Blog

A 401(k) projection answers a specific question: at a fixed contribution and a fixed return, what will the account be worth at retirement, and how much of that did the employer match create? The arithmetic is a future-value calculation with two additions a year, yours and the match.

FV = B(1+r)^t + (C + M) x [((1+r)^t - 1) / r]

A balloon loan makes small payments for a few years and then demands the rest in one lump. The size of that lump is just the outstanding balance of an ordinary amortising loan on the balloon date.

B = P x (1 + i)^n - PMT x ((1 + i)^n - 1) / i

The 50/30/20 budget is the easiest framework for people who hate budgeting. Instead of tracking dozens of line items, it sorts every dollar of take-home pay into three buckets: needs, wants and savings.

Needs = Income x 0.50; Wants = Income x 0.30; Savings = Income x 0.20

A lease payment is the interest on the whole car plus the depreciation you consume while you drive it. Both halves are set by the price, the residual value and the money factor, so a small change in any of them moves the monthly figure.

Payment = depreciation + finance charge

A down payment is not only the cash you hand over — it sets the loan-to-value ratio, and that ratio decides whether mortgage insurance attaches to every payment for years. Choosing between 10% and 20% down is therefore a cash decision and an insurance decision at the same time.

Down Payment = Purchase Price x Down Payment Percentage

A mortgage payment is more than principal and interest. Property tax, home insurance, HOA fees and private mortgage insurance all land on the same bill, and together they can add several hundred dollars a month.

P&I = Loan x monthly rate / (1 - (1 + monthly rate)^-n); Total = P&I + tax + insurance + PMI + HOA

A raise has three numbers that matter: the new salary, the extra pay per period, and whether the increase beats inflation. A nominal raise that falls below inflation is a cut in real terms, even though the paycheck is bigger.

New salary = old salary x (1 + raise%)

Gross pay is what you earn; net pay is what reaches your bank account. The gap is income tax, payroll taxes and any retirement or benefit contributions, and it is often large enough to change how you budget.

Net pay = gross - income tax - payroll tax - contributions

A personal loan is priced like any other amortising loan: the same formula that sets a mortgage payment sets a personal loan payment. What changes is the term, the rate and how much interest the term adds.

Payment = Loan x r / (1 - (1 + r)^-n), where r is APR / 12 and n is months

The price to earnings ratio is the market's asking price for each unit of profit, and it is the quickest way to put two share prices on the same footing.

P/E = share price / earnings per share

A language model bill is two multiplications and a scale factor. The token counts drive the input side and the output side separately, because they are priced differently.

cost = (prompt tokens / 1M x input price) + (completion tokens / 1M x output price)

An amortization schedule is the month-by-month ledger of a fixed-rate loan: every payment is the same size, but the split between interest and principal shifts a little each month. Early on almost everything goes to interest; near the end almost everything goes to principal. The monthly payment comes from one formula, and from it you can read off the first month's interest, the principal that payment retires, and the balance left afterwards.

M = P x r / (1 - (1 + r)^-n)

An emergency fund is cash set aside to cover essential spending if income stops or a large bill lands. The standard target is three to twelve months of essentials, depending on how secure your income is.

Fund = monthly essential expenses x months of cover

An annuity is a series of equal payments made at regular intervals, such as a monthly contribution into a retirement account or a monthly withdrawal from it. The future value tells you what the whole stream grows to by the end of the term, the present value tells you what that stream is worth in today's money, and the gap between them is the effect of compounding.

FV = PMT x ((1 + i)^n - 1) / i

Given a balance, a return and a number of years, the payment that exactly exhausts the account is one algebraic step away. The hard part is not the formula — it is recognising that a level payment for a fixed period is a very specific promise that real markets do not make.

PMT = PV x r x (1 + r)^n / ((1 + r)^n - 1)

The APR is the interest rate that would produce your loan's cash flows if the fee were rolled into the rate. It answers a fairer question than the advertised nominal rate: what single annual rate, applied to the money you actually received, reproduces the payments you actually make?

Solve for i: P - F = sum of pmt / (1 + i)^k

The annual percentage yield is what a deposit account actually returns in a year once compounding is counted. Comparing two accounts by their headline rates instead of their APYs is the most common way savers pick the slower one.

APY = (1 + r/n)^n - 1

State minimum liability limits are the legal floor, not a recommendation. Sizing cover around the assets and income you are protecting shows how much more than the minimum you may actually need.

Recommended liability = assets + 3 x annual income + 25,000 per dependant

A car loan is an amortizing loan like any other: a fixed payment retires the balance over the term. The car-specific wrinkle is the trade-off between payment size and total cost, and the effect of a trade-in and sales tax on the amount actually financed.

M = P x r / (1 - (1 + r)^-n)

A bond's yield is the income it pays divided by what you paid for it. Because bonds trade at prices above or below their face value, two bonds with the same coupon can yield very different amounts, and the price move at maturity is part of the return.

Current yield = annual coupon / price paid

CAGR compresses a messy multi-year history into one smooth annual rate. Because it reports the same kind of number for a seven-year stock hold and a five-year revenue line, it is the standard way to put two very different histories on one scale.

CAGR = (Ending Value / Beginning Value)^(1/n) - 1

The capitalization rate is the return a property would throw off if you owned it outright. Net operating income divided by value, with the mortgage left out of the picture entirely.

cap rate = net operating income / property value

When an asset is sold for more than it cost, the profit is a capital gain. Holding it over a year usually earns a lower long-term rate, while a shorter holding is taxed at the ordinary income rate.

Gain = proceeds - cost basis; Tax = gain x rate; After-tax profit = gain - tax

A higher deductible lowers the premium but raises what you pay when you claim. The right level is where the annual saving beats the expected extra cost of claiming.

Net benefit = (low-deductible premium - high-deductible premium) - extra exposure x claim frequency

A certificate of deposit pays a fixed rate for a fixed term in exchange for leaving your money untouched. The growth depends on whether interest compounds and on how early withdrawal penalties would bite, but the core number is simply the deposit grown at the APY.

A = P(1 + r)^t

A down payment is not the only cash a purchase needs. Closing costs run alongside it, and buyers who budget only for the deposit are regularly short at the table.

closing costs = loan fees + title charges + third-party fees + prepaid escrow

Compound interest is interest that earns interest. Once interest is added to your balance, that added amount starts earning too — which is why long horizons grow faster than people expect.

A = P(1 + r/n)^(nt)

A cost of living calculation adds up the bills you pay every month and compares them with what you earn. The gap between the two is the number that decides how much room there is for saving, debt repayment or anything else.

Total = Housing + Food + Transport + Utilities + Other; Leftover = Income - Total

Credit card interest is charged on the balance every day, then billed monthly. Converting the annual APR into a monthly and daily figure shows how much of a payment is consumed before a single dollar of principal is repaid.

Monthly = Balance x APR / 12; Daily = Balance x APR / 365

Paying a fixed amount every month retires a revolving balance on a predictable schedule — and that schedule can be written as a closed-form equation rather than simulated line by line. The catch is a boundary condition the formula cannot cross: pay only the interest and the balance never falls.

n = -ln(1 - (rm x B) / P) / ln(1 + rm)

Credit utilization is the one credit score factor you can move in a single billing cycle, which makes it the most actionable number in personal finance.

utilization = total balances / total credit limits

A crypto profit calculator shows what a trade actually keeps after fees. Enter the price paid, the price sold, the quantity and the fee rate, and the net profit and return on investment follow directly.

Net profit = Proceeds - Cost - Fees; ROI = Net profit / Cost x 100

Lenders on income producing property care about one question above all: does the property earn enough to make the payments. Debt service coverage ratio is that question turned into a single number.

DSCR = NOI / annual debt service

The debt-to-income ratio answers the question a lender is actually asking: not how much you owe, but whether the monthly payments fit inside your income. It takes two numbers and it decides mortgage approvals.

DTI = Total Monthly Debt Payments / Gross Monthly Income

Disability insurance replaces part of your income if illness or injury stops you working. Sizing the benefit means covering essential spending, not your whole salary, and knowing the premium for that protection.

Benefit = income x coverage percent (capped at the spending gap); Premium = benefit / 100 x rate

Dividend income is the easiest investment return to picture, because it arrives as cash on a date you can see. That makes it the easiest to over-count as well.

annual income = shares x dividend per share; yield = dividend per share / price

Your effective tax rate is what you actually paid, expressed as one number. It is always lower than your headline bracket on the same tax — and often badly misleading if you stop at federal income tax and ignore payroll.

Effective Tax Rate = Total Tax / Gross Income

An equated monthly instalment is the fixed payment that repays both principal and interest over a set term. It stays the same every month, but the split between interest and principal shifts steadily in the borrower's favour.

EMI = P x r x (1+r)^n / ((1+r)^n - 1)

An expense ratio looks tiny and is not. It is charged every year, so it compounds against the portfolio in exactly the way the return compounds for it.

net return = gross return - expense ratio; value = investment x (1 + net)^years

A fixed deposit pays a headline rate that is not quite what you earn, because interest is compounded during the year. This calculator shows the maturity value, the interest earned and the effective annual yield for any compounding frequency.

