flight time of a launched object.Density, mass & volumeAny one of the three from the other two.Pressure from force or depthPressure under a load, and water pressure at depth.Wave speed, frequency & wavelengthRelates the three properties of any wave.Work, power & efficiencyEnergy transferred, the rate of transfer and what is lost.Momentum & impulseMomentum of a mass and the force needed to change it.Ideal gas lawPressure, volume, temperature and moles of a gas.Torque & rotational powerTurning force, and the power it delivers at a given speed.Centripetal force & circular motionForce, acceleration and period of an object moving in a circle at constant speed.Hooke's law & spring energyRestoring force and stored energy of a stretched or compressed spring.Pendulum periodSwing period of a simple pendulum for small angles.Elastic & inelastic collisionFinal velocities of two colliding masses, in a straight line.Capacitor charge & energyStored charge and energy of a capacitor at a given voltage.Inductor stored energyMagnetic energy stored in an inductor carrying current.RC circuit charge & dischargeHow a capacitor charges or discharges through a resistor over time.Carnot efficiencyMaximum possible efficiency of a heat engine between two temperatures.Linear thermal expansionHow much a material lengthens or shortens with a temperature change.Heat energy & specific heatHeat needed to change the temperature of a mass of material.Snell's law of refractionAngle of refraction as light passes between two media.Thin lens equationImage distance and magnification for a lens of known focal length.Doppler effect (sound)Observed frequency of a sound source moving relative to a listener.Sound intensity level (decibels)Decibel level of a sound from its intensity, referenced to the hearing threshold.Escape velocityMinimum speed to escape a body’s gravity from a given distance, ignoring drag.Circular orbital velocitySpeed and period needed to hold a circular orbit around a mass.Kepler's third lawOrbital period of a body from its orbit size and the mass it orbits.Buoyancy (Archimedes’ principle)Upward force on an object from the fluid it displaces, and whether it floats.Terminal velocityMaximum falling speed once drag balances gravity.Angular momentum (point mass)Rotational momentum of a point mass moving on a circular path.Bernoulli's equationPressure at one point in a steady, frictionless flow from conditions at another point.Reynolds numberRatio of inertial to viscous forces in a flow, and whether it is laminar or turbulent.Coulomb's lawElectrostatic force between two point charges.Electric field of a point chargeField strength and potential at a distance from a single point charge.Magnetic field of a straight wireField strength at a distance from a long straight wire, from Ampère's law.Newton's law of universal gravitationGravitational force and potential energy between two masses.Time dilationHow much slower a clock moving at speed v runs, compared to a stationary observer.Length contractionHow much shorter a fast-moving object appears along its direction of motion.Mass-energy equivalenceEnergy contained in a rest mass, from Einstein’s E = mc².Mass-spring oscillator periodPeriod, frequency and peak speed of a mass bouncing on an ideal spring.Moment of inertia of standard shapesRotational inertia of common rigid shapes, and the resulting rotational kinetic energy.RLC circuit resonant frequencyResonant frequency, Q factor and bandwidth of a series RLC circuit.Photon energyEnergy of a single photon from its wavelength or frequency.De Broglie wavelengthWavelength associated with the momentum of any moving particle.Relativistic kinetic energyKinetic energy of an object moving fast enough that Newtonian mechanics breaks down.Wien's displacement lawPeak emission wavelength of a blackbody at a given temperature.Stefan-Boltzmann lawTotal power radiated by a hot surface, from its temperature and emissivity.Mach numberSpeed of an object relative to the local speed of sound in air.Simple harmonic motionPosition, velocity and energy of a mass-spring oscillator at a given moment.Beat frequencyHow fast two close sound frequencies pulse in and out of phase.AC power & power factorReal, reactive and apparent power in an AC circuit from RMS values and phase angle.Lensmaker's equationFocal length of a lens from its surface curvatures, thickness and refractive index.Schwarzschild radiusEvent-horizon radius of a black hole of a given mass, from general relativity.Mean free path of a gasAverage distance a gas molecule travels between collisions with other molecules.Wheatstone bridgeUnknown resistance from three known resistors at the balance point.Electron drift velocityAverage speed of charge carriers moving through a current-carrying wire.Redshift & blueshiftDoppler shift of light and the recession or approach velocity it implies.Compton scatteringWavelength shift of a photon scattered off