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A Mode C/S transponder reports pressure altitude referenced to 1013.25 hPa; dividing by 100 gives the flight level, independent of the local altimeter setting.

Revenue & SalesBasic

Total Revenue

TR = P \times Q

Total revenue is the selling price multiplied by the number of units sold.

Revenue & SalesBasic

Net Revenue

NR = GR - R - D

Net revenue is gross revenue less returns, refunds and discounts — the amount actually recognised.

Revenue & SalesBasic

Average Revenue

AR = \dfrac{TR}{Q}

Average revenue is total revenue divided by units sold — equal to price per unit in a single-price market.

Revenue & SalesIntermediate

Marginal Revenue

MR = \dfrac{\Delta TR}{\Delta Q}

Marginal revenue is the extra revenue earned from selling one additional unit.

Revenue & SalesBasic

Revenue Growth Rate

g = \dfrac{R_1 - R_0}{R_0} \times 100\%

The percentage change in revenue from one period to the next.

Revenue & SalesIntermediate

Monthly Recurring Revenue

MRR = N \times ARPU

Monthly recurring revenue is the number of active subscribers times average revenue per user per month.

Revenue & SalesBasic

Annual Recurring Revenue

ARR = MRR \times 12

Annual recurring revenue annualises the monthly recurring revenue of a subscription business.

Revenue & SalesBasic

Average Revenue Per User

ARPU = \dfrac{TR}{N}

Average revenue per user divides total revenue over a period by the number of users.

Revenue & SalesAdvanced

Customer Lifetime Value

CLV = \dfrac{ARPU \times m}{c}

Lifetime value estimates the gross profit a customer generates over their lifetime, using margin and churn.

Revenue & SalesIntermediate

Churn Rate

c = \dfrac{C_{lost}}{C_{start}} \times 100\%

Churn rate is the share of customers lost during a period relative to those at the start.

Revenue & SalesAdvanced

Net Revenue Retention

NRR = \dfrac{M_0 + E - C - D}{M_0} \times 100\%

Net revenue retention measures recurring revenue kept from existing customers, including expansion, contraction and churn.

Revenue & SalesIntermediate

Break-Even Revenue

R_{be} = \dfrac{FC}{CMR}

Break-even revenue is fixed costs divided by the contribution-margin ratio — the sales needed to cover all costs.

Music & AcousticsIntermediate

Sound Intensity Level (dB)

L = 10 \log_{10}\!\left(\frac{I}{I_0}\right)

Sound intensity expressed in decibels relative to the hearing threshold.

Music & AcousticsAdvanced

Combining Two Sound Levels

L = 10 \log_{10}(10^{L_1/10} + 10^{L_2/10})

Total level when two sound sources play together.

Music & AcousticsAdvanced

Sabine Reverberation Time

RT_{60} = 0.161\frac{V}{A}

Time for sound to decay 60 dB in a room.

Sports & Motion AnalyticsBasic

Hang Time

t = \frac{2v\sin\theta}{g}

Total time a projectile spends in the air on level ground.

Sports & Motion AnalyticsIntermediate

Expected Goals (xG)

xG = \sum_{i} p_i

Sum of scoring probabilities of every chance a team creates.

Cryptography & SecurityIntermediate

Brute-Force Time

t = \frac{K}{r}

Time to try every key at a given guess rate.

Photography & ImagingIntermediate

Exposure Value (EV)

EV = \log_2\!\left(\frac{N^2}{t}\right)

Combines aperture and shutter speed into one exposure figure.

Photography & ImagingAdvanced

Hyperfocal Distance

H = \frac{f^2}{N c} + f

Focus distance that keeps everything from H/2 to infinity sharp.

Automotive & EnginesIntermediate

EV Range

R = \frac{E}{c}

Electric-vehicle range from battery capacity and consumption.

Automotive & EnginesBasic

EV Charging Time

t = \frac{E}{P}

Hours to charge a battery at a given power.

Automotive & EnginesBasic

Tyre Rolling Circumference

C = \pi d

Distance covered in one wheel revolution.