Maturity = P x (1 + r / n)^(n x t)

Forex profit is the price move converted into pips and multiplied by the value of a pip at your position size. On a USD-quoted pair a standard lot is worth about ten dollars per pip, so the whole trade is a short piece of arithmetic once the direction is fixed.

Profit = (Exit - Entry) x 10,000 pips x (10 x Lots) x Direction

Future value of an annuity projects what a stream of equal contributions becomes. Because every deposit compounds for a different length of time, this is the formula retirement plans actually depend on — not the lump-sum one.

FV = PMT x (((1 + r)^n - 1) / r)

Gratuity is a lump sum an employer pays for long service. The statutory formula gives fifteen days of salary for every completed year, using only your basic pay and dearness allowance, not your full cost to company.

Gratuity = (15 / 26) x monthly salary x completed years

The gross rent multiplier is the fastest way to compare rental properties. Divide the price by the annual rent and you get a single number that ranks deals before any expenses are considered.

GRM = property price / annual gross rent

Goods and services tax is a single tax applied to most supplies of goods and services. Adding it multiplies the base price; removing it divides, and the same rate can be split between national and regional shares.

GST = base x rate

A health insurance subsidy, or premium tax credit, caps the share of your income you pay for a benchmark plan. The subsidy is the gap between that benchmark premium and your expected contribution.

Subsidy = benchmark premium - income x applicable percentage

Lenders judge affordability with two ratios. Housing should stay under twenty-eight percent of gross income and total debt under thirty-six percent, and the tighter of the two caps your mortgage payment.

Housing cap = 0.28 x gross monthly income; Debt cap = 0.36 x income - other debts

Lenders cap your monthly payment using two ratios: the front-end limits housing cost to a share of income, and the back-end limits total debt. The stricter one sets the loan you can qualify for.

Front limit = income x front ratio; Back limit = income x back ratio - debts; Loan from the lower limit

Homeowners insurance should insure the cost to rebuild the home, not its market value. Add contents and liability cover, then apply a rate per thousand to estimate the annual premium.

Dwelling = square feet x rebuild cost per square foot; Premium = (dwelling + contents) / 1,000 x rate

An hourly rate becomes a salary once you decide how many hours and weeks it will be paid for. The multiplication is simple, but the assumptions about paid leave and hours are what determine the answer.

Annual = Hourly x Hours per week x Weeks per year

A savings goal has three moving parts — what you already have, what you add each month, and what compounding quietly adds on top. The third one is the reason rounding your monthly figure to a nice number can push your finish line out by years.

n = ln((FV x r + PMT) / (PV x r + PMT)) / ln(1 + r)

The House Rent Allowance exemption is capped by three separate limits, and the smallest one applies. This calculator works out the exempt amount, the taxable balance and the annual figures from your salary, HRA and rent.

Exempt = min(HRA, rent - 10% of salary, 50%/40% of salary)

Income tax is progressive: each slice of income is taxed at its own rate, so the whole amount is never taxed at the top rate. The effective rate is what you actually pay; the marginal rate is what the next dollar costs.

Tax = sum of each bracket amount x its rate; Effective rate = tax / income

Indian income tax is charged in slabs, so only the income within each band is taxed at that band's rate. This calculator applies the slabs, the standard deduction, the section 87A rebate and the four percent cess for the new and old regimes.

Tax = (slab tax - 87A rebate) x 1.04

Inflation quietly erodes the buying power of money. A sum held in cash loses value every year even though its number stays the same. This calculator converts between nominal amounts and real, inflation-adjusted amounts so you can compare figures honestly across time.

Real value = nominal / (1 + i)^t

Total return measures how much an investment gained in percentage terms. Annualised return converts that into a per-year rate, which is the only fair way to compare a three-year holding with a ten-year one.

Total return = (final value - initial) / initial; Annualised = (1 + total)^(1/years) - 1

The internal rate of return is the discount rate that makes an investment's net present value exactly zero. It is the break-even rate the project must beat, and it lets you compare projects of different sizes on a single percentage scale.

NPV = 0 at the rate r, so r is the IRR

The DIME method is a quick way to size life insurance: Debt, Income, Mortgage and Education. Add the four, subtract what you already have, and the remainder is the cover your family would need.

DIME = debts + income x years + mortgage + education; Need = DIME - existing cover - savings

An amortizing loan keeps the payment fixed while the split between interest and principal changes every month. The formula that produces that payment — and the amortization behind it — explains why shortening the term saves so much.

M = P x r(1 + r)^n / ((1 + r)^n - 1)

Loan to value compares what you borrow with what the property is worth. Lenders use it to set pricing, and it decides whether mortgage insurance is required.

LTV = loan amount / property value

Long-term care insurance pays for help with daily living when age or illness makes it necessary. Sizing it means comparing the total cost of care with the benefit a policy would actually pay.

Out of pocket = total care cost - total benefit paid

Most cards set the minimum as the greater of a percentage of the balance or a flat amount, and interest accrues on the rest. Paying only that minimum can keep a balance alive for years.

Minimum = max(Balance x Percent, Flat); Months = -ln(1 - r x B / P) / ln(1 + r)

Every extra dollar paid against a mortgage removes principal that would otherwise be charged interest for the rest of the term. A modest extra payment can cut years off the loan and save tens of thousands.

Months with extra = -ln(1 - rate x balance / (payment + extra)) / ln(1 + rate)

Net worth is the single number that summarizes a balance sheet: everything owned minus everything owed. It takes about ten minutes to compute and it answers a question no income figure can — whether you would be solvent if every debt came due today.

Net Worth = Total Assets - Total Liabilities

Net present value asks a single sharp question: after discounting every future cash flow back at the rate you require, does the investment create or destroy value? It is the most reliable single test in capital budgeting because it works in money rather than in ratios.

NPV = -C0 + sum of CF / (1 + r)^t

PayPal takes a percentage of every payment plus a small fixed fee, and the fixed part makes the effective rate higher than the headline percentage on small transactions.

Fee = Amount x Percentage + Fixed; Net = Amount - Fee

A defined-benefit pension is built from years of service and an accrual rate. Each year of service earns a percentage of your final average salary, and the total is your annual pension for life.

Annual = years x accrual rate x final salary

Pet insurance trades a monthly premium for reimbursement of vet bills. Whether it is worth it depends on the premium, the deductible and how much care your pet actually needs in a year.

Reimbursed = (vet cost - deductible) x reimbursement percent; Net cost = premium + your share + routine costs

The Public Provident Fund is a long-term, tax-free savings account. A fixed amount is contributed each year, interest is credited annually and compounds, and the whole balance is free of tax at maturity.

Maturity = C x ((1 + r)^n - 1) / r x (1 + r)

Present value runs the time value of money backwards: instead of growing today's money forward, it tells you what a future payoff is worth right now. Every bond price, lease valuation and capital budgeting decision rests on this one operation.

PV = FV / (1 + r)^n

Property tax is charged on an assessed value that may be a fraction of market value, minus any exemptions, multiplied by the local rate. The effective rate on the price is often lower than the headline rate.

Annual tax = (market value x assessment ratio - exemptions) x tax rate

Purchasing power translates an amount of money into what it can actually buy. Every long-term plan denominated in fixed dollars is quietly wrong until this adjustment is applied.

Real Value = Nominal Amount / (1 + i)^n

Refinancing replaces one loan with another, usually at a lower rate. The monthly saving is the obvious gain, but closing costs and a possibly longer term mean the break-even point and the lifetime interest matter just as much.

Saving = old payment - new payment; Break-even = costs / monthly saving

Rent versus buy is not a comparison of rent with a mortgage payment. Owning costs the down payment, the loan, taxes, insurance and maintenance, but it returns the equity and any appreciation at sale.

Net cost to own = down payment + payments + tax + maintenance + insurance - sale proceeds

Rental yield is the return a property produces before financing, and the gap between the gross and net figures is where most landlords discover their real return.

gross yield = annual rent / property value

Renters insurance covers your belongings, your liability and somewhere to live if your rental becomes uninhabitable. The landlord's policy covers the building, not your things.

Total cover = contents + liability + loss of use; Premium = contents / 1,000 x rate + liability base

Retirement planning comes down to two numbers: how large your nest egg will grow by the time you stop working, and how much income that pot can safely pay you each year.

Nest egg = savings x (1 + r)^n + contribution x ((1 + r)^n - 1) / r

Return on equity is the single number that says how hard a company is working with the money its owners have actually put in.

return on equity = net income / shareholders equity

Return on investment expresses what you gained as a percentage of what it cost you. That single normalization is why ROI survives every criticism leveled at it: it works on a stock, a rental property, or a training program with exactly the same arithmetic.

ROI = (Current Value - Cost) / Cost x 100%

A Roth IRA is funded with money that has already been taxed, and qualifying withdrawals are tax free. The benefit is not a deduction today but decades of compounding that the tax authority never touches.

Future value = sum of contributions x (1+r)^t, plus existing balance grown

A salary is quoted annually but lived monthly and often compared hourly. Converting between the two uses a standard working year, and the assumptions about hours and weeks are what move the answer.

Hourly = Annual / (Weeks x Hours per week)

Sales tax and VAT calculations have one common trap: the rate applies to the net price, not to the gross. Adding tax to a net figure means multiplying; removing tax from a gross figure means dividing, and using the wrong operation gives an answer that is close enough to look right but wrong by the tax amount.