a free electron at a given angle.RMS speed of gas moleculesRoot-mean-square speed of molecules in an ideal gas from temperature and molar mass.Malus's lawLight intensity transmitted through a polarizer at an angle to the polarization axis.Aerodynamic drag forceDrag force on an object moving through a fluid, from its shape and speed.Lever & mechanical advantageEffort force needed to balance a load on a simple lever, from the arm lengths.Center of mass (two bodies)Balance point of two masses along a line, and its distance from each.Heisenberg uncertainty principleMinimum combined uncertainty in the position and momentum of a quantum particle.Capacitors in series & parallelCombined capacitance of up to four capacitors, wired in series or in parallel.Incline with frictionForces and acceleration of an object sliding down a ramp with friction.Young's ModulusA material's stiffness from an applied force, cross-section area, and resulting elongation.Poiseuille's LawVolumetric flow rate of a viscous fluid through a cylindrical pipe under laminar flow.Brewster's AngleAngle of incidence where reflected light becomes fully polarized, from two refractive indices.Solenoid Magnetic FieldMagnetic field strength inside a long, tightly-wound solenoid carrying a steady current.Force Between Parallel WiresMagnetic force per unit length between two long, straight, parallel current-carrying wires.Radiation PressurePressure an electromagnetic wave, like sunlight, exerts on an absorbing or reflecting surface.Angular VelocityRotational speed from the angle turned over time, plus linear speed at a given radius.RMS Voltage & CurrentRMS, average and peak-to-peak values of a sinusoidal AC voltage or current from its peak value.Ideal Rocket EquationVelocity change a rocket can achieve from its exhaust velocity and the propellant mass burned.Bragg's LawDiffraction angle for X-rays reflecting off crystal lattice planes at a given spacing.Rydberg EquationWavelength of light from an electron jumping between energy levels in a hydrogen-like atom.Stokes' Law Terminal VelocityFalling speed of a small sphere through a viscous fluid once drag balances its weight.Cyclotron FrequencyOrbital frequency of a charged particle circling in a uniform magnetic field.Distance to the HorizonGeometric sight distance to the horizon from an observer height above a sphere.Angular AccelerationRate of change of angular velocity over a time interval, from spin-up or spin-down.G-Force from AccelerationHow many g's a change in velocity over time represents, relative to standard gravity.Magnetic Dipole MomentStrength of the magnetic moment produced by a current flowing around a coil loop.Thermal DiffusivityHow quickly a material equalises temperature, from conductivity, density and specific heat.Poisson's RatioHow much a material bulges sideways when stretched or squeezed lengthways.Shear ModulusStiffness of a material against sideways (shear) deformation from an applied force.Specific ImpulseHow efficiently a rocket or jet engine converts propellant into thrust.Froude NumberRatio of flow inertia to gravity in open-channel or ship-hull flow.Hall CoefficientCharge-carrier density and sign in a conductor from a measured Hall voltage.Hoop StressCircumferential and longitudinal stress in a thin-walled pressure vessel.Synodic PeriodTime between successive alignments of a body with Earth, or another chosen reference body.Skin DepthDepth below a conductor surface where AC current density falls to 1/e of its surface value.Boyle's LawPressure or volume of a fixed gas amount after an isothermal change, from the other values.Photoelectric EffectKinetic energy and stopping voltage of electrons ejected from a metal by light.Blackbody RadiationSpectral radiance a hot object emits at one wavelength, from Planck's law.Hubble's LawRecession speed of a distant galaxy from its distance and the Hubble constant.Black Hole TemperatureHawking temperature and evaporation timescale of a black hole from its mass.Drake EquationOrder-of-magnitude estimate of communicating civilisations in the galaxy from seven factors.Hydrogen-Like AtomElectron energy level and orbital radius in a one-electron atom or ion, from the Bohr model.Diffraction GratingAngle at which a given order of light diffracts off a grating of known line spacing.Friction ForceFriction force resisting motion on a flat or inclined surface, from mass and coefficient.Angle Of ReposeSteepest stable slope angle for a granular pile from its static friction coefficient.Coriolis EffectCoriolis acceleration and force on a moving mass from its speed and latitude on Earth.Gear RatioOutput speed and torque of a gear pair from the tooth counts and the input speed.Latent HeatHeat energy absorbed or released during a phase change from mass and the specific latent heat.Weight On Other PlanetsHow much a mass would weigh on the Moon or another planet, from each body’s surface gravity.

