Every equation used on this site, with the meaning of each variable. Each entry links back to the full page, where the formula is worked through with real numbers step by step.
329 formulas across 120 calculations.
FV = B(1+r)^t + (C + M) x [((1+r)^t - 1) / r]
The first term compounds the opening balance; the second is the future value of a level annual deposit of your contribution plus the match.
M = C x matchRate
A 50% match pays half of what you contribute, up to any cap the employer sets.
Dep = (price - residual) / term
The value consumed over the lease, spread evenly across the term.
Fin = (price + residual) x moneyFactor
Interest on the average outstanding balance of price and residual.
Down Payment = Purchase Price x Down Payment Percentage
The headline cash figure. Divide percentages by 100 — ten percent is 0.10, not 10.
LTV = Loan Amount / Purchase Price
Loan divided by value. Above 0.80 on a conventional loan triggers private mortgage insurance and often a higher rate.
Monthly PMI = Loan Amount x Annual PMI Rate / 12
The premium applies only while LTV exceeds 80%. Typical annual rates run from roughly 0.19% to 1.5% depending on credit score and LTV band.
New = old x (1 + raise)
Multiply the current salary by one plus the raise percentage.
Real = (1 + raise) / (1 + inflation) - 1
The exact gain in purchasing power after inflation is stripped out.
Gross = salary (or hourly rate x hours)
The full amount earned before any deduction.
Net = gross - income tax - payroll tax - contributions
Subtract each deduction from gross to find take-home pay.
M = P x r / (1 - (1 + r)^-n)
P is the amount borrowed, r the monthly rate and n the number of monthly payments. The denominator is the present-value factor of an annuity, which is why the same formula covers every fixed-rate loan.
I1 = P x r ; Pr1 = M - I1
Interest for a month is the opening balance times the monthly rate; whatever is left of the payment reduces the balance.
B1 = P - Pr1
The balance falls by exactly the principal portion of the payment.
PMT = PV x r x (1 + r)^n / ((1 + r)^n - 1)
The present value of an annuity solved for the payment. Every symbol is either something you set or something you already have.
PMT = PV x r
The limit as the horizon becomes infinite. It provides a lower bound: any finite-horizon payment must exceed pure interest income, because principal is being returned.
Total = PMT x n; Growth = PMT x n - PV
Total tells you the gross stream; growth isolates how much of it came from investment returns rather than your original money.
P - F = sum_{k=1..n} pmt / (1 + i)^k ; APR = 12 x i
The net cash you receive equals the present value of the payments when discounted at the monthly rate i. There is no closed form; the rate is found by trial or a financial calculator.
APY = (1 + r/n)^n - 1
Divide the nominal rate by the compounding frequency, add one, raise to that frequency, and subtract one. The result is the true one-year return.
APY = e^r - 1
The upper bound as compounding frequency approaches infinity. On a 5% nominal rate this is 5.12711% — only marginally above daily compounding.
Ending Balance = Principal x (1 + APY)
Once APY is known, one year of growth is a single multiplication. This is what makes two offers directly comparable.
M = P x r / (1 - (1 + r)^-n)
P is the amount financed, r the monthly rate and n the term in months.
Interest = M x n - P
Total payments minus the amount financed is the interest paid.
CY = coupon / price
Annual coupon income divided by the price paid.
Income = coupon x years
Coupons received while the bond is held.
Break-Even Units = Fixed Costs / (Price - Variable Cost Per Unit)
Divide total fixed costs by the contribution margin earned on each unit. Round up — you cannot sell a fraction of a unit.
Break-Even Revenue = Fixed Costs / ((Price - Variable Cost) / Price)
Fixed costs divided by the contribution margin ratio. This is the form to use for service businesses and mixed product lines.
Required Units = (Fixed Costs + Target Profit) / Contribution Margin
Treat the desired profit as an additional fixed cost and divide again. The slope never changes, which is the whole insight.
CAGR = (EV / BV)^(1/n) - 1
Take the total growth factor, raise it to 1/n to spread it evenly across every year, and subtract 1 to convert the factor back into a rate.
EV = BV x (1 + CAGR)^n
The same relationship solved for the endpoint. Useful for checking your work and for projecting what a given constant rate produces over n years.
A = P(1 + r)^t
The deposit grows at the APY for the number of years. Because the APY already includes compounding, no further adjustment is needed.
Interest = A - P
The maturity value minus the original deposit.
A = P (1 + r/n)^(nt)
The general form. Divide the annual rate by compounding periods per year, add 1, raise it to the number of years times periods per year, multiply by principal. Interest earned equals A − P.
A = P · e^(rt)
The limiting case as the compounding frequency grows without bound. It gives the maximum possible result for a given nominal rate, and is used in derivatives pricing and theoretical finance rather than consumer accounts.
n = -ln(1 - (rm x B) / P) / ln(1 + rm)
Requires P > rm x B. Round the result up, because a partial final month is still a full calendar month of payments.
P = (B x rm) / (1 - (1 + rm)^-n)
The same relationship solved for payment instead of time. Pick the payoff deadline and read off what that costs each month.
Total Interest = (P x n) - B
Everything paid beyond the original balance is interest. Compare two payment sizes here — the difference is startling.
DTI = Total monthly debt payments / Gross monthly income
Add every contractual monthly obligation and divide by monthly income before tax. Multiply by 100 for the percentage lenders quote.
Front-end = Monthly housing expense / Gross monthly income
Housing alone. Conventional underwriting typically holds this near 28%.
Headroom = (Target ratio x Gross income) - Current debt payments
How much additional monthly obligation fits before hitting a threshold — or how much must be cut if already over.
Effective Tax Rate = Total Tax / Gross Income
Add every tax actually paid for the period, divide by income before any deduction, and multiply by 100. Specify which taxes you included — the answer changes enormously.
Marginal Burden = Tax Increase on the Raise / Raise Amount
Price each incremental amount separately. Different taxes stack, so the real cost of extra income widely exceeds the headline bracket.