Renewable EnergyAdvanced

Levelised Cost of Energy

LCOE = \frac{C + \sum \frac{M_t}{(1+r)^t}}{\sum \frac{E_t}{(1+r)^t}}

Lifetime cost per unit of energy produced.

Actuarial & InsuranceIntermediate

Pure Premium

PP = f \times S

Expected claim cost per policy: frequency times severity.

Robotics & ControlBasic

Stepper Motor Step Angle

\theta = \frac{360}{S}

Angle turned per step from steps per revolution.

Agriculture & SoilAdvanced

Crop Water Use (ET)

ET_c = K_c \times ET_0

Crop water demand from a crop coefficient and reference ET.

Game Theory & DecisionBasic

Expected Value

EV = \sum_i p_i x_i

Probability-weighted average outcome of a decision.

Game Theory & DecisionAdvanced

Bayes' Theorem

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Update a belief's probability given new evidence.

Game Theory & DecisionAdvanced

Value of Perfect Information

VPI = EV_{info} - EV_{no\,info}

How much better decisions become with full information.

Game Theory & DecisionAdvanced

St. Petersburg Expected Value

EV = \sum_{k=1}^{\infty} \tfrac{1}{2^k} 2^{k-1}

A game with infinite expected value yet little real worth.

Game Theory & DecisionBasic

Break-Even Probability

p = \frac{L}{L + W}

Win probability needed for a bet to break even.

Real Estate & MortgagesAdvanced

Break-Even Occupancy

BE = \frac{OE + D}{GPI}

Occupancy needed to cover expenses and debt.

Culinary & Baking MathBasic

Baker's Percentage

\% = \frac{m_{ingredient}}{m_{flour}} \times 100

Every ingredient expressed relative to the flour weight — the backbone of bread recipes.

Culinary & Baking MathBasic

Recipe Scaling

m_{new} = m_{old} \times \frac{n_{desired}}{n_{original}}

Scales every ingredient up or down for a different number of servings.

Culinary & Baking MathBasic

Beverage Cost Percentage

BC = \frac{C_{drink}}{P_{drink}} \times 100

Cost of a drink's ingredients relative to its selling price.

Culinary & Baking MathIntermediate

Boiling Point vs Altitude

T_b \approx 100 - \frac{h}{285}

Approximate drop in water's boiling point with elevation (h in metres).

HVAC & RefrigerationIntermediate

Latent Heat Load

Q = m L

Heat absorbed or released when moisture condenses or evaporates — the hidden load air conditioners fight.

Logistics & Supply ChainBasic

Reorder Point

ROP = d L

Stock level that triggers a new order, covering demand during the lead time.

Logistics & Supply ChainAdvanced

Safety Stock

SS = Z \sigma \sqrt{L}

Buffer inventory that absorbs demand variability during lead time.

Surveying & GeomaticsBasic

Horizontal Distance

H = L \cos\theta

Level distance from a slope distance and its inclination angle.

Surveying & GeomaticsBasic

Vertical Distance

V = L \sin\theta

Elevation change from a slope distance and angle.

Surveying & GeomaticsAdvanced

Stadia Distance

D = 100 \, s \cos^2\theta

Horizontal distance from stadia hair readings on a level rod.

Surveying & GeomaticsAdvanced

Earth Curvature Correction

C = 0.0675 \, K^2

Drop due to Earth's curvature over a sight distance (metres, K in km).

Surveying & GeomaticsAdvanced

Area by Simpson's Rule

A = \frac{h}{3}(y_0 + 4\Sigma_{odd} + 2\Sigma_{even} + y_n)

Area under an irregular boundary using parabolic strips.

Telecom & NetworkingIntermediate

Nyquist Data Rate

C = 2B \log_2 M

Maximum data rate for a noiseless channel with M signal levels.

Telecom & NetworkingBasic

Power Decibels

dB = 10 \log_{10}\frac{P_1}{P_2}

Ratio of two power levels expressed logarithmically.