Home & build

Quantities and sizing for renovation, landscaping and trade work: paint, tiles, concrete, roofing and more.

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Paint requiredLitres of paint for walls, allowing for doors and windows.Flooring & tilesBoxes of tiles or boards needed, including wastage.Concrete volumeCubic metres for a slab, footing or set of posts.Mulch, soil & gravelBulk material for a garden bed at a given depth.Roof pitch & rafter lengthAngle, slope factor and true roof area.Fencing & decking materialsPosts, rails and boards for a run of fence or deck.Air conditioner sizingCooling capacity for a room, adjusted for sun and occupancy.Tank & pool volumeCapacity of a rectangular or round tank in litres.Drywall sheets neededPlasterboard sheets for a room, allowing for openings and waste.Bricks or blocks neededMasonry units for a wall, from unit size and mortar joint thickness.Stair rise & runStep count, riser height and comfort check for a straight staircase.Pavers & bedding sandNumber of pavers and bedding sand volume for a patio or path.Roof shingle bundlesBundles of shingles needed, accounting for slope and waste.Board feet of lumberBoard feet and cost for a batch of dimensional lumber.Baluster / spindle spacingEven spindle count and gap for a stair or deck railing.Wallpaper rolls neededRolls required for a room, accounting for pattern repeat.Wall stud countStuds needed for a wall on a chosen centre spacing, plus openings.Ladder angle & base distanceSafe base distance and length for a leaning ladder, using the 4-to-1 rule.Insulation thickness & coverageThickness needed to hit a target R-value, and rolls required for an area.Air changes per hour (ACH)Ventilation rate for a room, from fan airflow and room volume.Door / window header sizeMinimum header depth for a rough opening, using the standard light-framing rule of thumb.Fence post hole depthHow deep to set a fence or deck post below grade, and the total post length to buy.Post hole concreteConcrete needed to set posts in round holes, allowing for the post itself.Board and batten sidingBattens needed to cover a wall at a chosen spacing, plus base board coverage.French drain materialsGravel and pipe needed for a French drain trench.Rebar for a concrete slabBar count and total length for a two-way rebar grid in a slab or footing.Tile grout neededWeight of grout for a tiled area, from tile size and joint dimensions.Ramp slope & lengthSlope ratio, angle and ramp length for a given rise, checked against the ADA maximum.Retaining wall blocksNumber of retaining wall blocks for a wall from its length, height and block size.Deck footing countNumber of footings for a deck frame at a chosen maximum footing spacing.Well water volumeWater volume held in a well casing from its diameter and the water column depth.Roof truss countNumber of roof trusses for a building length at a chosen on-centre spacing.Board-on-board fenceBoards, posts and rails for a board-on-board fence, from the board overlap.Rolling offset (pipefitting)True offset, travel and run for a pipe that must dodge an obstacle both sideways and up.Concrete block core fillGrout volume to fill the hollow cores of concrete masonry blocks.Floor area ratio (FAR)Ratio of building floor area to lot size, and the max floor area a zoning FAR limit allows.Paver sand neededVolume and weight of bedding or joint sand needed to cover a paver area at a given depth.Arch radius from rise and spanRadius of the circular arc that gives a doorway or window arch its rise and span.Sheet metal bend allowanceFlat pattern length added by a bend, from thickness, radius, angle and the K-factor.Sheet metal punch forceForce needed to punch a hole through sheet metal, from the hole perimeter and material.Bolt circle spacingStraight-line spacing between adjacent bolt holes evenly spaced on a bolt circle.Air changes per hourHow many times an hour a ventilation system replaces a room's air.Bathroom mirror sizeRecommended mirror width and height from your vanity and ceiling.Furnace size (BTU)Heating capacity needed for a home, from floor area, climate band and insulation.Metal bar & sheet weightWeight of a metal plate, bar or round rod from its dimensions and material density.Lumber weightWeight of dimensional lumber or a stack of boards, from species density and count.

Everyday

Practical comparisons for shopping, travel, cooking and time: the small calculations that add up.