Payroll Tax = (6.2% x eligible earnings) + (1.45% x all earnings)
Flat rates from the first dollar, unlike progressive income tax brackets. Social Security stops at an annual earnings ceiling; Medicare does not.
EMI = P x r x (1+r)^n / ((1+r)^n - 1)
The level payment whose present value equals the principal borrowed.
Interest = EMI x n - P
Everything paid beyond the principal is interest.
FV = PMT x (((1 + r)^n - 1) / r)
The standard form. Contributions are made at the end of each period, so the final contribution earns no interest at all.
FV(due) = PMT x (((1 + r)^n - 1) / r) x (1 + r)
Every contribution arrives one period earlier and therefore compounds once more. Multiply the ordinary result by (1 + r).
GST = base x rate
The tax is the base price multiplied by the rate.
Total = base x (1 + rate)
Add the tax to the base to get the price the customer pays.
Real = nominal / (1 + i)^t
Discount a future amount back to today's purchasing power using inflation as the discount rate.
Future = present x (1 + i)^t
Grow today's amount forward at the inflation rate to see what it will take to buy the same things later.
Total = (final - initial) / initial
The percentage gain over the whole holding period.
Annualised = (final / initial)^(1/years) - 1
The constant annual rate that produces the same end value.
0 = -C0 + sum CF / (1 + IRR)^t
The rate that discounts every cash flow back to the initial cost.
factor = C0 / CF
For level flows, the ratio of cost to annual cash pins down the rate.
M = P x r(1 + r)^n / ((1 + r)^n - 1)
Standard amortization. Produces the level payment that retires principal exactly over n periods at a periodic rate r.
B = P(1 + r)^k - M x (((1 + r)^k - 1) / r)
Grow the original principal forward k periods, then subtract the accumulated value of the payments made so far. This is how the interest/principal split is computed month by month.
Net Worth = Assets - Liabilities
Total every asset at current market value, total every liability at its payoff amount, and take the difference. The result can be negative; that is meaningful, not an error.
Liquid Net Worth = Liquid Assets - Liabilities
Restricts assets to what is reachable without penalties, taxes or a sale. Usually far smaller — and often negative even when total net worth is healthy.
Debt Ratio = Liabilities / Assets
Expresses leverage as a proportion. 0.48 means creditors have a claim on roughly half of everything owned.
NPV = -C0 + sum_{t=1..n} CF / (1 + r)^t
Discount each year's cash flow and subtract the upfront cost. The present value of a level annuity simplifies the sum.
PV = FV / (1 + r)^n
The inverse of compounding. Divide the future amount by the growth factor it would have accumulated over n periods at rate r.
PV = sum of CF_t / (1 + r)^t for t = 1..n
Discount each payment individually by the number of periods until it arrives, then add them up. Payments further out shrink more, which is what produces realistic valuations for irregular streams.
Real Value = Nominal / (1 + i)^n
Raise one plus the annual inflation rate to the number of years, then divide the nominal amount by that factor. This restates future or past money in terms of today's prices.
Years = ln(2) / ln(1 + i)
How long until rising prices erase half of what an amount buys. At 3% this is 23.45 years — the rule of 72 gives 24 as an approximation.
Real Return = (1 + nominal return) / (1 + inflation) - 1
The two rates combine multiplicatively, not by subtraction. A 5% return with 3% inflation gives 1.9417% real, not 2%.
Pmt = balance x r x (1+r)^n / ((1+r)^n - 1)
The annuity formula gives the payment for a given balance, rate and term.
Break-even = costs / monthly saving
How long the monthly saving takes to recover the closing costs.
ROI = (Current Value - Cost) / Cost x 100%
Net return divided by cost basis. A negative numerator means the investment lost money, and ROI comes out negative.
Annualized ROI = (1 + ROI)^(1/t) - 1
Converts a multi-year return onto a per-year equivalent so opportunities with different horizons can be ranked directly. Only valid when the interim income can be assumed to compound at the same rate.
FV = balance x (1+r)^t + contribution x ((1+r)^t - 1) / r
Grow the current balance and the annual contributions to the retirement date.
Growth = FV - balance - total contributions
The portion of the final balance that is investment growth, withdrawn tax free.
Gross = net x (1 + rate)
Multiply the pre-tax price by one plus the rate.
Net = gross / (1 + rate)
Divide the gross price by one plus the rate to recover the pre-tax amount.
Balance = principal x (1 + rate/n)^(n x t)
Grow the principal at the nominal rate compounded n times a year.
Effective = (1 + rate/n)^n - 1
The true annual yield once compounding is taken into account.
n = ln((FV x r + PMT) / (PV x r + PMT)) / ln(1 + r)
The compound growth formula solved for the number of periods. Use the monthly rate and a monthly deposit so n comes out in months.
PMT = (FV - PV x (1 + r)^n) x r / ((1 + r)^n - 1)
The inverse question: given a fixed number of months, what monthly deposit closes the gap. This is the form to use when the deadline is not negotiable.
Growth = FV - PV - PMT x n
Total contributions subtracted from the target. Anything left over was earned by compounding, which tells you how much of the plan depends on the return assumption.
I = P x r x t
Multiply principal by the annual decimal rate and by time in years. The units of r and t must agree: if r is annual, t is in years; convert months or days into fractions of a year.
A = P + I = P(1 + rt)
The combined form avoids computing interest separately when all you need is the final figure.
Pmt = P x r x (1+r)^n / ((1+r)^n - 1)
The level payment that repays the balance over the term.
Interest = payment x n - P
Everything repaid beyond the balance is interest.
Tax = sum over brackets of (slice x rate)
Each slice of income is taxed at its bracket's rate and the results are added.
Effective = tax / income
The average rate across all income, always at or below the marginal rate.
Years = 72 / rate percent
Divide 72 by the annual percentage return to estimate the doubling time.
Years = ln(2) / ln(1 + rate)
The precise time to double from the compound growth equation.
Gross = net x (1 + rate)
Multiply the pre-tax price by one plus the rate.