Digital Marketing & AnalyticsIntermediate

Return on Ad Spend

ROAS = \frac{\text{Revenue from Ads}}{\text{Ad Spend}}

Revenue generated for every currency unit spent on advertising.

Digital Marketing & AnalyticsIntermediate

Customer Lifetime Value

LTV = ARPU \times \frac{1}{\text{Churn Rate}}

Predicted total revenue a single customer will generate over their relationship with a business.

Digital Marketing & AnalyticsBasic

Average Revenue Per User

ARPU = \frac{\text{Total Revenue}}{\text{Total Users}}

Mean revenue generated per active user in a period.

Epidemiology & Public HealthBasic

Prevalence

P = \frac{\text{Existing Cases}}{\text{Total Population}} \times k

Proportion of a population that has a particular condition at a point in time.

Epidemiology & Public HealthIntermediate

Herd-Immunity Threshold

H = 1 - \frac{1}{R_0}

Minimum fraction of the population that must be immune to prevent sustained spread.

Epidemiology & Public HealthBasic

Attack Rate

AR = \frac{\text{Cases during outbreak}}{\text{Population exposed}} \times 100

Proportion of an exposed population that develops the illness during an outbreak.

Epidemiology & Public HealthIntermediate

Relative Risk

RR = \frac{\text{Risk in Exposed}}{\text{Risk in Unexposed}}

How many times more likely the exposed group is to develop the disease compared to the unexposed group.

Pharmacology & DosageIntermediate

Loading Dose

LD = \frac{C_{target} \times V_d}{F}

Initial higher dose to rapidly achieve the desired plasma concentration.

Scientific Laws & PrinciplesIntermediate

Le Chatelier's Principle

Q < K \Rightarrow \text{shift forward}, \quad Q > K \Rightarrow \text{shift reverse}

When a system at equilibrium is disturbed by a change in concentration, temperature or pressure, it shifts to partially counteract the change.

Math Tricks (56)

Doubling and Halving

When one factor is even, halve it and double the other to reach an easier multiplication.

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Quick Tip Calculation

Find 10% first, then build 15% or 20% from it in your head.

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Rule of 72 (Doubling Money)

Estimate how many years it takes an investment to double: divide 72 by the annual interest rate.

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Quick Celsius ↔ Fahrenheit

A fast approximation: to go C→F, double and add 30. For exact values use ×9/5 + 32.

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Miles ↔ Kilometres Fast

Multiply miles by 1.6 for km, or divide km by 1.6 for miles. Consecutive Fibonacci numbers are handy approximations.

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Compare Unit Prices

To find the better deal, divide price by quantity to get the price per unit and compare.

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Estimate a Square Root

Bracket the number between two perfect squares, then refine with a quick average step.

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Split a Bill Evenly

Round the total up to a friendly number, divide by the number of people, then adjust down slightly.

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Sum of First N Even Numbers = N(N+1)

Add up 2 + 4 + 6 + … up to the Nth even number instantly with a single multiplication.

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Cube Root of a Perfect Cube (In Your Head)

Find the cube root of a perfect cube between 1 and 1,000,000 mentally using the last digit and the leading group.

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Reverse & Subtract Always Gives a Multiple of 9

Reverse the digits of any two-digit number and subtract — the result is always 9 times the difference of the digits.

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Multiply by 1.5 (Add a Half)

Multiplying by 1.5 is simply adding half of the number to itself — handy for recipes and time-and-a-half pay.

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The 9 Times Table on Your Fingers

A foolproof way for anyone to get any 9 times table answer (9×1 to 9×10) using just their ten fingers.

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Scale a 3-4-5 Triangle for Right Angles

Builders use multiples of the 3-4-5 triangle to mark a perfect right angle — no protractor needed.

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Average of Evenly Spaced Numbers

The average of any evenly spaced (arithmetic) sequence is simply (first + last) ÷ 2. No need to add all terms.