Gross = net x (1 + rate) ; Net = gross / (1 + rate)

Savings account interest depends on the rate and how often it compounds. Monthly compounding earns a little more than annual compounding at the same headline rate, and regular deposits add to the balance faster than the interest alone.

Balance = principal x (1 + rate/n)^(n x t)

When you work for yourself you pay both halves of Social Security and Medicare, which is 15.3 percent of net profit. The calculation applies a 92.35 percent base and a wage cap, and half of the result is deductible.

SE tax = 12.4% x min(base, wage base) + 2.9% x base, where base = profit x 92.35%

Simple interest pays only on the original principal and never reinvests itself. That makes it the easiest interest calculation to verify by hand — and the right model for most short-term loans, bonds and treasury bills.

I = P x r x t

A Systematic Investment Plan invests a fixed amount in mutual funds every month. Each instalment compounds for the months remaining, so the corpus at the end is larger than the sum of your contributions by the gains.

Corpus = P x ((1 + i)^n - 1) / i

Social Security and Medicare tax, together called FICA, comes out of every paycheck. Social Security applies only up to a wage base, while Medicare has no cap and adds an extra charge above a high-income threshold.

FICA = min(Wages, Base) x 6.2% + Wages x 1.45% + max(0, Wages - Threshold) x 0.9%

Stock profit is the difference between what the shares sold for and what they cost, minus the commission paid to the broker. It sounds trivial, but the fee is easy to forget and it is the difference between a gain and a loss on a small move.

Net profit = Shares x (Sell - Buy) - Fees

A student loan repaid on the standard plan uses the same annuity formula as any other instalment loan. The balance, the rate and the term set the payment, and the term is usually the most powerful lever on total interest.

Payment = P x r x (1+r)^n / ((1+r)^n - 1)

A progressive tax system taxes each slice of income at its own rate. Your bracket is the rate on the last dollar you earn, not the rate on all of it, and the difference between the two is the source of most confusion about taxes.

Tax = sum of (income in each bracket x that bracket's rate)

Term life insurance pays a lump sum if you die during a fixed period, usually 10 to 30 years. The two questions that matter are how much cover your family actually needs and roughly what it costs.

Cover = income x years replaced + debts + education + final costs - existing cover - savings

Term and whole life both pay a death benefit, but they are priced very differently. Comparing them over the same number of years, and investing the difference, shows what the permanent cover really costs.

Difference invested = (whole premium - term premium) x future value of an annuity

Break-even is the unit count at which contribution from sales finally covers every fixed cost. Below it each additional unit loses money; above it each one contributes its full margin to profit — which is why the number is more useful as a threshold than as a forecast.

Break-Even Units = Fixed Costs / (Price - Variable Cost Per Unit)

The rule of 72 is a mental shortcut for doubling time: divide 72 by the annual return and you get roughly how many years it takes for money to double. It is fast, memorable and close enough for nearly every rate you will meet.

Doubling time = 72 / annual return percent

Your edge in a trade is the price move minus friction. Fees, spreads and funding quietly shrink every trade, so a small directional move can still lose money after costs are counted.

Net = (Exit - Entry) x Qty x Direction - (Entry x Qty + Exit x Qty) x Fee rate

Value added tax is charged on the price of goods and services and is normally included in the price you see. Adding it is a multiplication; removing it is a division, and mixing the two up is the most common VAT error.

Gross = net x (1 + rate)

Every source of money a company uses has a price. WACC is the single blended price, weighted by how much of each source the company actually relies on.

WACC = (E / V) x Re + (D / V) x Rd x (1 - T)

The FIRE number is your annual spending divided by the withdrawal rate you consider sustainable. At 4% that means twenty-five times a year of expenses — a figure that is easy to compute and very easy to compute on the wrong expenses.

FIRE Number = Annual Spending / Safe Withdrawal Rate

Two loan offers can be compared on the monthly payment or on the total cost. The two rankings are often different, and the difference is exactly what the fee is for.

payment = principal x r / (1 - (1+r)^-n); total cost = payment x n - principal + upfront fee

The textbook answer divides salary by 2,080 hours. The honest answer divides it by the hours you actually surrender to the job, which for most professional roles is a substantially larger number and a substantially smaller rate.

Hourly Rate = Annual Salary / (Hours Per Week x Paid Weeks Per Year)

Dollar-cost averaging is the discipline of investing a fixed amount on a fixed schedule, whatever the market is doing. It replaces a timing decision with a calendar.

value = initial x (1+r)^n + payment x ((1+r)^n - 1) / r

Debt payoff looks like a division problem and is not. Interest is charged on what is left each month, so the balance falls more slowly than a naive division suggests.

months = -ln(1 - r x balance / payment) / ln(1 + r)

The 50/30/20 rule splits after-tax income into three buckets and nothing else. It is popular because it needs no categories, no spreadsheet and no tracking of individual purchases.

bucket amount = take-home pay x bucket percentage

A confidence interval turns a single sample estimate into a range, and the width of that range is the honest measure of how much the sample actually tells you.

confidence interval = mean +/- z x (standard deviation / sqrt(n))

A cube root asks which number, multiplied by itself twice more, produces the value you started with. It is the exact inverse of cubing, and unlike a square root it has a single real answer for every real input, negative numbers included.

cube root of x = x^(1/3), with the sign of x carried separately

The geometric mean is the average you use when values multiply rather than add. Multiply all the numbers together and take the root that matches how many there are.

Geometric mean = (x1 x x2 x ... x xn)^(1/n)

The harmonic mean is the average for rates. Instead of adding the values and dividing, you flip each one, average the flipped numbers, then flip the answer back.

H = n / (1/x1 + 1/x2 + ... + 1/xn)

A logarithm asks what power a base must be raised to in order to produce a given number, and it turns multiplication into addition.

log_b(x) = ln(x) / ln(b)

A determinant collapses a square matrix into a single number that tells you how much the matrix stretches space and whether it collapses it flat.

2x2: det = ad - bc. 3x3: sum the three down-right diagonals and subtract the three down-left ones.

The midpoint of two points sits exactly halfway between them in every direction. You get it by averaging the x coordinates and averaging the y coordinates.

midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Every percentage question is one of three sentences wearing different clothes. Once you can identify which one you are looking at, the arithmetic is a single multiplication or division, and the common mistakes all come from picking the wrong one.

Part = Whole x Percentage / 100

A ratio is a division you have decided to keep as a pair, and the two things you do with it are comparing sizes and splitting a whole.

ratio value = first quantity / second quantity; share = total x quantity / (sum of all quantities)

A weighted average answers "what is the typical value?" when some values count more than others. Grades, portfolio returns, inventory costs and survey results are almost never plain averages, which is why the distinction matters.

Weighted Average = Sum of (Value x Weight) / Sum of Weights

A z-score restates a value as how many standard deviations it sits from the mean. That rescaling removes the original units, which is exactly what lets you compare things that were never commensurable.

z = (x - mean) / standard deviation

Expected value is the long run average of a gamble. Weight every outcome by how likely it is, add the results, and you have the number that decides whether a bet is worth taking.

E = p1 x x1 + p2 x2 + ... + pn x xn

Exponential growth happens when a quantity is multiplied by the same factor every period. Interest, population and radioactive decay all follow the same shape once the numbers are set up.

A = P x (1 + r)^t, where r can be negative for decay

Fraction arithmetic is ordinary arithmetic with one extra step: the denominators have to agree before the numerators can be combined. Once that is handled, the four operations reduce to a handful of rules.

a/b + c/d = (a x d + c x b) / (b x d)

Linear interpolation fills the gaps between two points you already know. Work out how far along the gap your target sits, then walk the same fraction along the values.

y = y1 + (x - x1) x (y2 - y1) / (x2 - x1)

Logistic growth is exponential growth with a ceiling. Everything grows quickly at first, then the growth rate fades as the population presses against what the environment can support.

P(t) = K / (1 + ((K - P0) / P0) x e^(-rt))

Percent decrease is a drop divided by the level it fell from. The denominator decides everything: 120 to 90 is a fall of 25.00%, and the same 30 measured against 90 gives 33.3333%, which answers a different question entirely.

Percent Decrease = (Original Value - New Value) / Original Value x 100

Percent increase is a gap divided by where you started. Get the denominator right and the rest is arithmetic; get it wrong and every figure downstream is quietly answering a different question.

Percent Increase = (New Value - Original Value) / Original Value x 100

Percentage change is a difference divided by a starting point. That denominator is where almost every mistake lives — including the one that makes a 90% collapse look recoverable.

Percentage Change = (New Value - Original Value) / Original Value x 100

Probability is the number of ways an event can happen divided by the number of equally likely outcomes. Everything else, from complements to combined events, is built from that one fraction.

P = favourable outcomes / total outcomes

Slope is how much a quantity changes vertically for one unit of horizontal movement. It turns two data points into a rate, and once you have it, the whole line follows.

Slope = (y2 - y1) / (x2 - x1)

Standard deviation is the typical distance of observations from their mean, measured in the original units. The squaring is what makes it algebraically useful — and the n-1 is what makes it honest for a sample.

SD = sqrt(Sum of (xi - Mean)^2 / (n - 1))

Circle area is pi times the radius squared. Everything difficult about the topic sits in two decisions: which length is the radius, and how much precision you carry.