Net = gross / (1 + rate)
Divide the tax-inclusive price by one plus the rate to recover the base.
FIRE Number = Annual Spending / Safe Withdrawal Rate
Spending divided by the withdrawal rate. Equivalent to multiplying spending by the reciprocal of the rate — 25x at 4%.
Multiple = 1 / Withdrawal Rate
Twenty-five is not a magic number — it is 1 divided by 0.04. At 3.5% the multiple is 28.57 and at 3% it is 33.33.
Years = ln(1 + 25 x r x (1 - s) / s) / ln(1 + r)
Income cancels out of this expression because savings rate determines both the contribution and the target. Assumes a 25x target and a constant real return.
Hourly Rate = Salary / (Hours Per Week x Paid Weeks)
The baseline conversion. Use 2,080 only when the role really is forty hours across fifty-two paid weeks with no additional expected time.
True Rate = Salary / (Worked Hours + Commute Hours + Prep Hours)
Add every hour the job consumes but does not pay for. This is the version to use when comparing roles with different commutes or unpaid setup expectations.
Required Salary = Target Hourly Rate x Actual Annual Hours
Run the conversion backwards before negotiating. It converts a desired standard of living into a concrete ask.
Deficit = maintenance calories - intake calories
The gap between the energy you use and the energy you take in. A positive result is a deficit; zero or below means no deficit at all.
Pounds lost = deficit x days / 3500
Applies the common approximation that 3500 kcal corresponds to about one pound of body fat. The result is a linear estimate, not a guarantee.
Kilograms = pounds x 0.45359237
Converts the model's pound figure to kilograms using the exact mass conversion factor.
BMI = kg / m^2
The standard form. Convert height to metres, square it, then divide weight by that square.
BMI = 703 x lb / in^2
The 703 folds in both the pound-to-kilogram and inch-to-metre conversions, so the result matches the metric form.
weight = targetBMI x height^2
Rearranging the definition gives the weight that lands on a particular index, which is how the healthy range is turned into kilograms.
BMR = 10w + 6.25h - 5a + s
The most widely used predictive equation. Ten times the weight in kilograms, plus 6.25 times the height in centimetres, minus five times the age in years, plus a sex constant.
BMR = base + s, where base = 10w + 6.25h - 5a
The same three terms feed both versions; only the constant differs. A male adds 5, a female subtracts 161, a gap of 166 kcal for identical body data.
TDEE = BMR x activity factor
Multiplying the resting rate by an activity factor gives the energy actually burned in a day. The factor, not the formula, is usually where the error enters.
BF% = 495 / (1.0324 - 0.19077 x log(waist - neck) + 0.15456 x log(height)) - 450
The logarithm is base 10 and all circumferences are in centimetres. The subtraction of 450 converts the ratio into a percentage.
BF% = 495 / (1.29579 - 0.35004 x log(waist + hip - neck) + 0.22100 x log(height)) - 450
The female equation adds the hip to the circumference term and uses a higher base, because the reference population stores fat differently.
lean mass = weight x (1 - BF% / 100), fat mass = weight - lean mass
Once the percentage is known, it applies directly to body mass to give the two components in kilograms.
IBW = base + 2.3 x (height in inches - 60)
The version written for drug dosing. The base is 50 kg for men and 45.5 kg for women, and the increment applies only to inches above five feet.
IBW = base + 1.9 x (height in inches - 60)
A later revision with a higher base and a smaller increment. The base is 52 kg for men and 49 kg for women.
Low = 18.5 x height_m^2, High = 25 x height_m^2
Returns a band rather than a point. At 175 cm it runs from 56.65625 to 76.5625 kg, which is a more honest answer than any single figure.
1RM = w x (1 + reps/30)
The simpler of the two forms. The load is multiplied by a factor that grows linearly with the number of repetitions completed.
1RM = w x 36 / (37 - reps)
A ratio rather than a linear factor. Because the denominator shrinks as reps rise, it diverges from Epley at higher rep counts, which is why the two rarely agree.
Load = 1RM x share, Reps = 30 x (1/share - 1)
Turns the estimate into training numbers: the load for a chosen percentage of the maximum, and the number of reps that percentage typically supports.
Protein (g) = bodyweight (kg) x grams per kilogram
The standard form. Multiply bodyweight in kilograms by the rate you have chosen. The rate carries all the judgement; the multiplication is trivial.
Calories (kcal) = protein (g) x 4
Protein supplies about 4 kcal per gram, so its contribution to a daily energy budget is the mass in grams multiplied by four.
Per meal (g) = daily protein (g) / meals
Dividing the daily total by the number of meals gives a practical per-serving figure. It does not change the total, only how it is delivered.
Target = Resting + (Max - Resting) x Intensity
The reserve-based method. Take the share of the reserve first, then add the resting rate back as the baseline.
Max = 220 - Age (Tanaka: Max = 208 - 0.7 x Age)
A population average, not an individual measurement. The Tanaka variant is flatter and gives a lower figure at younger ages.
Target = Max x Intensity
The simpler convention. It ignores the resting rate entirely, so it sits lower than Karvonen whenever resting rate is below roughly a fifth of the maximum.
WHR = waist / hip
The definition. Divide one circumference by the other; because both are lengths, the result is dimensionless.
waist = targetRatio x hip
Rearranging the definition gives the waist that would land exactly on a threshold, which is how the 0.90 line is turned into centimetres.
inches = centimetres / 2.54
Converts a tape reading from centimetres to inches. Because the factor cancels in the ratio, the two unit systems return the same WHR.
V = w x r
The core rule. Multiply weight in kilograms by the millilitres-per-kilogram guideline; 35 gives the familiar 2450 mL at 70 kg.