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Trip fuel costFuel and time for a journey, split between passengers.Unit price comparisonWhich pack is actually cheaper per unit.Recipe scalerRescales quantities and adjusts baking time.Aspect ratio & screen sizeResize while keeping proportions, or find screen dimensions.Pizza & round-item valueArea per dollar: the honest way to compare sizes.Reading & speaking timeHow long a word count takes to read aloud or silently.Password strengthEntropy in bits and how long a brute force would take.Coffee brew ratioGrounds and water for a chosen strength.Hours of work to buy somethingPrices a purchase in hours of your life.Battery charge timeHow long a device takes to charge from its current level to a target level.Typing speed (WPM)Gross and net words per minute, and accuracy, from a timed typing sample.Luggage linear inchesChecks a bag against an airline’s combined length + width + height limit.Cost of a meetingWhat a meeting actually costs in attendee salary time.True cost per mile to driveAll-in running cost of a vehicle, spread across the miles driven.Weight of a pile of cashHow much a dollar amount weighs in bills or coins.2-stroke fuel mix ratioOil to add to a volume of petrol for a chosen mix ratio.Generator sizingGenerator capacity needed to cover a set of appliances, including starting surge.LED bulb savingsElectricity and replacement-bulb savings from switching to LED lighting.Event tables & seatingTables needed for a guest count, for round or rectangular tables.Pomodoro session plannerHow many focused work intervals a task will take, breaks included.Money saved by quitting smokingWhat not buying cigarettes adds up to over time.Roll length remainingHow much material is left on a roll, from its diameter and material thickness.Cost of a showerWater and water-heating energy cost for one shower.EV home charging cost & timeCost and time to charge an electric vehicle from one charge level to another.Binge-watch time plannerTotal time to finish a TV series, and how many days that takes.Moon phase finderThe approximate Moon phase on any date, from the lunar synodic cycle.Weighted blanket sizeRecommended blanket weight from body weight, using the common 10% rule.Mobile data usageDaily and monthly data use from your everyday activities, and how long a data cap will last.Carpool cost sharingSplits a shared commute's fuel cost fairly among riders.Commute cost & time comparisonCompares driving to public transit for a regular commute, in cost and time.Toilet paper: how long a pack lastsHow many days a pack of toilet roll lasts a household, from usage per person.Firewood needed for the seasonHow many cords of firewood a heating season needs, from stove burn rate and heating days.Tire size & speedometer differenceOverall diameter, circumference and speedometer error between two tire sizes.Crosswind & headwind componentSplits a wind reading into the part blowing across your path and the part blowing along it.Packing boxes estimateHow many moving boxes a household will need, from room count plus kitchen and fragile items.Belt sizeBelt size to buy from a waist measurement, using the standard buckle allowance.Standing desk heightStanding desk surface height from your height, using the elbow-height ergonomic ratio.Screen viewing distanceRecommended seating distance from a screen size and target field of view.Sunscreen amountSunscreen to apply per full-body application, from the dermatology 2 mg/cm² rule.Hammock hang angleRidgeline length and hang-point drop for a comfortable hammock sag between two anchors.Engine compression ratioStatic compression ratio for an engine cylinder from bore, stroke and clearance volume.Cross-stitch fabric sizeFabric size to buy for a cross-stitch pattern, from stitch count, fabric count and margin.Bean bag fillingFiller needed for a bean bag chair from its filled dimensions, using the standard bead density.Clay shrinkageFired size of a clay piece after drying and firing shrinkage, from its wet (greenware) size.Sidereal timeLocal sidereal time for a date, UTC time and longitude, the star clock astronomers use.Balloon arch balloon countBalloons needed for a single-row or garland balloon arch from its size and balloon diameter.Cost per minuteBreaks a total cost down to a rate per minute, for calls, rentals, or metered time.Price per weightUnit price of an item by weight, so you can compare packages sold in different sizes or units.Tablecloth sizeTablecloth dimensions needed for a table, from its size and how far you want the drop.Curtain fabric neededFabric width and panel count for curtains, from window size and desired fullness.Pool salt neededHow much salt to add to a saltwater pool to reach a target concentration.Bath vs. shower water useCompares the water used filling a bath against taking a shower of a given length.Hat sizeHat size number from head circumference, plus the letter size.Glove sizeGlove size number and letter size from a hand circumference measurement.Paper sheet & ream weightWeight of a sheet or a stack of paper from its GSM and dimensions.Helium balloon liftHow many balloons of a given size are needed to lift a payload.Winch sizeRecommended winch pulling capacity from vehicle weight and a safety factor.Lumens to wattsPower draw needed for a target brightness, by bulb technology.Event tent sizeFloor area and footprint needed for a party or event tent, by seating style.Toothpaste tube lifeHow many days a toothpaste tube lasts a household, from tube size and brushing habits.

Growth

Growth rates for revenue, users and any other metric that moves period to period: YoY, MoM, QoQ and beyond, each with its own dedicated page.

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Year-over-year (YoY) growthGrowth between this period and the same period a year ago.Month-over-month (MoM) growthGrowth between this month and last month, with the annualised equivalent.Quarter-over-quarter (QoQ) growthGrowth between this quarter and the previous quarter, with the annualised equivalent.Week-over-week (WoW) growthGrowth between this week and last week, with the annualised equivalent.Day-over-day (DoD) growthGrowth between today and yesterday, with the annualised equivalent.Period-over-period (PoP) growthGrowth between any two comparable periods you define, with a custom annualising cadence.Compound monthly growth rate (CMGR)The smooth monthly rate that links a start and end value over several months.Annualised run rateExtrapolates a single period of revenue to a full-year figure.Rule of 40Checks whether growth rate plus profit margin clears the 40% SaaS health benchmark.User & customer growth rateNet growth rate for a user base once new signups and churned users are both counted.Revenue growth & target plannerGrowth achieved between two periods, and the revenue needed next period to hit a target rate.