Area = pi x r^2

A regular hexagon splits into six equilateral triangles, which makes its area a short formula involving the square root of three. One side length is enough to find everything else.

A = 1.5 x sqrt(3) x s^2, or A = 0.5 x perimeter x apothem

Parallelogram area looks like rectangle area until you notice the sides are slanted, and using the slanted side instead of the height is the mistake almost everyone makes once.

area = base x perpendicular height

Rectangle area is the first formula most people learn and the one most often got wrong, because the units, the perimeter and the diagonal all get mixed into the same sentence.

area = length x width

A sector is the wedge-shaped piece of a circle bounded by two radii and the arc between them. Its area is a simple fraction of the whole circle, with the fraction set by the central angle.

sector area = (angle / 360) x pi x r^2, or 0.5 x r^2 x theta in radians

A trapezoid has two parallel sides of different lengths, and its area is the average of those two sides multiplied by the distance between them.

A = (a + b) / 2 x h

A triangle covers exactly half of whatever parallelogram shares its base and height. Nearly every wrong answer comes from one of two slips: a height that was never perpendicular, or a base and a height borrowed from two different sides.

Area = 1/2 x base x height

An ellipse is a circle that has been stretched along one axis, and its area is the circle area with one radius swapped for the other.

A = pi x a x b, where a is the semi-major axis and b is the semi-minor axis

Asking for the average is really asking which of three different summaries you want, and they disagree whenever the data is lopsided.

mean = sum / count

Circumference is the distance around a circle, and it is the one circle measurement you can take with a piece of string, which makes it the easiest to check.

circumference = 2 x pi x radius

The distance formula is Pythagoras in disguise, and seeing that connection is what stops it being a formula you have to memorise.

distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

The inverse of a matrix is the matrix that undoes it, and for the two-by-two case the recipe is shorter than the explanation.

A inverse = (1 / (ad - bc)) x [ d, -b; -c, a ]

The median is the number that sits in the middle, and it is the statistic that survives an outlier when the average does not.

median = middle value once the data is sorted

Perimeter is the fence, the skirting and the picture frame: the distance around the edge, which is always a plain length rather than a square measure.

perimeter = 2 x (length + width)

The surface area of a cone is the flat circular base plus the curved side that tapers to the point. The only extra quantity you need beyond radius and height is the slant height, the distance down the outside of the cone.

total surface area = pi x r x l + pi x r^2, where l = sqrt(r^2 + h^2)

A cube is the one solid where every face is identical, so the total surface collapses to a single multiplication once you know the edge.

surface area = 6 x s squared, where s is the edge length

A cylinder is the easiest curved solid to understand, because unrolling its side turns it into a plain rectangle whose width is the circle's circumference.

surface area = 2 x pi x r x h + 2 x pi x r^2

A sphere's surface area is exactly four of its great circles, a result Archimedes was so pleased with that he asked for it on his tombstone.

surface area = 4 x pi x radius^2

A cone holds exactly one third of the cylinder that would fit around it, which is where the one third in the formula comes from and why it surprises people.

volume = pi x radius^2 x height / 3

A cube is the one solid where a single measurement determines everything, which makes it the cleanest place to see how length, area and volume scale differently.

volume = side^3

Cylinder volume is the area of the circular base multiplied by the height. Nearly every wrong answer comes from one of two places: a diameter used where a radius belongs, or units that were never made consistent before being cubed.

V = pi x r^2 x h

A pyramid holds exactly one third of the prism built on the same base with the same height, whatever the shape of that base.

volume = base area x height / 3

Volume of a rectangular prism is length times width times height. Nearly every error on this page comes from somewhere else: mismatched units, external instead of internal measurements, or a cubic answer read as a liquid one.

V = l x w x h

A sphere is the only solid whose volume and surface area both follow from a single measurement. Measure the radius once and every other quantity is a formula away.

V = (4/3) x pi x r^3

A triangular prism is a length of constant triangular cross-section. Its volume is the area of that triangle times the length, and the triangular face comes from Heron's formula when you only know the three sides.

volume = triangle area x length, where triangle area = sqrt(s(s-a)(s-b)(s-c))

Significant figures are the digits in a number that carry real information. They exist because a measurement is only as precise as the instrument that produced it, and reporting extra digits implies a precision that was never measured.

rounded = round(x, sigFigs - 1 - floor(log10(|x|)))

In a geometric series each term is the previous one multiplied by a fixed ratio. That single change turns polynomial growth into exponential growth — and turns the sum formula into something with a constraint worth respecting.

S = a1 (1 - r^n) / (1 - r), and S-infinity = a1 / (1 - r) when |r| < 1

An arithmetic series adds the terms of a sequence that changes by a fixed amount each step. The sum has a closed form, which means adding a thousand terms costs the same effort as adding five.

S = n/2 x (a1 + an)

A proportion says two ratios are equal, which means one unknown journey holds all the information you need. The entire skill is deciding which quantity sits above which in the division.

a / b = c / d, so d = c x b / a

Every quadratic equation has a closed-form solution. The discriminant tells you what kind of answer to expect before you compute anything, which is the part people skip.

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Two equations with two unknowns have a single solution whenever the two lines cross at exactly one point, and Cramer's rule finds that point with nothing more than determinants.

D = a1 b2 - a2 b1

A scientific calculator is only as reliable as the assumptions you make about it. Precedence and angle units cause more wrong answers than the arithmetic does.

brackets first, then powers, then multiply and divide, then add and subtract

For right triangles, the squares of the two shorter sides add exactly to the square of the longest. That single relation turns any two known sides into the third.

c^2 = a^2 + b^2, so c = sqrt(a^2 + b^2)

Anything moving in a circle needs a force pointing at the centre, and its size depends on speed squared, mass and how tight the turn is.

F = m x v squared / r

Every pair of masses in the universe pulls on each other, and the strength of that pull is one multiplication, one division and one constant.

F = G x m1 x m2 / r squared

Stretch a spring and it pushes back in proportion to how far you stretched it. That single sentence is Hooke's law, and it is the reason springs appear in suspension, in scales, in mattresses and in the shock absorbers that keep a door from slamming. The force is easy to compute; the subtlety is knowing how far the linear rule actually holds.

F = -k x

The thin lens equation relates the focal length of a lens to where an object sits and where its image forms. Magnification then tells you how big the image is and whether it is flipped.

1/f = 1/do + 1/di, rearranged as di = f x do / (do - f)

Acceleration is how quickly velocity changes. One subtraction and one division turn a starting speed, an ending speed and a time into the rate at which the motion is changing.

a = (v - u) / t

Escape velocity is the speed at which an object's kinetic energy exactly cancels the gravity holding it down, and it depends only on the body you are leaving.

v = sqrt(2 x g x r), where g is surface gravity and r is the body's radius

Battery life in hours is the stored energy divided by the power the load draws. Amp hours give the charge, volts multiply that into watt hours, and the load in watts sets how fast it drains.

runtime hours = amp hours x volts / watts

Buoyancy is the upward push a fluid gives anything placed in it. It depends on how much fluid is pushed aside, not on how heavy the object is, which is why a steel ship floats and a steel nail does not.

buoyant force = fluid density x displaced volume x gravity

Coulomb's law is the electric equivalent of Newton's law of gravity. It says the force between two point charges is proportional to both charges and falls off with the square of the distance between them.

F = k x q1 x q2 / r^2, with k = 8.9875 x 10^9 N m^2 / C^2

Decibels compress an enormous range of sound pressures into a manageable scale. The formula is logarithmic, which is why adding three decibels roughly doubles the intensity.

dB = 20 x log(pressure / reference pressure)

Density is mass divided by volume, and nearly every practical difficulty comes from units: knowing which pair you are holding and which pair the answer should be in.

Density = mass / volume (rho = m / V)

Diluting a solution adds solvent without changing how much solute is present. That single fact gives one equation relating the stock solution to the dilute one.

C1 x V1 = C2 x V2

Efficiency is the ratio of what you get out to what you put in, and it is the number that decides whether a machine is worth running at all.

efficiency = useful output / total input

Force is mass times acceleration, but the numbers only make sense once mass and weight are kept apart. A 1200 kilogram car needs 3600 newtons to accelerate at 3 metres per second squared, yet it weighs 11767.98 newtons because gravity is pulling on it constantly.

F = m x a

Free fall is what an object does when gravity is the only force acting on it. Air resistance is ignored, which makes the mathematics exact and the answers a good approximation for anything dense.

t = sqrt(2h / g) and v = g x t

Frequency counts how many complete wave cycles pass a point each second. It is the bridge between the speed of a wave, its wavelength, and the time one cycle takes, and it is quoted in hertz.

f = v / lambda, or f = 1 / T

A gear ratio is the number of teeth on the driven gear divided by the number on the driving gear. It sets both the change in speed and the change in torque, in opposite directions.

ratio = driven teeth / driving teeth

A half life is the time it takes for half of a sample to decay. Three quantities describe the whole process, and any two of them fix the third.

N = N0 x (1/2)^(t / t1/2)

Impulse explains why a long soft stop hurts less than a short hard one: the same change in momentum, delivered over more time, needs far less force.

impulse = force x time

Kinetic energy is one half times mass times the square of speed. The square is what makes the formula interesting: it is why doubling your speed quadruples the energy you have to lose, and why momentum tells a different story.