L = V / 1000, cups = V / 236.5882365, fl oz = V / 29.5735295625
The same volume in the units people actually drink from. Only the conversion factors are exact; the requirement itself is not.
target = base + sweat loss - fluid from food
The baseline is only a starting point. Add losses from heat, exercise and altitude, then subtract what food and other drinks already contribute.
sale = original x (1 - rate)
Combines the reduction and the subtraction in one multiplication, which is the form to use once the rate is settled.
saving = original x rate
One multiplication gives the money removed from the price, which is the figure a tag advertises as the saving.
rate = (original - sale) / original
Recovers the rate from the two prices alone, which is how you compare a discount quoted in currency against one quoted in per cent.
tip = bill x rate
One multiplication, applied to the bill rather than to anyone's share. The rate is the percentage written as a decimal, so eighteen per cent is 0.18.
total = bill + tip
The tip is added to the bill before any division, so the total is the figure that actually gets split.
per person = total / people
Divide the tip-inclusive total, not the bare bill, so the percentage stays intact for each person.
tip = bill x rate
One multiplication. The rate is the percentage written as a decimal, so eighteen per cent is 0.18.
total = bill x (1 + rate)
Combines the bill and the tip in one multiplication, which is the form to use once the rate is settled.
per person = total / people
Divide the tip-inclusive total, not the bare bill, so the percentage stays intact for each person.
kWh = watts x hours / 1000
Turn a power rating and a duration into the energy the meter records. The division by 1000 converts watts into kilowatts.
Cost = kWh x rate
Price the energy once you have it. This is the step where the rate from the bill enters.
Cost = watts x hours x rate / 1000
Folds the kilowatt conversion and the pricing into a single expression, useful when only the final figure matters.
Litres = Distance x Consumption / 100
The distance supplies how far, the consumption supplies how much fuel per 100 kilometres, and the division by 100 turns that rate into a per-kilometre one.
Cost = Litres x Price per litre
Multiply the volume you computed by the pump price. The price unit must match the volume unit, or the result is meaningless.
mpg = 235.214583 / (L/100km)
One constant converts either way, because the two units are reciprocal. The constant applies to US gallons.
L/100km = litres / distance x 100
Divide the fuel burned by the distance it covered, then scale to 100 kilometres so the figure is comparable across trips of any length.
mpg = 100 / (L/100km) x 3.785411784 / 1.609344
Take the reciprocal, then apply two unit conversions. Litres per 100 km falls as efficiency rises, so mpg must be formed by dividing rather than multiplying.
cost per km = litres x price / distance
The litres times the pump price gives the money, and dividing by distance turns it into a rate that multiplies cleanly by any journey length.
Regular pay = min(H, T) x R
Cap the hours at the threshold and pay them at the base rate. If hours worked never reach the threshold, the whole week is standard pay.
Overtime pay = max(0, H - T) x R x M
Only the hours above the threshold attract the multiplier. The max ensures a week under the threshold produces no overtime at all.
Total = Preg + Pot, effective rate = Total / H
Add the two bands, then divide by all hours worked. The effective rate rises above the base rate whenever overtime is present.
Total area = wall area x number of coats
Coverage is quoted per coat, so the surface is multiplied by the coat count before any division happens.
Litres = (wall area x coats) / coverage per litre
Dividing the total painted area by the per-litre coverage gives the volume of paint, before any rounding to tins.
Tins = litres / tin size
Dividing litres by the tin size gives the count, which is then rounded up to the next whole tin because part-tins are not sold.
Final = Price x (1 - a) x (1 - b)
Convert each rate to the share it leaves behind, then multiply. For 200, 0.20 and 0.10 the result is 200 x 0.80 x 0.90 = 144.0.
Effective = 1 - (1 - a)(1 - b) = a + b - a x b
The true single rate that matches the chain. For 0.20 and 0.10 it is 0.28, not the 0.30 of a plain sum.
Overstated = (a + b) - (a + b - a x b) = a x b
The exact amount by which adding the rates overshoots. For 0.20 and 0.10 it is 0.02, printed as 0.020000000000000018.
A = L x W
Length times width, with both measurements in one consistent unit. Subtract permanent obstructions only if they will never be tiled behind.
a = s x s
For a square tile, multiply the side by itself. Rectangular tiles use length times width instead, so a 0.3 by 0.6 metre tile covers 0.18 square metres.
N = (A / a) x (1 + w)
Divide the surface area by the tile area, then multiply by one plus the wastage rate written as a decimal. Round the result up to a whole tile.
Unit price = price / quantity
The core division. The result carries units of currency per unit of goods, so 4.50 over 750 g gives 0.006 per gram.
Unit price per 100 g = (price / quantity) x 100
Scaling the base rate to a fixed reference is what makes packs of different sizes directly comparable.
Saving = (dearer unit price - cheaper unit price) / dearer unit price x 100
Expresses the gap as a percentage of the higher rate, so a 0.04 difference on 0.6 reads as 6.6666667%.
factor = desired servings / original servings
The single ratio that drives everything else. A recipe for four scaled to six gives 6 / 4 = 1.5.
new amount = original amount x scale factor
Applied once per proportional ingredient. With a factor of 1.5, 250 g becomes 375 g.
area ratio = (new diameter)^2 / (old diameter)^2
Compare tins by area, never by diameter. Ten squared over eight squared is 1.5625.
Part = Whole x Percentage / 100
Type one. Convert the percentage to a decimal by dividing by 100, then multiply by the base.
Percentage = Part / Whole x 100
Type two. Divide the part by the base and multiply by 100 to convert the ratio back into a percentage.
Whole = Part / (Percentage / 100)
Type three. Divide rather than multiply — this is the step people most often invert.
Weighted Average = (v1 x w1 + v2 x w2 + v3 x w3) / (w1 + w2 + w3)
Sum every value times its weight, then divide by the sum of weights. Weights may be percentages, counts, or ratios — only their relative size matters.
Plain Average = (v1 + v2 + v3) / n
The equal-weights case. Comparing it with the weighted result isolates how much the weights actually changed the answer.