Chemistry

Concentrations, pH, dilutions and reaction yields: the maths behind the lab bench.

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Molarity from massConcentration of a solution made from a known mass of solute.Solution dilution (C₁V₁ = C₂V₂)Stock volume and diluent needed to hit a target concentration.pH from hydrogen ion concentrationConversion between [H⁺] and pH, with pOH and [OH⁻] at 25 °C.Henderson–Hasselbalch (buffer pH)Buffer pH from the acid dissociation constant and the base/acid ratio.Percent yieldHow much product a reaction actually gave versus the theoretical maximum.Mass percent concentrationSolute mass as a percentage of total solution mass, plus ppm.Radioactive decay & half-lifeAmount of a substance remaining after a given elapsed time.Beer–Lambert law (absorbance)Absorbance and transmittance of a solution from its concentration.Osmotic pressure (van’t Hoff)Pressure needed to stop osmosis across a semipermeable membrane.Molality of a solutionMoles of solute per kilogram of solvent, a concentration measure independent of temperature.Acid-base titration (equivalence point)Unknown acid or base concentration from the titrant volume at the equivalence point.Freezing point depressionHow much a solvent’s freezing point drops when a solute is dissolved in it (a colligative property).Boiling point elevationHow much a solvent’s boiling point rises when a solute is dissolved in it (a colligative property).Buffer capacity (Van Slyke equation)How resistant a buffer solution is to pH change, from its total concentration and pKa.Arrhenius equationReaction rate constant at a given temperature from activation energy and pre-exponential factor.Mole fraction of a mixtureFraction of the total moles in a mixture contributed by each component.Water hardness (as CaCO₃)Total water hardness as calcium carbonate equivalent, from calcium and magnesium ions.Serial dilution seriesFinal concentration after a series of equal-factor dilutions, and the concentration at each step.Raoult’s law (vapor pressure of a solution)Vapor pressure of an ideal solution containing a non-volatile solute.Weak acid/base pH from Ka or KbpH of a weak acid or base solution from its dissociation constant and starting concentration.Activation energy from two rate constantsActivation energy from rate constants at two temperatures (two-point Arrhenius equation).Kp ↔ Kc equilibrium constant conversionConvert between pressure-based and concentration-based equilibrium constants for a reaction.Degree of unsaturation (DBE)Number of rings and pi bonds in a molecule from its molecular formula.Empirical formula from mass percentSimplest whole-number atom ratio of a compound from percent composition and atomic masses.Ionic strength of a solutionIonic strength from the concentration and charge of every ion present in a solution.Graham's law of effusion/diffusionRelative rate at which two gases effuse or diffuse, from their molar masses.Radioactive activity (Bq / Ci)Decay rate of a radioactive sample from its amount and half-life, in becquerels and curies.Rate constant from concentration vs. timeReaction rate constant k from concentration data, for zero, first or second order kinetics.Atom economyShare of reactant mass that ends up in the desired product versus all products formed.Average atomic mass from isotopesWeighted average atomic mass of an element from its isotope masses and natural abundances.Nernst equation (cell potential)Cell potential under non-standard conditions from the standard potential and reaction quotient.Gibbs free energyReaction spontaneity from the change in enthalpy, entropy and temperature.Partial pressure (Dalton's law)Partial pressure of each gas in a mixture from its moles, volume and temperature.Bond order (MO theory)Bond order and bond stability of a molecule from its bonding and antibonding electron counts.Diffusion coefficient (Stokes-Einstein)Diffusion rate of a spherical particle in a fluid, from its size, viscosity and temperature.Young-Laplace pressure (capillary)Pressure difference across a curved liquid interface, from surface tension and curvature.Isoelectric point (pI)pH at which a molecule such as an amino acid carries no net electric charge.Grams to molesConverts a mass in grams to an amount in moles using the substance’s molar mass.Molar mass of a gasFinds a gas’s molar mass from its mass, pressure, volume and temperature (ideal gas law).Normality of a solutionConverts molarity to normality using the number of equivalents per mole.pKa from KaCalculates pKa from the acid dissociation constant Ka using pKa = -log10(Ka).Alligation mixing ratioFinds the ratio and volumes of two concentrations that mix to a target concentration.Crude protein from nitrogenEstimates crude protein percentage from measured nitrogen content and a conversion factor.PPM to molarityConverts a concentration in ppm (mg/L) to molarity using the solute molar mass.Reaction quotient (Q)Calculates the reaction quotient Q for aA + bB ⇌ cC + dD from current concentrations.Combustion analysisFinds the empirical formula of a CHO compound from the CO2 and H2O produced by combustion.Saponification valueDetermines a fat or oil’s saponification value from a KOH back-titration with HCl.Moles to atomsConverts an amount in moles to a number of atoms or particles using Avogadro’s number.Vapor pressure (Clausius–Clapeyron)Finds vapor pressure at a new temperature from a known point and enthalpy of vaporization.Avogadro's number (particles from moles)Number of particles and sample mass from an amount in moles.Atom composition (protons, neutrons, electrons)Subatomic particle counts from atomic number, mass number and charge.Percent ionic characterIonic character of a bond estimated from the electronegativity difference.Lattice energy (Kapustinskii equation)Ionic crystal lattice energy from ion charges, radii and formula-unit count.Cubic unit cell (packing efficiency)Edge length, atoms per cell and packing efficiency for SC, BCC and FCC lattices.Miller indices from axial interceptsMiller indices (hkl) of a crystal plane, plus interplanar spacing for a cubic lattice.Effective nuclear charge (Slater's rules)Net nuclear charge felt by an electron after shielding, using Slater's rules.Combustion reaction balancerBalanced coefficients for complete combustion of a CxHyOz compound in oxygen.Stoichiometric air-fuel ratioTheoretical air-to-fuel mass ratio for complete combustion of a CxHy hydrocarbon.STP conversionCorrects a measured gas volume to STP and finds the equivalent moles.