KE = 1/2 x m x v^2

A light year is a distance, not a time: it is how far light travels through vacuum in one year. That makes it a convenient yardstick for the gap between stars, and it converts cleanly into parsecs and parallax angles.

d = c t

Molar mass is the mass of one mole of a substance, and it converts between the grams you weigh on a balance and the moles every chemistry formula expects.

M = m / n

Molarity is the concentration chemists reach for first: moles of solute per litre of solution. Everything difficult about it lives in two distinctions — moles rather than grams, and solution rather than solvent.

M = n / V

Momentum is mass multiplied by velocity, and it is the quantity that decides how hard something is to stop. It is a vector, so direction matters, and unlike kinetic energy it grows in a straight line with speed.

p = m v

A hot object cools fastest when it is hottest, because the rate follows the temperature gap. That single idea produces an exponential curve.

T(t) = Ta + (T0 - Ta) x e^(-kt)

Ohm's law is one relationship among three quantities, and any two of them determine the third. Everything else in the topic is either a rearrangement of that single equation or a consequence of it.

V = I x R

Percent error turns the gap between a measurement and an accepted value into a single number that can be compared across experiments of different sizes.

percent error = |experimental - theoretical| / |theoretical| x 100

Percent yield compares what a reaction actually produced with what the balanced equation says it should have produced. It is the standard measure of how well a synthesis went.

percent yield = actual mass / theoretical mass x 100

pH is the negative base ten logarithm of the hydrogen ion concentration. That single logarithm is why a hundredfold change in concentration moves the scale by only two units.

pH = -log10[H+]

Lifting an object stores energy in the gravitational field. Mass, gravity and height are all it takes to work out how much.

PE = m x g x h

Power is the rate at which energy is transferred, calculated by dividing work by time. Almost every error in this topic comes from mixing up a rate with a quantity — watts with watt-hours — before any arithmetic begins.

Power = work / time

Pressure is force divided by area, and almost everything interesting about it comes from the area in the denominator. The same push concentrated on a knife edge and spread across a snowshoe produces pressures that differ by orders of magnitude.

P = F / A

Resistance describes a particular object; resistivity describes the material it is made of. Turning one into the other is a matter of geometry.

rho = R x A / L

Specific heat is the price of a temperature rise. Every material charges a different number of joules per kilogram per degree, and that number explains why a beach cools faster than the sea beside it.

Q = mass x specific heat x temperature change

Speed is distance divided by time, and almost every error in the topic comes from the units rather than the division. Get the units consistent and the arithmetic is a single line.

s = d / t

A falling object stops accelerating when the air pushing up matches its weight. Terminal velocity is the speed where those two forces balance.

v = sqrt(2 x m x g / (rho x Cd x A))

The Doppler effect is the change in observed frequency caused by motion of the source, the observer, or both. Approaching motion raises the pitch, receding motion lowers it, and one formula covers every case.

f' = f x (c + vo) / (c - vs)

The ideal gas law ties pressure, volume, amount and temperature into one equation, and the only genuinely awkward part is choosing the right value of R for the units you are holding.

PV = nRT

The Moon's phase is a position in a repeating cycle rather than a property of the calendar, so finding it means counting days from a known new moon and dividing by the length of the cycle.

age = (JDN - epoch) mod 29.530588853

A pendulum's period depends on just two things: how long it is and how strong gravity is. Neither the mass hanging on the end nor a small swing angle changes the answer.

T = 2 x pi x sqrt(L / g)

Most materials get longer as they get warmer. The change is small, predictable, and large enough to buckle a bridge or jam a piston if nobody allowed for it.

delta L = alpha x L x delta T

Time dilation is the special relativity result that a moving clock ticks more slowly. The effect is tiny at everyday speeds and dramatic once you get close to the speed of light.

t = t0 / sqrt(1 - v^2/c^2), where the reciprocal of the root is the Lorentz factor

Torque is the turning effect of a force, and the lever arm matters as much as the force: the same push does far more with a longer spanner.

torque = force x lever arm

Total resistance is the single resistor that behaves exactly like a whole network. Series connections add, parallel connections combine reciprocally, and mixing the two is where most mistakes happen.

Series: R = R1 + R2 + R3. Parallel: 1/R = 1/R1 + 1/R2 + 1/R3

Voltage drop is the loss along a cable as current flows through its resistance. It is proportional to the length of the run, the current, and the resistance of the conductor, and it is counted twice because the current must return.

V_drop = 2 x length in km x resistance per km x current

Wave speed is frequency multiplied by wavelength, written v = f times lambda. The arithmetic is trivial; the insight is that the two factors are locked together by the speed of the medium, so changing one forces the other to compensate.

v = f x lambda

Work has a strict meaning in physics that is narrower than everyday usage: no movement in the direction of the force, no work done, however tired you get.

work = force x distance

A caloric surplus is maintenance calories plus a daily amount sized to a weekly weight goal.

daily surplus = weekly goal x calories per unit / 7

A calorie deficit is subtraction: maintenance calories minus the calories you eat. The arithmetic is simple, but the figure you subtract from is a moving estimate rather than a fixed constant, and that is where most of the difficulty lives.

Daily deficit = maintenance calories - intake calories

A due date is forty weeks counted from the first day of the last menstrual period, and the arithmetic is a single addition of 280 days.

due date = last period + 280 days

Estimated blood alcohol content is a mass balance question: how much ethanol is in how much water.

BAC = (standard drinks x 14) / (r x body weight in kg x 10) - 0.015 x hours

Body mass index divides weight by height squared and returns a single number that is easy to compute and easy to misread. The arithmetic is trivial. The difficulty is in knowing what the number cannot say about muscle, fat distribution, age and population.

BMI = weight in kilograms / (height in metres)^2

Basal metabolic rate is the energy your body spends staying alive while you do nothing at all. The number is easy to compute and easy to misuse, because it is deliberately lower than anything you would ever eat.

BMR = 10w + 6.25h - 5a + s

Body fat percentage splits the body into fat and everything else, and estimates the fat share from a few tape measurements. The US Navy circumference method is the most widely used of these estimates. Its arithmetic is short; its limits are worth understanding before you trust the number.

BF% = 495 / (1.0324 - 0.19077 x log(waist - neck) + 0.15456 x log(height)) - 450 for men

Body surface area scales drug doses and burn estimates to the person rather than to a flat number. Height and weight are all it takes, and the Mosteller formula is the clinical standard.

BSA = sqrt( height in cm x weight in kg / 3600 )

Caffeine half life tells you how long a dose stays active. The amount halves every five hours or so, which is why an afternoon coffee can still be in your system at bedtime.

remaining = dose x 0.5 ^ (hours / half life)

Every activity has a MET value, and one MET is what you burn sitting still. Multiply that by your weight and the minutes, and you have the energy cost of the session.

calories per minute = MET x 3.5 x weight in kg / 200

Gestational age is counted from the last period, so it is a day count divided into weeks and days.

weeks = floor(days since last period / 7)

Ideal body weight is a formula, not a fact about health. It was written in 1974 to help clinicians dose drugs, and every version of it returns a slightly different number.

IBW = base + increment x (height in inches - 60)

Lean body mass splits a scale reading into the part that is fat and the part that is not, which is often the more useful half.

LBM = 0.407W + 0.267H - 19.2 (men)

Macros are just a calorie budget divided three ways, and the only arithmetic that matters is the four-four-nine rule.

grams = calories x share / caloriesPerGram

A one rep max is the heaviest load you can lift once. Testing it directly is risky, so most lifters estimate it from a lighter set — and the whole topic is really about how much you should trust that estimate.

Epley: 1RM = w x (1 + reps/30) | Brzycki: 1RM = w x 36 / (37 - reps)

Ovulation cannot be felt reliably, so it is estimated from the cycle. The standard method counts backwards from the next expected period, because the interval after ovulation is far more consistent than the interval before it.

Ovulation day = cycle length - luteal phase length

Protein needs are quoted as a rate, not a fixed number: grams per kilogram of bodyweight per day. That single multiplication hides most of the real questions, which are about which rate applies to you and how the daily total is best spread.

Protein (g) = bodyweight (kg) x grams per kg

Pace is time divided by distance, and getting the division the right way round is most of the skill.

pace = total minutes / distance

Target heart rate turns an intensity you can name into a pulse you can check. The arithmetic is short; the interesting part is that almost every input is an estimate.

Target HR = Resting HR + (Max HR - Resting HR) x Intensity

Total daily energy expenditure is the number of calories you burn in a day, made up of your resting metabolic rate plus everything else you do. Estimating it starts with the Mifflin-St Jeor equation for the resting rate, then scales that figure by an activity factor.

TDEE = BMR x activity factor, BMR = 10w + 6.25h - 5a + s

The clock decides how you feel in the morning as much as the hours do. Counting backwards in 90 minute cycles from your alarm finds the bedtimes that land between cycles instead of inside one.

bedtime = wake time - (15 + 90 x cycles)

VO2 max measures the ceiling on your aerobic engine. You can estimate it from a twelve minute run, or from the ratio between your maximum and resting heart rates.