Required v = (Target - Sum of known v x w) / w for the missing item
Subtract everything already locked in from the target, then divide by whatever weight remains. Use it to answer what score or return is still needed.
z = (x - mean) / SD
Recentre then rescale. The result carries no units because the numerator and denominator share them.
x = mean + z x SD
The same relationship inverted, for turning a target z back into a raw threshold.
|z| = |x - mean| / SD
How unusual the value is, regardless of whether it sits above or below the mean.
% Decrease = (Original - New) / Original x 100
The drop over the starting value, scaled to a percentage. The original is always the divisor, never the new value.
New = Original x (1 - p)
The forward form. Multiply the original by the share that survives: a 25% decrease keeps the factor 0.75.
Original = New / (1 - p)
The backward form, used when you know the reduced price and the discount rate. Undefined when p is 1.
% Increase = (New - Original) / Original x 100
The increase divided by the starting value, scaled to a percentage. The original is always the divisor, whatever else the problem offers you.
New = Original x (1 + p)
The multiplier form. A rise of 25.00% means multiplying by 1.25, and chained rises multiply rather than add.
Original = New / (1 + p)
Divide rather than subtract. Removing 25.00% from 100 is not the same as undoing a 25.00% rise, which is why this form is worth memorising.
% Change = (New - Original) / Original x 100
Difference over the starting value, scaled to a percentage. The original is always the divisor.
Growth Factor = New / Original
The multiplicative form. A factor of 1.25 is the same information as +25%, and chained factors multiply.
Required Change = (Original - New) / New x 100
Swap the roles: to return to the starting point, the base becomes the current value. Equivalent to x/(1+x) in magnitude.
m = (y2 - y1) / (x2 - x1)
Rise divided by run. Take both differences in the same direction, or the sign reverses.
b = y1 - m x x1
Rearranged from y = mx + b using either known point. This completes the line equation.
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
The straight-line separation, from Pythagoras. Related to slope but answering a different question.
s = sqrt(Sum(xi - Mean)^2 / (n - 1))
Use when the data is a sample drawn from a larger population. Dividing by n-1 removes the downward bias in estimating that population's spread.
sigma = sqrt(Sum(xi - mu)^2 / n)
Use when the values are the entire population of interest rather than a sample from it — every student in the class, every item in the batch.
CV = Standard Deviation / Mean x 100
Spread expressed as a percentage of the centre. Unitless, so it can compare variability across quantities measured in different units.
A = pi r^2
The standard form. Square the radius first, then multiply by pi.
A = pi d^2 / 4
Equivalent to halving first. Use whichever matches the measurement you actually have.
Sector A = pi r^2 x angle/360, arc L = 2 pi r x angle/360
Both are simply the corresponding whole-circle quantity scaled by the fraction of the turn.
A = 1/2 x base x height
The standard form. Multiply the two lengths first and halve afterwards, so the intermediate figure can be compared with the bounding parallelogram.
A = sqrt(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2
Use when no height and no angle are available. Halve the perimeter first, then subtract each side from that half-sum before multiplying.
A = 1/2 x a x b x sin C
The base-and-height formula with the height replaced by b x sin C. The angle must be the one between the two known sides.
V = pi x r^2 x h
The standard form: square the radius, multiply by pi to get the base area, then multiply by the height. Only the radius is squared.
V = pi x d^2 x h / 4
Equivalent to halving the diameter first. Use it when the measurement you hold was taken across the object.
Lateral A = 2 pi r h, total A = 2 pi r h + 2 pi r^2
Both come from the same inputs as the volume but are measured in square units. The two ends are the base area counted twice.
V = l x w x h
Multiply the three measurements, all in the same unit. The result is in that unit cubed: centimetres give cubic centimetres.
SA = 2(lw + lh + wh)
The six faces pair into three equal couples, so add the three distinct face areas and double. Measured in square units, not cubic ones.
h = V / (l x w)
Divide the known volume by the area of the known base. The same rearrangement finds length or width if either is the unknown instead.
an = a1 x r^(n - 1)
Each step applies one more factor of r, so the nth term carries n - 1 of them — the first term has none.
S = a1 (1 - r^n) / (1 - r)
Subtracting the series multiplied by r cancels every middle term. Valid for any r except exactly 1.
S-infinity = a1 / (1 - r), valid only for |r| < 1
As n grows, r^n tends to zero, so the finite formula collapses to this. Outside that condition there is no limit to quote.
an = a1 + (n - 1) x d
The (n - 1) rather than n trips people constantly: the first term has had zero steps applied to it, so the last of n terms has had n - 1.
S = n/2 x (a1 + an)
Pair the series with its reverse: every column sums to a1 + an, there are n columns across both rows, so one row is half.
S = n/2 x (2a1 + (n - 1)d)
Substitute an = a1 + (n-1)d into the previous formula. Use this when you hold the step size rather than the endpoint.
d = c x b / a
Multiply the known quantity by the other half of the complete pair, then divide by the remaining half. Cross multiplication a x d = b x c gives the same result.
Unit Rate = b / a; Answer = Unit Rate x c
Compute what one unit costs first, then scale. Slower, but it keeps units visible and catches inverted setups.
a x d = b x c
Multiply the diagonals. Equality confirms the proportion was set up consistently, and inequality means one side was inverted.
x = (-b +/- sqrt(b^2 - 4ac)) / (2a)
Derived by completing the square. The plus gives one root, the minus the other; their midpoint is always the vertex.
D = b^2 - 4ac
Positive for two real roots, zero for one repeated root, negative for none. Compute it before anything else.
Sum of roots = -b/a, product = c/a
A two-line verification that catches most transcription errors without redoing the main calculation.
c = sqrt(a^2 + b^2)
Square each leg, add, then take the square root. The result must exceed both inputs.
a = sqrt(c^2 - b^2)
Rearranged from the same relation, so it subtracts. The result must be shorter than the hypotenuse.
Right angle if and only if a^2 + b^2 = c^2
The converse. If measured sides satisfy the relation, the included angle is ninety degrees — the basis of squaring corners.
PR = count below / (n - 1)
Divides by one less than the group size, treating the group as a sample. For 12th out of 250 this gives 238 / 249 = 0.9558232931726908.