Biology

Population genetics, enzyme kinetics, growth curves and lab concentration formulas.

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Hardy–Weinberg equilibriumExpected genotype frequencies in a non-evolving population from allele frequency.DNA/RNA concentration from A₂₆₀Nucleic acid concentration from a spectrophotometer absorbance reading.Log reduction (microbial kill)How many orders of magnitude a treatment reduces a microbial count.Michaelis–Menten kineticsEnzyme reaction rate at a given substrate concentration.Bacterial/cell growth (doubling time)Population size after exponential growth at a fixed doubling time.Q₁₀ temperature coefficientHow much a biological or chemical rate changes per 10 °C rise.Plant water potentialTotal water potential from solute and pressure potential in a plant cell.Allele frequency from genotype countsDirect-count allele frequencies for a two-allele gene from observed genotype numbers in a sample.Multi-gene cross probabilityProbability of a given number of recessive traits among offspring of a multi-gene cross.DNA copy number from concentrationDNA or RNA molecules per microlitre from concentration and fragment length.PCR primer melting temperature (Tm)Estimated primer melting temperature from base composition (Wallace/Marmur-Doty).qPCR amplification efficiencyPCR amplification efficiency from the slope of a standard curve (log₁₀ starting quantity vs. Ct).Vapor pressure deficit (VPD)Gap between saturation and actual vapor pressure, the driving force for plant transpiration.Daily light integral (DLI)Total photosynthetic light a plant receives per day, from light intensity and photoperiod.Mutation frequencyFraction of a tested population carrying a mutation, reported per cell or organism tested.Growing degree units (GDU)Accumulated heat units driving crop development, from daily high/low temperature and a base.Ligation molar ratio (insert:vector)Insert mass needed for a target insert-to-vector molar ratio in a DNA cloning ligation.Cricket chirp thermometerEstimate air temperature from a cricket’s chirp rate using Dolbear’s law.PCR annealing temperature (Ta)Optimal PCR annealing temperature from primer and product melting temperatures.Feed conversion ratio (FCR)Efficiency of converting feed into body weight gain for livestock or aquaculture.Corn yield estimatorEstimated corn yield in bushels per acre using the ear-count yield component method.Acres per hourField work rate from implement width and travel speed, accounting for overlap.Dry matter basis conversionConverts an as-fed nutrient percentage to a dry-matter basis using moisture content.DNA to mRNA transcriptionmRNA transcript from a DNA template strand using base-pairing rules.Tree height (clinometer method)Tree height from the angle to its top and your distance from it.Tree diameter at breast height (DBH)Trunk diameter and cross-sectional area from a measured circumference.Grain bin storage capacityBushel capacity of a cylindrical grain bin from its dimensions.Fertilizer application rateProduct needed to hit a target nutrient rate given the label percentage.Dihybrid cross ratiosGenotype and phenotype ratios from a cross of two heterozygous two-gene parents.Swine gestation (farrowing date)Expected farrowing date from the breeding date, using the average 114-day sow gestation.Water-soluble fertilizer dosingFertilizer weight needed to reach a target nutrient concentration in a given volume of water.

Ecology

Population and diversity formulas for the field: carrying capacity, species diversity and predator-prey dynamics.