VO2max = (distance - 504.9) / 44.73

Waist-to-hip ratio divides one circumference by another and returns a single unit-free number. The arithmetic takes seconds; the judgement lies in measuring at the right landmarks and in reading the result as a screening signal rather than a diagnosis.

WHR = waist / hip

Water intake is estimated from body weight: the common guideline is 35 millilitres for every kilogram you weigh, which is 2450 mL at 70 kg. The number is a starting point rather than a rule, because food, other drinks, heat and exercise all shift it.

mL = bodyweight (kg) x mL per kg

A1C is a percentage; the glucose meter gives milligrams per decilitre. The estimated average glucose formula translates one into the other so the two numbers can be compared directly.

eAG = 28.7 x A1C - 46.7

Your phone counts steps, not calories. The bridge between the two is distance: steps times stride length gives how far you walked, and distance times weight times an activity factor gives the energy it cost.

calories = steps x stride x weight x factor

A reorder point is the inventory level at which you place the next order. Set it too low and you stock out before the delivery lands; set it too high and you tie up cash in shelves.

reorder point = daily usage x lead time + safety stock

Amazon profit is the selling price minus everything Amazon and your supply chain take before the money reaches you. The referral fee alone is usually eight to fifteen percent, and FBA fees, inbound freight and pay-per-click spend stack on top.

Net per unit = Price - COGS - Referral - Fulfilment - Ad spend - Other

Burn rate is the speed at which a business spends its cash, and runway is how long that cash lasts at that speed. Both come from one subtraction and one division — and almost every argument about them is really an argument about which numbers belong in the subtraction.

Net Burn = Monthly Expenses - Monthly Revenue; Runway = Cash Balance / Net Burn

Cash flow is net income restated as cash, which means adding back non-cash charges and adjusting for working capital.

operating cash flow = net income + depreciation - increase in receivables + increase in payables

Churn rate is customers lost divided by customers at the start of the period. The division is trivial; the trap is everything that follows, because a monthly rate compounds into a yearly one that is far lower than twelve times its size.

Churn Rate = Customers Lost / Customers at the Start

Contribution margin is what each sale leaves after its own variable costs, and it is the number that decides how many sales you need to break even.

contribution margin = (price - variable cost per unit) / price

Conversion rate is conversions divided by visitors. The arithmetic is one division; the discipline is deciding what counts as a visitor, what counts as a conversion, and whether both were counted over the same stretch of time.

Conversion rate = conversions / visitors

Customer acquisition cost divides everything you spent to win demand by the number of customers who actually paid. The arithmetic is trivial; the judgement sits in deciding which costs belong in the numerator and which signups belong in the denominator.

CAC = (Marketing spend + Sales spend) / New paying customers

Customer lifetime value estimates how much contribution one customer leaves behind over their whole relationship, not just in the first month. With 50 of monthly revenue, a 70 per cent gross margin and 4 per cent monthly churn, that comes to 875.0.

LTV = ARPU x gross margin / churn

Days sales outstanding is the average number of days a business waits to be paid after making a sale. It comes from a single ratio and a single multiplication, yet it decides how much cash is trapped inside receivables — and it is routinely computed with the wrong denominator.

DSO = (Accounts Receivable / Revenue) x Days

Straight-line depreciation spreads an asset's cost evenly across its useful life. It is the simplest and most common method, and it turns a large purchase into a predictable annual expense.

Annual = (Cost - Salvage) / Life

Dropshipping has no inventory cost, so the ad that wins the customer is usually the biggest line item. Profit lives or dies on the gap between what the customer pays and what the supplier, the platform, the payment processor and the ad network take.

Net per sale = Retail - Supplier cost - Shipping - Platform fee - Payment fee - Ad spend

Every eBay sale loses a slice to fees. The final value fee is the biggest, but payment processing and insertion fees stack on top, and together they can take more than a sixth of what the buyer pays.

Net payout = (Price + Shipping) - Final value fee - Payment processing - Insertion fee

EBITDA strips out financing costs, tax and the two big non-cash charges to show what the operating business actually earned.

EBITDA = net income + tax + interest + depreciation + amortisation

List price is not profit. Between the marketplace's commission, the payment processor, shipping and the ad that brought the customer in, the money a store actually keeps on an order is a fraction of what the buyer pays.

Net per order = Sale price - Product cost - Platform fee - Payment fee - Shipping - Advertising - Other

Employee turnover measures how fast people leave. Divide separations by the average headcount and you get the rate that HR benchmarks against the industry.

turnover rate = separations / average headcount

Gross margin is gross profit as a share of revenue, and the most common error in using it is confusing it with markup.

gross margin = (revenue - cost of goods sold) / revenue

Inventory turnover is cost of goods sold divided by average inventory, and it answers one question: how many times did the shelf empty during the period? Everything contentious about it comes from which inventory figure sits in the denominator.

Inventory turnover = COGS / average inventory

Market share answers a question revenue growth cannot: are you growing because you are winning, or because the whole market is rising and carrying you with it.

Market share = your revenue / total market revenue

Markup is the amount loaded on top of cost to reach a selling price, expressed as a percentage of that cost. Margin divides the very same profit by the price instead, which is why a 40% markup on a cost of 60 produces a price of 84 but a margin of only 28.5714%.

Selling price = Cost x (1 + markup), markup = (Price - Cost) / Cost

Monthly recurring revenue is the number a subscription business is actually valued on. It is not total revenue: it strips out one-off sales and spreads annual contracts evenly across twelve months.

MRR = monthly plans + annual plans / 12 + expansion - churn

The Net Promoter Score is a single number that summarises how likely customers are to recommend you. It is the percentage of promoters minus the percentage of detractors, and it ranges from minus one hundred to plus one hundred.

NPS = % promoters - % detractors

Payback period answers the simplest question in capital budgeting: how long until I get my money back? It ignores everything that happens afterwards, which is both its weakness and its appeal.

payback period = initial investment / annual cash inflow

Elasticity measures how much buyers change their behaviour when the price moves. It is the number that tells you whether a price rise earns more money or less.

E = (% change in quantity) / (% change in price)

Profit margin is the share of revenue left over after cost, computed as revenue minus cost divided by revenue. Everything contentious about the topic comes from a single choice: which number sits on the bottom of the fraction.

Profit margin = (Revenue - Cost) / Revenue x 100%

The quick ratio asks whether a business can settle its short-term bills without having to sell its stock, and it answers by excluding inventory from the numerator.

quick ratio = (cash + receivables) / current liabilities

Reselling profit is the spread between where you buy and where you sell, minus the marketplace's cut and the shipping you pay on both ends. Thin spreads need volume to matter.

Net/unit = Sell - Buy - (Sell x Fee%) - Ship to acquire - Ship to buyer - Other

Revenue growth is a gap divided by where the business started. The annualised figure is a different question with a different answer, and collapsing the two into one number is why most growth reports quietly overstate themselves.

Revenue Growth = (Ending Revenue - Starting Revenue) / Starting Revenue

Revenue per employee is the bluntest useful measure of how much output a business gets from its people. It is easy to compute and easy to misread, so it rewards care with the inputs.

Revenue per employee = annual revenue / number of employees

Safety stock is the buffer sized so demand does not run out during a lead time.

safety stock = z x demand standard deviation x sqrt(lead time)

A time card records the hours worked on each day of a week. Turning those hours into pay means adding them up, splitting the total into regular and overtime hours at the forty-hour threshold, and multiplying each group by the right rate.

Gross pay = regular hours x rate + overtime hours x rate x 1.5

Wholesale margin looks fat at the top, but inbound freight, storage, labour and marketplace fees all land on the unit. Spreading a shipping bill across a small order count can quietly gut the margin.

Profit per unit = Retail - Unit cost - Inbound/unit - Overhead - Fee

Working capital is what is left of the short-term balance sheet after the short-term bills. It answers a blunt question: if everything due within a year came at once, would you still be solvent?

working capital = current assets - current liabilities

A curve changes the scale, not the ranking. Three methods cover almost every real curve: proportional, square root, and simply adding the same points to everyone.

curved = score / highest x 100

A final grade is two numbers and a weighting, and the weighting is where every mistake happens.

final grade = current average x current weight + exam score x exam weight

A percentile rank tells you what fraction of a group sits below a value. For 12th out of 250 there are 238 below, and the two standard conventions give 0.9558232931726908 and 0.952 — answers that differ by less than one person, yet must be kept distinct from the percentage score.

Percentile rank = count below / (n - 1), or count below / n

A test score is the marks you earned divided by the marks available. The arithmetic is one division, but the answer is reported in three different currencies: the raw mark, the percentage, and the letter grade. Only the first two are yours to compute.

Percentage score = (correct / total) x 100

A weighted GPA on the school scale adds bonus points to difficult courses before averaging, so an AP A counts as 5.0 and an honours A as 4.5 rather than 4.0. The arithmetic is trivial; the difficulty is keeping this weighting distinct from the credit weighting used elsewhere, and knowing which figure a reader actually wants.

Weighted GPA = Sum of (grade points + bonus) / number of courses

Most courses do not count every assignment equally. A weighted grade applies each category's share of the total before adding the scores up.

course grade = score1 x weight1 + score2 x weight2 + ...