PR = count below / n
Divides by the group size itself, treating the group as the whole population. For the same data this gives 238 / 250 = 0.952.
top share = rank / n
The complement of the rank, reporting the fraction of the group at or above the position rather than below it.
Percentage = (correct / total) x 100
The core division. Forty-two correct out of fifty gives 42 / 50 = 0.84, which reads as 84 per cent; forty-three gives 0.86 and forty-one gives 0.82.
Raw score = correct x pointsEach
When questions are not all worth one point, the raw total is the correct count scaled by the point value. Forty-two correct at two points each is 84 out of a maximum of 100, leaving the percentage at 0.84.
Points lost = (total - correct) x pointsEach
The complement of the raw score. With 50 questions, 42 correct and one point each, the loss is (50 - 42) x 1 = 8 points, which is 8 questions wrong.
Weighted GPA = (gp1 + b1 + gp2 + b2 + ... + gpn + bn) / n
Add each course's bonus to its grade points, sum the results, and divide by the number of courses. The worked set gives 21.5 / 5 = 4.3 against an unweighted 3.8.
Lift = total bonus points / number of courses
The bonus total is diluted by the course count. A 2.5 bonus total over five courses lifts the mean by 0.5; the same 2.5 over ten courses would lift it by only 0.25.
Gap = weighted GPA - unweighted GPA
The gap measures exactly what the bonus policy added. In the worked set it is 4.3 - 3.8 = 0.5, and it equals the lift from the bonus points.
rate = present / possible
Divide the days attended by the days enrolled. With 172 attended and 180 possible the rate is 0.9555555555555556, which is 95.56%.
maxAbsences = possible x (1 - threshold)
The number of missed days a given attendance threshold permits. At 90% over 180 days, 180 x 0.1 = 18 absences are tolerated.
percentage = 100 x present / possible
The same division scaled by 100. 100 x 172 / 180 = 95.56%, and 100 x 170 / 180 = 94.44%.
Cost per credit = tuition / credits
The core division. It turns a lump tuition sum into a rate that can be compared across institutions and terms.
Course cost = cost per credit x course credits
Scales the per-credit rate to the size of a single course; a three-credit course at 400 costs 1200.
Cost per contact hour = tuition / contact hours
Divides the same tuition by scheduled instruction time instead of credits, giving 26.666666666666668 per contact hour for 12000 over 450 hours.
GPA = (gp1 x c1 + gp2 x c2 + gp3 x c3 + gp4 x c4) / (c1 + c2 + c3 + c4)
Weight each course's grade points by its credits before averaging. With 41.3 quality points across 12 credits the result is 3.4416667, whereas the unweighted mean of the four grade points is 3.5.
Quality points = grade points x credits
The quantity that accumulates across courses and terms. A 3-credit A earns 12.0 quality points and a 2-credit A- earns 7.4; sum these first and divide only at the end.
Required grade points = (Target x credits after - quality points so far) / new credits
Multiply the target by the credits you will hold, subtract what you have already banked, then divide by the credits still to be earned. From 41.3 points over 12 credits, reaching 3.50 across 16 credits needs (3.50 x 16 - 41.3) / 4 = 3.675, just above an A-.
Needed on final = (Target - Current x (1 - Final weight)) / Final weight
Subtract the coursework contribution from the target, then divide by the final's weight. From 82 with a 30 per cent final, (0.90 - 0.574) / 0.30 = 1.0866666666666667, which exceeds 1 and so is unattainable.
Locked in = Current x (1 - Final weight)
The part of the grade that no exam performance can change. With 82 standing and a 30 per cent final, 0.82 x 0.70 = 0.574, which is 57.4 per cent of the course.
Max possible = Current x (1 - Final weight) + Final weight
Set the exam score to a perfect 1.0. From 82 with a 30 per cent final, 0.574 + 0.30 = 0.874, so 87.4 per cent is the ceiling and a target of 90 is 2.6 points beyond it.
Net Burn = Monthly Expenses - Monthly Revenue
The amount drawn from the bank each month. Use cash collected, not invoiced, and include every outflow line including payroll taxes and benefits.
Runway (months) = Cash Balance / Net Burn Per Month
Cash divided by the monthly draw. Because the burn figure sits in the denominator, runway moves in inverse proportion to it.
Additional Monthly Revenue = Monthly Expenses - Monthly Revenue
The gap that must be closed before net burn reaches zero. It is numerically identical to current net burn, which is why a 50,000 burn reads equally as a 50,000 revenue target.
c = Lost / Start
The customers lost during the month divided by the customers present at the start of it. This is the base figure every other formula on the page is built from.
Annual = 1 - (1 - c)^m
Compound the retention factor across the months in the year, then subtract from one. With c = 0.04 and m = 12 this gives 0.38729024267023293, not 0.48.
Lifetime = 1 / c
The reciprocal of the monthly churn rate gives the expected number of months a customer stays. It feeds straight into customer lifetime value.
Conversion rate = conversions / visitors
The one division at the centre of the topic. Multiply by 100 to state it as a percentage; 1200 / 40000 = 0.03, which is 3%.
Visitors = conversions / rate
The same relation rearranged. Useful when a target is fixed and the question is how much traffic it takes to reach it.
Overall rate = stage rate 1 x stage rate 2 x ... x stage rate n
Stage rates multiply because each acts only on the survivors of the previous one. Here 0.20 x 0.15 = 0.03, so 40000 visitors yield 1200 conversions.
CAC = (Marketing spend + Sales spend) / New customers
Total the period's sales and marketing spend, then divide by the customers who actually paid in that period. Omitting sales spend is the standard error and understates the result by the whole sales cost per customer.
Payback months = CAC / (Monthly revenue per customer x Gross margin)
Divides the acquisition cost by the monthly contribution the customer leaves behind. Use contribution, not gross revenue: with 80 of revenue at 75 per cent margin the contribution is 60, giving 5.33 months rather than the misleading 4 months.