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Logistic population growthPopulation size over time under a fixed carrying capacity (Verhulst model).Shannon diversity indexBiodiversity of a community from species counts.Lotka–Volterra predator–prey ratesInstantaneous population growth rates for a classic predator–prey pair.Kaya identity (CO₂ emissions)Global CO₂ emissions decomposed into population, wealth, energy and carbon factors.Tree basal areaCross-sectional trunk area at breast height (DBH), the standard forestry measure of stand density.Animal mortality ratePercentage of a population lost to death over a period, accounting for births during that period.Wind turbine power outputElectrical power generated by a wind turbine from rotor size, wind speed and efficiency.Hydroelectric power outputPower generated by a dam turbine from head, flow rate and efficiency.

Sports

Performance and scoring maths for training and competition, from gear ratios to rating systems.

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Bike gear ratio & speedGear ratio, development and road speed for a given chainring, cog and cadence.Bike frame sizeRecommended frame size from inseam length, by bike type.Stride length & distance from stepsStride length from height, and a step count turned into distance.Hull speed (displacement boat)The theoretical top speed of a displacement-hull boat from its waterline length.Wilks score (powerlifting)Bodyweight-adjusted strength score for comparing lifters across weight classes.Elo rating changeHow a single result moves a player or team’s Elo rating.NFL passer ratingThe 0–158.3 quarterback rating from completions, yards, touchdowns and interceptions.Race time predictorPredicts a finish time at a new distance from a recent race result, using Riegel’s formula.Hiking time (Naismith’s rule)Estimated walking time for a hike from distance and total ascent.Bruce protocol VO₂maxEstimated maximal oxygen uptake from total time on a standard Bruce treadmill test.Magic number (pennant race)Combined wins by your team and losses by the nearest rival needed to clinch a spot.Net run rate (cricket)The run-rate differential used to separate teams level on points.E-bike rangeEstimated riding range from battery capacity and energy use per distance.Fishing reel line capacityHow much line of a new diameter fits on a reel, from a known spool capacity.Pool table room sizeMinimum room dimensions so a cue has room to swing on every shot.Swim pacePace per 100 m/yd and average speed for a swim, from distance and time.Cycling power-to-weightWatts per kilogram from your power output and body weight, with a rough training category.Golf course handicapCourse handicap for a round, from Handicap Index, Slope Rating and Course Rating.Ape indexHow your arm span compares to your height, a rough climbing-reach indicator.Bike chain lengthChain length in links for a bike from chainstay length and the largest chainring and cog.Fish weight from lengthEstimated fish weight from length and girth, or length alone, using standard angler formulas.Baseball WHIPWalks plus hits per inning pitched, the standard measure of baserunners a pitcher allows.Winning percentageWin rate for a team or player, with ties counted as half a win.Batting average (baseball or cricket)Hitting average from hits and at-bats, or runs and dismissals for cricket.Slugging percentageTotal bases per at-bat, a power-hitting measure in baseball and softball.Goalie save percentageShare of shots faced that a goalkeeper stopped, hockey or soccer.Effective field goal percentage (eFG%)Basketball shooting efficiency that weights three-pointers as worth more.Cricket bowling averageRuns conceded per wicket taken, lower is better for a bowler.Earned run average (ERA)A pitcher’s earned runs allowed per nine innings pitched, baseball or softball.Calories burned by heart rateEnergy burned during exercise from average heart rate, age, weight and sex.Vertical leap needed to dunkMinimum vertical jump required to dunk a basketball, from your standing reach and rim height.Rowing ergometer pace to wattsConverts a rowing machine 500 m split time to power output, the Concept2 formula.Triathlon total timeAdds swim, transition, bike and run splits into a total triathlon finish time.On-base percentage (OBP)How often a batter reaches base safely per plate appearance, baseball or softball.Fielding percentageShare of fielding chances handled cleanly, baseball or softball defense.Bench press one-rep maxEstimated one-rep max (1RM) bench press from a weight and rep count, using the Epley formula.NBA game scoreJohn Hollinger’s single-game performance rating from a box score line.True shooting percentageBasketball shooting efficiency that accounts for three-pointers and free throws.MET-minutes per weekWeekly physical activity volume in MET-minutes, checked against the WHO activity guideline.Goals against average (hockey)A goaltender’s goals allowed per 60 minutes of play, the standard hockey goaltending stat.Cricket batting strike rateRuns scored per 100 balls faced, the standard measure of scoring pace in cricket.

Food

Kitchen and brewing ratios: baker’s percentages, dilutions, yields and the maths behind a recipe that scales.