An attendance rate is the days a student actually attended divided by the days they were enrolled for. The division is trivial; the two decisions around it are not, namely which days belong in the denominator and what the resulting percentage is being measured against.

Attendance rate = Days attended / Days enrolled

Class rank turns a raw position into a percentile, and the percentile is the number that travel.

rank percentile = (class size - rank) / class size x 100

Cost per credit is the tuition divided by the credits it buys, and it is the cleanest single rate for comparing one programme against another. Tuition of 12000 spread over 30 credits gives 400 per credit, which scales to 1200 for a three-credit course and 48000 across a 120-credit degree — but only if the tuition line is the whole bill.

Cost per credit = tuition / credits

A cumulative GPA combines every term you have taken into one credit-weighted average. Because it is credit-weighted, a strong term in a heavy load moves it further than the same grades in a light one.

cumulative GPA = total quality points / total credits

Extra credit only helps if you add it to the denominator as well as the numerator. Do that and the new grade is a single division away.

new grade = (points earned + extra earned) / (points possible + extra possible)

Readability formulas turn two measurements into a score: how long your sentences are and how many syllables your words have. Both are easy to count and both predict how hard a text feels.

ease = 206.835 - 1.015 x (words / sentences) - 84.6 x (syllables / words)

A grade point average is a credit-weighted average of grade points, not an average of letter grades. Every difficulty with it comes down to two things: which grade point table applies to you, and whether the credits were actually used as weights.

GPA = Sum of (grade points x credits) / Sum of credits

Reading time is one division away. Count the words, divide by how many you read per minute, and the answer is how long the piece will take.

minutes = words / words per minute

Study time is not guesswork. The credit hour already encodes an expected workload, and multiplying it out turns a vague intention into a weekly number you can plan around.

hours = credits x hours per credit

The grade you need on the final is not a percentage to aim for but an unknown inside a weighted average. Solve it directly and a comfortable-looking target can turn out to be arithmetically impossible.

Needed on final = (Target - Current x (1 - Final weight)) / Final weight

Words per minute is the simplest rate there is: a count divided by a time.

words per minute = words / minutes

A GPA only means anything once the scale is known, so conversion is a division by that scale.

percentage = GPA / scale

A grade scale is the table that turns a percentage into a letter. The cut-offs differ between schools, so the first step is always to check which scale you are being marked against.

letter = the band the percentage falls into

Cost per mile is the number that tells you what driving actually costs, and it is almost always higher than the fuel receipt alone suggests.

cost per mile = total cost / miles driven

A discount is a percentage taken off the price before tax, so the sale price and the money saved are two separate numbers that the tag never prints. On an original price of 120 with 25 per cent off, the saving is 30.0, the sale price is 90.0, and with 8 per cent tax the bill is 97.2.

sale price = original x (1 - rate); saving = original x rate

A grocery budget is easiest to build from a rate rather than a total. Work out what one person's food costs for one week, multiply by household size, and every other figure follows by arithmetic.

Weekly budget = household size x spend per person per week

Pizzas are priced by diameter but you eat area, so the bigger size is almost always better value and the arithmetic shows by how much.

value = (pi x (diameter / 2) squared) / price

Pool volume is the number every other pool calculation depends on, because chemicals, heating and pump run times are all dosed per litre rather than per pool.

volume = length x width x average depth

A security deposit is a multiple of the monthly rent, hold interest in some jurisdictions, and comes back minus whatever the landlord lawfully deducts.

deposit = monthly rent x months required

A split bill divides one restaurant bill between several people, and nearly every disagreement about it comes from the order of operations rather than from the arithmetic. The tip is a percentage of the pre-tip bill, it has to be added before anything is divided, and the split only stops being trivial once the shares stop being equal.

tip = bill x rate; total = bill + tip; per person = total / people

A tip is just a percentage of a bill, yet most of the arguments around it come from two decisions rather than from arithmetic: what you take the percentage of, and at what point you split it.

Tip = bill x rate; total = bill x (1 + rate)

Age is the distance between a date of birth and a reference date, and it can be expressed in several ways: whole years, total months, weeks, days, or hours. The calculation is a subtraction once both dates have been converted to a day count.

Days alive = day number of reference date - day number of birth date

Air conditioners are rated in BTU per hour, which is a heat removal rate rather than a weight or a size. Getting the number right stops you buying a unit that cannot keep up, or one that is so oversized it never runs long enough to dehumidify.

BTU = area x 20 x (ceiling height / 8) x climate factor x kitchen factor

Counting bricks is an area problem. Work out how much wall there is, subtract the holes, and divide by the area one brick occupies once the mortar joint is included.

bricks = (wall area - openings) x 144 / ((brick length + joint) x (brick height + joint)) x (1 + waste)

Calorie burn from walking is a product of weight, duration and how hard you are walking, and the weight term is what most people leave out of the rough estimates they have seen.

calories = 0.0175 x MET x weight in kg x minutes

A commute is a repeating expense, which makes it one of the few household costs that can be forecast accurately. Distance, frequency and three unit costs are enough to turn it into an annual figure.

Annual cost = fuel + parking + tolls

Concrete is ordered by volume and bought by the bag, so every slab, footing and path begins with the same three-number multiplication: length times width times thickness.

V = L x W x T

Counting the days between two dates, or adding a number of days to a date, is a calendar arithmetic problem. Both operations reduce to converting each date to a day number, doing simple addition, and converting the result back to a calendar date.

Days between = day number of later date - day number of earlier date

A deck is covered one row at a time. Work out how many rows fit across the width, multiply by the length, and you have the linear feet to buy.

rows = ceil(12 x deck width / (board width + gap)), boards = ceil(rows x deck length x (1 + waste) / board length)

A single overtime multiplier prices a simple week. A week with daily double time, weekly overtime and a holiday needs each band priced separately before the total means anything.

weekly pay = straight hours x rate + OT hours x rate x 1.5 + double time hours x rate x 2

Download time is file size divided by connection speed, provided both are expressed in the same unit. The trap is that file sizes are quoted in bytes while connection speeds are quoted in bits.

time = size in megabits / speed in megabits per second

Drywall is sold in whole sheets, so the job is to add up every surface, take off the openings, divide by the sheet area and round up with a waste allowance.

sheets = ceil((wall area + ceiling area - openings) / sheet area x (1 + waste))

Electricity cost is power multiplied by time, converted into the unit the bill actually uses, then priced. Almost every wrong answer comes from confusing the two quantities — watts and kilowatt-hours — before any arithmetic begins.

Cost = watts x hours / 1000 x rate per kWh

Fence materials come from one division and a few additions. The post count is the length divided by the spacing plus one, and everything else follows from the number of sections.

posts = fence length / post spacing + 1, rounded up

Ordering flooring is an arithmetic problem with a rounding trap at the end, because you cannot buy a fraction of a box.

boxes = (room area x (1 + waste allowance)) / pack coverage

Fuel cost is two multiplications and one division: distance into litres, litres into money, and money into a rate you can reuse. The only real traps are the units and a consumption figure that runs backwards.

Fuel cost = (distance x consumption / 100) x price per litre

Gas mileage is a rate, and it arrives in two scales that run in opposite directions: litres per 100 kilometres, where lower is better, and miles per gallon, where higher is better. Converting between them is a reciprocal, not a single multiplication.

mpg = 100 / (L/100km) x 3.785411784 / 1.609344

Grass seed is sold by weight but spread by area, so the number of bags you need depends on two things: how big the lawn is and whether you are starting from scratch or thickening an existing sward.

pounds = (area / 1000) x seeding rate, bags = ceil(pounds / bag size)

Household water use is dominated by a small number of fixtures. Measure showers, toilets and laundry and you have captured most of what goes through the meter indoors, without needing to read the bill and guess.

Daily gallons = shower use + toilet use + laundry use

Framing a wall comes down to counting studs at a fixed spacing, adding the plates that tie them together, and buying a little extra for the cuts that go wrong.

n = ceil(L / s) + 1

Ordering too few boxes means a second trip to the shop mid-move, and ordering too many leaves you storing empty cardboard. The count scales with the number of bedrooms, which is the standard shorthand for household size.

boxes = bedrooms x 20, split 10 small, 6 medium, 3 large and 1 wardrobe

Working out mulch is a volume problem. Multiply the bed area by the depth you plan to spread, then convert that volume into the bags or yards the supplier actually sells.

cubic feet = length x width x (depth in inches / 12)

Overtime pay is not a single multiplication. It is a split: hours up to a threshold are paid at the base rate, hours beyond it are paid at the base rate times a multiplier, and the two bands are added.

Total pay = min(H, T) x R + max(0, H - T) x R x M

Paint coverage is quoted per litre per coat, so the coat count multiplies the area before you divide. The arithmetic is one multiplication and one division; the judgement lies in reading the coverage figure honestly and turning litres into tins you can actually buy.