LTV : CAC = (Monthly contribution x Lifetime months) / CAC
Compares what a customer returns over their whole life against what it cost to win them. Around 3 to 1 is the usual floor; below 1 means acquisition loses money on every sale.
LTV = (ARPU x gross margin) / churn
Divide the monthly contribution by the monthly churn rate. The margin keeps only the revenue the business actually keeps, and the churn rate converts into an average lifetime.
Lifetime months = 1 / monthly churn
Churn and lifetime are reciprocals. At 4 per cent a month the average life is 25.0 months; at 2 per cent it is 50 months.
LTV : CAC = LTV / CAC
Compares the contribution a customer returns against the cost of winning them. Around 3 to 1 is the usual floor; below 1 destroys value.
DSO = (Accounts Receivable / Revenue) x Days
The standard form. The ratio of receivables to revenue gives the unpaid share of sales, and multiplying by the days in the period converts that share into a number of days.
Receivables Turnover = Days / DSO = Revenue / Accounts Receivable
The reciprocal view of DSO, expressed as the number of times the receivables balance is collected in a period. A DSO of 30.416666666666664 days corresponds to a turnover of 12.000000000000002 times a year.
Revenue Per Day = Revenue / Days
The daily pace of sales, which is what one day of DSO is worth in cash. Dividing revenue of 1,800,000 by 365 gives 4931.506849315068.
T = COGS / AVG
The standard form. Divide the cost of the stock sold by the average value of the stock held to get the number of turns.
AVG = (opening + closing) / 2
The two-point mean used when only the start and end of the period are known. Averaging removes the timing artefact of a single snapshot.
D = days / T
Converts the ratio into time. It is the same fact as turnover, expressed as how many days of stock are on hand.
P = C x (1 + m)
The working form. Turn the percentage into a decimal, add one, and multiply by cost. A cost of 60 at a 40% markup gives 60 x 1.4 = 84.
m = (P - C) / C
The audit form: take any price already in the market and recover the loading that produced it. Dividing the same profit by P instead of C gives the margin, so the two are always one substitution apart.
m = g / (1 - g)
Use this when the target arrives as a margin. It returns the larger markup needed to hit that margin, and the price follows from C / (1 - g). A 40% margin needs a markup of 0.40 / 0.60 = 66.6667%.
m = (R - C) / R
The standard form. Profit left after cost, expressed as a share of what the customer paid. Multiply by 100 to read it as a percentage.
u = (R - C) / C, and m = u / (1 + u)
Markup divides the same profit by cost instead of revenue. The second half converts a markup back into the margin it actually delivers.
P = C / (1 - m)
The inversion used when a margin has been decided and the price is unknown. Never add the margin percentage to cost; divide by what is left.
Total Growth = (End - Start) / Start
The increase divided by where the business began. This is the figure to quote when the question is how much revenue grew in total.
g = (End / Start)^(1/n) - 1
Take the growth multiple, raise it to one over the number of periods to spread it evenly, then subtract one. This is the figure to quote when comparing spans of different length.
End = Start x (1 + g)^n
The same relationship solved for the endpoint. Use it to check a computed rate by compounding it back, or to project what a given rate produces over n periods.
rho = m / V
The defining relationship. Divide the mass by the volume it occupies; the units of the answer follow from the units you feed in.
rho[kg/m3] = rho[g/cm3] x 1000
A factor of 1000, never 100. Going the other way you divide by 1000 instead of multiplying.
SG = rho / rho_water
Divide the density by the density of water. The units cancel, leaving a dimensionless number that is 1 for water itself.
F = m a
The core relation. Multiply mass by acceleration to get the net force. Both inputs must be in SI units, kilograms and metres per second squared, for the answer to appear in newtons.
W = m g
Weight is the force gravity exerts, so it is the second law applied to the gravitational acceleration. It gives 11767.98 N for a 1200 kilogram car on Earth.
F_net = F_applied - F_friction
The force that actually accelerates the object is what is left after opposing forces are subtracted. To keep 3600 N of net force against 1000 N of friction, apply 4600 N.
P V = n R T
The full equation of state. Pressure times volume on one side, amount times gas constant times absolute temperature on the other.
n = P V / (R T)
The rearrangement used most often. Pressure and volume multiply on top; the gas constant and temperature divide below.
V m = V / n = R T / P
The volume one mole occupies. At 0 C and 101.325 kPa it equals 22.4139695 litres per mole.
KE = 1/2 m v^2
The standard form. Square the speed first, then multiply by mass and halve. The speed must be in metres per second.
p = m v
The linear companion to kinetic energy. Momentum grows in proportion to speed rather than its square, and it is a vector.
W = F d = Delta KE
Net work equals the change in kinetic energy. It is the bridge between a force acting over a distance and the energy an object ends up carrying.
M = n / V
The defining relation. Divide moles of solute by litres of solution — the volume measured after dissolving, not the solvent added.
n = m / Mr
The bridge from the balance to the mole. Molar mass is the sum of atomic masses, in grams per mole.
M1 x V1 = M2 x V2
Adding solvent leaves the moles of solute unchanged, so the concentration-volume product is conserved.
V = I x R
The core relationship. Rearrange to I = V / R for current or R = V / I for resistance; all three are the same equation.
P = V x I = I^2 x R = V^2 / R
Three equivalent forms. Use whichever matches the two quantities you already hold.
R_series = R1 + R2 + ..., 1 / R_parallel = 1 / R1 + 1 / R2 + ...
Series resistances add; parallel resistances add as reciprocals. Conductance G = 1 / R turns the parallel rule into plain addition.
P = W / t
Divide the energy by the interval to get the rate. With joules and seconds the result is already in watts.
P = F x v
The same relationship for steady motion, since work is force times distance and distance over time is velocity.
hp = P / 745.6998715822702
Divide the watt figure by the exact conversion constant to restate the same power in horsepower.
P = F / A
The defining relation. Divide the perpendicular force by the area that carries it, and the result is in pascals.