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Baker’s percentage & hydrationExpresses each dough ingredient as a percentage of the flour weight, as bakers do.Brine strengthSalt needed to make a brine of a given strength, by weight of the water.ABV from gravity readingsAlcohol by volume of a home-brewed beer or wine from hydrometer readings.Alcohol dilutionWater needed to cut a spirit down from one ABV to a weaker target ABV.Yeast type converterA quantity of yeast converted between active dry, instant and fresh (cake) forms.Self-rising flour from all-purposeBaking powder and salt to turn plain flour into self-rising flour.Rice to water ratioWater needed to cook a given amount of rice, and the approximate cooked yield, by rice type.Dry to cooked pasta weightHow much a portion of dry pasta weighs once boiled, given its typical expansion factor.Party drink plannerTotal drinks needed for a party, split across beer, wine and spirits.Cake servingsHow many servings a round or rectangular cake yields at a given slice size.Ice cream overrunHow much the volume of a churned ice cream mix has grown from the air whipped into it.Cups to grams (ingredient weight)Converts a volume measure of a common ingredient to weight, using typical ingredient density.Turkey roasting & thawing plannerOven time and refrigerator/cold-water thawing time for a turkey, from its weight.Cake pan size converterRescales a recipe's batter amount when swapping to a different pan shape or size.BBQ / party food plannerProtein, sides and drinks to buy for a barbecue or party, per guest.Ham per personHow much ham to buy for a group, by bone-in or boneless serving guidelines.Priming sugarSugar to add for bottle-carbonating a home-brewed beer to a target CO2 level.Cake pricingA sale price for a custom cake from ingredient cost, labour and a target profit margin.Fresh to dried herb conversionDried herb amount to substitute for a fresh herb quantity in a recipe, or the reverse.Garlic form converterEquivalent amount of minced, powdered or granulated garlic for a number of fresh cloves.Turkey thawing timeTime needed to safely thaw a turkey in the refrigerator or in cold water, by weight.Bacon curing ratiosDry-cure salt, pink curing salt and (for wet brining) water for a pork belly, by weight.Molecular SO2 in wineThe antimicrobially active molecular SO2 fraction, from free SO2 and wine pH.Drink chilling timeTime for a drink to cool to a target temperature, using Newtons law of cooling.Ham cooking timeOven roasting time for a ham, from its weight and whether it is fully cooked or fresh.Mashed potatoes per personHow much mash to make and how many raw potatoes to buy for a group.BBQ grill sizeGrilling surface area needed for a cookout, and an equivalent round grill diameter.Egg boiling timeBoiling time for an egg by size and doneness, scaled from a large-egg baseline.Tea steeping guideRecommended water temperature and steep time by tea type, and leaves needed for a batch.Oil to butter converterConverts a recipe quantity between cooking oil and butter using the standard baking ratio.Popcorn yieldPopped popcorn volume from a quantity of unpopped kernels, and servings it makes.

Pets

Sizing, feeding and age-conversion maths for the animals in your household, from crate size to daily calories.

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Dog age in human yearsA dog's age converted to a human-equivalent age using the epigenetic-clock formula.Cat age in human yearsA cat's age converted to an equivalent human age using the standard veterinary chart.Pet gestation & due dateExpected birth date range for a common pet or farm animal, from the breeding date.Dog & cat daily caloriesDaily energy needs from resting energy requirement and an activity factor.Dog daily water intakeRoughly how much water a dog needs per day, by activity level.Dog crate sizeMinimum crate dimensions so a dog can stand, turn around and lie stretched out.Turtle tank sizeMinimum water volume for a pet turtle based on shell length.Chicken coop & run sizeMinimum floor space for a flock, depending on how much outdoor access they get.Dog chocolate toxicity riskTheobromine dose from chocolate eaten, checked against published toxicity thresholds.Rabbit cage sizeMinimum enclosure dimensions based on a measured hop, body length and standing height.Dog daily food amountDaily feeding amount as a percentage of body weight, by life stage.Cost of owning a petFirst-year and ongoing costs from your own monthly and one-off figures.Guinea pig & hamster ageHuman-equivalent age for a guinea pig or hamster from published veterinary age charts.Fish tank stockingRoughly how many inches of fish a freshwater community tank can hold.Pet bird ageHuman-equivalent age for a pet bird, scaled by its species’ typical lifespan.Horse weight estimateEstimated adult horse body weight from a heart-girth and body-length tape measurement.Round pen panel countFence panels needed to build a round training pen from its diameter and panel length.Dog harness sizeRecommended harness size from chest girth, the measurement harnesses are actually fitted to.Dog heat cycle predictorEstimated stage dates and next heat window from the last cycle's start date.Dog onion toxicity riskOnion dose eaten checked against published toxicity thresholds for a dog's weight.Fish weight estimateEstimated live weight of a fish from its length and girth, without using a scale.