Litres needed = (wall area x coats) / coverage per litre

Shingle quantity starts from the roof area, not the floor area. Steeper roofs cover more surface, so the pitch multiplier has to be applied before anything is ordered.

squares = roof length x roof width x pitch multiplier x (1 + waste) / 100

Shipping cost is rarely just weight times rate, and understanding the base fee and the zone multiplier is what stops a parcel quote coming as a surprise.

cost = (base fee + weight x rate per kg) x zone multiplier

Sizing a solar array is a division problem. Work out how much energy one panel produces where you live, then divide your daily consumption by that number.

panels = ceil(daily kWh / (panel kW x peak sun hours x (1 - losses)))

Stacked discounts multiply rather than add. A price of 200 cut by 20% and then by 10% falls to 160.0 and then to 144.0, an effective discount of 0.28, not the 0.30 that adding the two rates would suggest.

Final Price = Price x (1 - first) x (1 - second)

Take home pay is the number that actually reaches your bank account, and it is always smaller than the salary figure you were quoted for a reason worth understanding.

net pay = gross pay - taxes - other deductions

Tile count is the surface area divided by the area of one tile, with a wastage allowance added and the answer rounded up. The only real decisions are how much wastage to allow and which direction to round.

Tiles = (room area / tile area) x (1 + wastage)

Viewing distance is a trade-off between immersion and comfort. Sit too far away and the picture loses detail; sit too close and your eyes have to sweep to follow the action. The distance scales with the screen height, not the diagonal.

distance = screen height x viewing multiple, screen height = 9 x diagonal / sqrt(337)

Unit price is the price divided by the quantity, expressed per gram, per litre or per 100 g, and it is the only figure that compares two packs of different sizes fairly. The arithmetic is one division; the discipline is applying it before you trust a label.

Unit price = price / quantity

Wallpaper is bought by the roll, but the room is measured in square feet. Convert the walls into an area, subtract the openings, then divide by what one roll actually covers.

rolls = (2 x (length + width) x ceiling height - openings) / coverage per roll

A hashtag is a label that joins a post to a feed. The labels everyone uses are the ones nobody sees, which is the whole problem in one sentence.

a set of n tags split roughly 25% broad, 45% mid and 30% niche, longest and most specific first

Changing the case of a piece of text is not a single operation. There are at least eight named styles, each with its own rules, and picking the wrong one is as much a mistake as getting the letters wrong.

Title Case = capitalise each significant word, lower-case the small ones

A time zone conversion is two arithmetic steps around a shared reference. The day shift that comes out of it is the part people get wrong, not the arithmetic.

target local time = local time - source offset + target offset

Character limits are the hard constraints of writing online. Unlike a word count, which is a guide, a character limit truncates, rejects or splits your text the moment you cross it.

characters = length of the string, including spaces

A word count looks trivial until two tools disagree about the same paragraph. The number depends on what counts as a word, and every editor makes that decision slightly differently.

reading time (minutes) = words / 200

Base64 takes any sequence of bytes and rewrites it using only letters, digits and two punctuation marks. The point is not secrecy but safety: the output survives channels that would otherwise mangle raw binary.

3 bytes -> 4 Base64 characters (with = padding to a multiple of 4)

A random number generator turns a range into a draw. The interesting part is not the numbers themselves but the statistics of the set you get back.

value = floor(min + u x (max - min)), where u is a uniform draw in [0, 1)

Resizing is one multiplication applied to every coordinate in the frame. The only decisions are which ratio to scale by and what to do about the shape that is left over.

scale = min(box width / source width, box height / source height) to fit inside, max(...) to fill and crop

Reversing text is the simplest string operation there is, and also one of the easiest to get subtly wrong once the text leaves plain English and starts containing emoji or accented letters.

reverse(s) = the characters of s in the opposite order

A rewrite is not a translation of meaning but of register. The facts stay where they are; what changes is how much space they take and who they sound like they are for.

rewrite = apply the strategy's substitution table in order, then delete its filler list

Scaling a recipe is one division followed by a great many multiplications. The division is trivial; knowing which ingredients should not be multiplied at all is the actual skill.

Scale factor = desired servings / original servings, then new amount = original amount x factor

A prompt is a specification rather than a question. The model cannot ask what you meant, so anything left out of it is something it will decide for you.

prompt = role + task + context + tone + format + constraints + a final-answer instruction

A subject line has one job and about forty characters to do it in. Everything else in the email is invisible until that job is done.

subject = one concrete noun + one reason to open, kept under 41 characters

An acre and a hectare both describe land, and both are casually called about the size of a football pitch, which is exactly why they get confused. The conversion itself is a single exact multiplication: one acre is 0.40468564224 hectare, so five acres is 2.0234282112 hectares.

ha = acres x 0.40468564224

Angle is measured in several units that all describe the same rotation. A converter moves between them through a single base unit, the degree, so every pair converts exactly.

Result = value x from-factor / to-factor

Two things are happening in the temperature conversion, and keeping them apart is the whole skill: one scales the size of a degree, the other moves where zero sits.

F = C x 9/5 + 32

Converting Celsius to kelvin looks almost too easy beside the Fahrenheit rule, and the reason is that only the zero point moves while the size of the degree stays exactly the same.

K = C + 273.15

A colour can be written in several ways, and each notation is convenient for a different job. Hex is compact for stylesheets, RGB maps directly to how a screen mixes light, and HSL is the one that lets you reason about a colour by eye.

Hex RRGGBB = (R x 65536) + (G x 256) + B

Recipes use spoons and cups in some countries and millilitres in others. A converter routes every measure through the millilitre so a recipe from anywhere becomes usable in your kitchen.

Result = value x from-factor / to-factor

Recipes measure dry ingredients by weight or by volume depending on the country. A converter ties ounces and pounds to grams and kilograms, and shows what a cup of flour or sugar actually weighs.

Result = value x from-factor / to-factor

A currency conversion is a single multiplication once you know the rate, but finding the right rate is the hard part. The rate depends on direction, on the day, and on whether you are looking at the mid-market figure or the one your bank actually gives you.

Amount in target = amount x (rate of target per USD / rate of source per USD)

Data sizes use two conventions that differ by about two and a half percent per step. A converter that offers both decimal and binary units shows exactly where the missing space on a drive goes.

Result = value x from-factor / to-factor

Energy, work and heat are the same physical quantity measured in many different units. A converter routes every value through the joule, the SI unit, so any pair converts consistently.

Result = value x from-factor / to-factor

Feet and metres are tied together by a fixed, exact number, so the whole task is one multiplication. Every difficulty that remains comes from how the original measurement was written down.

m = ft x 0.3048

Frequency counts cycles per second, but the same rate can be written as hertz, revolutions per minute or radians per second. A converter routes every value through hertz so the factors stay exact.

Result = value x from-factor / to-factor

Fuel economy units are not all the same kind of number. Most measure distance per unit of fuel, but litres per 100 kilometres measures fuel per unit of distance, so the relationship is inverted.

Result = value x from-factor / to-factor

One decision comes before the arithmetic — which gallon you have — and once that is settled the whole conversion is a single multiplication by an exact factor.

L = gal x 3.785411784

Inches to centimetres is the rare conversion where the factor is exact by definition rather than measured, so the whole skill reduces to applying it in the right direction and reporting no more precision than the measurement earned.

cm = in x 2.54

Kilograms to pounds is a single multiplication by 2.2046226218487757. The only real decision is whether you use that exact factor or the familiar 2.2 shortcut, which quietly runs 0.3235835 lb high on 70 kg — about 147 g of avoidable error.

lb = kg x 2.2046226218487757

Miles to kilometers is the one everyday conversion where the factor is exact rather than rounded for convenience, which means there is no excuse for a wrong digit and every real error comes from applying the right number in the wrong direction.

km = mi x 1.609344

Miles per hour to kilometres per hour is the simplest kind of rate conversion, because both rates are per hour and the hour cancels, leaving a single exact factor of 1.609344. Almost every mistake in the topic comes from rounding that factor or from changing the time unit when the question did not ask for it.

kph = mph x 1.609344

A number base is just a choice of how many digits to count to before starting a new column. Converting between bases means regrouping the same quantity into different columns, and the arithmetic is the same place-value arithmetic you already use in decimal.

value = sum of (digit x base^position)

Ounces to grams is a single multiplication by an exact factor, and almost every mistake people make comes from confusing the mass ounce with the fluid ounce that shares its name.

g = oz x 28.349523125

Pounds to kilograms is one multiplication. The only real decision is whether you use the exact factor, or a mental shortcut that quietly overshoots by an amount that grows with every pound you add.

kg = lb x 0.45359237

Power is the rate of doing work, and it is measured in watts, horsepower and BTU per hour depending on the field. A converter routes every value through the watt so the factors stay exact.

Result = value x from-factor / to-factor

Pressure is force per unit area, and it is written in pascals, bar, PSI or millimetres of mercury depending on the field. A converter routes every value through the pascal so the factors stay exact.

Result = value x from-factor / to-factor

Converting square feet to square metres is a single multiplication, but it is the one conversion where the most common mistake, using the linear factor once instead of squaring it, produces an answer that is wrong by a factor of 3.28.

m2 = sq ft x 0.09290304

Temperature is the one common conversion that is not a simple multiplication. Each scale has its own zero point as well as its own degree size, so every conversion is an affine transformation rather than a ratio.

°F = °C x 9/5 + 32, K = °C + 273.15, °R = (°C + 273.15) x 9/5

Torque is a turning force, measured in newton-metres, foot-pounds or kilogram-metres depending on the workshop. A converter routes every value through the newton-metre so the factors stay exact.

Result = value x from-factor / to-factor