P psi = P Pa / 6894.757293168 ; P bar = P Pa / 100000
Both conversions are single divisions by an exact factor. One psi is 6894.757293168 pascals and one bar is 100000 pascals.
P absolute = P gauge + P atmospheric
Gauge pressure is measured relative to the surrounding air, so absolute pressure adds the atmosphere back on.
s = d / t
The standard form. Convert distance and time to matching units, then divide.
d = s x t
The same relationship solved for distance. Use it when a speed is held for a known duration.
v(m/s) = v(km/h) / 3.6
Divide by 3.6, the exact number of kilometres per hour in one metre per second. Multiply to go the other way.
v = f x lambda
The standard form. Put the frequency in hertz and the wavelength in metres, then multiply to get metres per second.
T = 1 / f
The period is the time for one complete cycle. It is the reciprocal of the frequency, so a higher frequency means a shorter period.
lambda = v / f
The same relationship solved for wavelength. Use it when the medium is known and only the frequency is chosen.
ha = acres x 0.40468564224
One multiplication, no offset. The factor is exact, so the result is as precise as the input.
acres = ha / 0.40468564224 = ha x 2.471053814671653
Divide by the same factor, or multiply by its inverse. Never multiply by 0.40468564224 here.
sq ft = acres x 43560, m2 = acres x 4046.8564224, sq mi = acres / 640
The first two expand the area into smaller units; the third contracts it, so it is a division rather than a multiplication.
F = C x 9/5 + 32
Scale then shift. Multiplication first, addition second — reversing them changes the answer.
C = (F - 32) x 5/9
Undo the offset first, then undo the scaling. This ordering is what makes round-tripping exact.
Delta F = Delta C x 9/5 (no 32 added)
Differences carry no offset, because the zero point cancels when you subtract two readings.
K = C + 273.15
One addition and no multiplication, because the kelvin and the Celsius degree are the same size. Only the zero point shifts.
C = K - 273.15
The exact inverse of the direct rule. Subtracting 273.15 from 298.15 K returns 25 C with nothing left over.
R = K x 1.8
Rankine is the absolute scale with Fahrenheit-sized degrees, so the kelvin value is scaled by the ratio of the two degree sizes.
m = ft x 0.3048
One multiplication by an exact constant. Use it whenever the measurement is already in feet, or has been converted to decimal feet first.
m = (ft x 12 + in) x 0.0254
Fold everything into inches first, then apply the inch factor once. This avoids reading 6'2" as 6.2.
ft = m / 0.3048
The same relation inverted. Dividing by 0.3048 is the same as multiplying by 3.280839895.
L = gal x 3.785411784
One US liquid gallon is 231 cubic inches, and since the inch has an exact metric definition this factor is exact rather than measured. Multiply, then keep the digits.
L = gal x 4.54609
The Imperial gallon was historically the volume of ten pounds of water and is now fixed at 4.54609 L. It is 1.2009499255 times the US gallon, so one US gallon is only 0.8326741846 Imperial gallons.
L/100km = 235.214583 / mpg
Two units change at once, so the constant does both: 100 km divided by 1.609344 km per mile, times 3.785411784 liters per gallon. A rating of 30 mpg gives 7.8404861 L/100 km.
cm = in x 2.54
A single multiplication. The factor is exact by definition, so it never contributes error of its own.
in = cm / 2.54 = cm x 0.3937007874
Divide by 2.54, or multiply by its reciprocal. Both are the same operation, so use whichever your calculator handles cleanly.
total in = feet x 12 + inches
Always perform this step before touching 2.54. Converting the two parts separately gives the wrong answer.
lb = kg x 2.2046226218487757
The exact conversion. One multiplication, with a factor that carries no measurement error of its own.
kg = lb / 2.2046226218487757
The return leg of a round-trip check. Dividing by the same factor must reproduce the original mass.
oz = lb x 16, stone = lb / 14
Restating the same mass in the imperial units people actually quote for parcels and bodies.
km = mi x 1.609344
One multiplication, no offset. The factor is exact, so the result is as precise as the input.
mi = km / 1.609344 = km x 0.62137119
Divide by the same factor, or multiply by its inverse. Never multiply by 1.609344 here.
km/h = mph x 1.609344, m = mi x 1609.344, ft = mi x 5280
The same factor converts speeds, because only the distance part changes. The other two expand a distance into smaller units.
kph = mph x 1.609344
One multiplication, no time factor. The hour cancels because both rates are per hour, so only the distance unit changes.
mph = kph / 1.609344 = kph x 0.62137119
Divide by the same factor, or multiply by its inverse. Never multiply by 1.609344 in this direction.
m/s = mph x 1609.344 / 3600 = kph / 3.6
Two factors are needed here because both the distance unit and the time unit change. The chain collapses to 0.44704 m/s per mph, or to dividing kph by 3.6.
g = oz x 28.349523125
One multiplication, no offset. The factor is exact, so the result is as precise as the input.
oz = g / 28.349523125 = g x 0.03527396194958041
Divide by the same factor, or multiply by its inverse. Never multiply by 28.349523125 here.
kg = oz x 28.349523125 / 1000, lb = oz / 16
Convert to grams first, then shift the decimal point for kilograms. Pounds come straight from the sixteen-to-one definition.
kg = lb x 0.45359237
The exact conversion. One multiplication, with a factor that carries no measurement error of its own.
lb = kg x 2.2046226218
The reciprocal of the exact factor. Use this as the return leg of a round-trip check.
stone = lb / 14, oz = lb x 16
Restating the same mass in the imperial units people actually quote for bodies and parcels.
m2 = sq ft x 0.09290304
The standard form. Multiply the area in square feet by the squared factor once; never use the linear 0.3048 here.
sq ft = m2 / 0.09290304
The same relation inverted. Dividing by 0.09290304 undoes the conversion and is the natural round-trip check.
m2 = sq ft x 0.3048^2
The same calculation written from the length factor. Square it first, then multiply; skipping the square is the classic error.