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Logarithm Shortcuts

Use log properties to compute logs mentally: log(5) = log(10/2) = 1 − log(2). Memorize log(2) ≈ 0.301 and log(3) ≈ 0.477 to derive everything else.

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Stacking Multiple Discounts

Apply successive discounts by multiplying the "keep" fractions, not by adding the percentages. The result is always less than the sum of discounts.

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Rule of 114 (Tripling Time)

To find how many years it takes for money to TRIPLE at a given interest rate, divide 114 by the rate. Like the Rule of 72 but for tripling.

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Quick BMI Calculation

BMI = weight (kg) ÷ height (m)². Square the height first, then divide. A healthy BMI is 18.5–24.9.

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Estimate Trip Fuel Cost

Fuel cost = (distance ÷ fuel efficiency) × price per litre. Know your car does and estimate trips in seconds.

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Reverse Percentage (Find Original)

If a price increased by p% to become X, the original = X ÷ (1 + p/100). People often mistakenly subtract the percentage instead.

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Scale a Recipe

Scale any recipe by multiplying each ingredient by (desired servings ÷ original servings). Works for any ratio.

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Speed-Distance-Time Mental Math

Distance = Speed × Time. When time is in minutes, convert to hours by dividing by 60: e.g. 40 minutes = 40/60 = 2/3 hour.

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Square from the Previous Square

Build a square from the one before it: n² = (n−1)² + (2n − 1). Great for a running sequence.

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Estimate Division Quickly

Round the divisor and dividend to compatible numbers that divide evenly for a fast estimate.

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Solve a Linear Equation

Isolate x by undoing operations in reverse order: move constants first, then divide by the coefficient.

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Kilograms to Pounds

Multiply kilograms by 2.2 to get pounds (close enough for daily use).

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Pounds to Kilograms

Divide pounds by 2.2 to get kilograms.

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Inches to Centimetres

Multiply inches by 2.54 to get centimetres.

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Centimetres to Inches

Divide centimetres by 2.54 to get inches (or multiply by 0.39).

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Metres to Feet

Multiply metres by 3.28 to get feet.

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Litres to US Gallons

Divide litres by 3.785 to get US gallons (roughly divide by 3.8).

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Cups to Millilitres

One US cup ≈ 240 ml. Multiply cups by 240.

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Simple Interest

Interest = Principal × Rate × Time (rate as a decimal, time in years).

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Estimate Compound Growth

Each year multiply by (1 + rate). Over a few years, chain the multipliers.

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Add Sales Tax Fast

Find the tax with the 1% or 10% method, then add it to the price.

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Everyday Percentage Change

Change = (new − old) ÷ old × 100. Negative means a decrease.

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Retail Markup

Selling price = cost × (1 + markup%). A 40% markup means multiply cost by 1.4.

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Hourly Wage to Annual Salary

Quick rule: double the hourly rate and add “000” (assumes ~2000 work hours/year).

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Annual Salary to Monthly

Divide the annual salary by 12 to get gross monthly pay.

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Fuel Economy (MPG)

Miles per gallon = miles driven ÷ gallons used.

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Area of a Room

Area = length × width. Multiply the two sides for square footage/metres.

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Tiles Needed for a Floor

Divide the floor area by the area of one tile, then add ~10% for cuts and breakage.

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Paint Needed for a Wall

Litres ≈ wall area ÷ coverage per litre (about 10 m²/L for one coat).

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Average Speed for a Round Trip

Average speed over equal distances is the harmonic mean: 2ab ÷ (a + b), not the plain average.

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Simple Probability

Probability = favourable outcomes ÷ total outcomes.

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Odds to Probability

Odds of a:b convert to probability a ÷ (a + b).

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Binary to Decimal

Add the place values (1, 2, 4, 8, ...) wherever there is a 1.

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Decimal to Binary

Repeatedly divide by 2 and record remainders bottom-up.

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Read Roman Numerals

Add values left to right; when a smaller numeral precedes a larger one, subtract it.

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Elapsed Time Across the Hour

Count up to the next hour, then add the remaining whole hours and minutes.

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Find the Unit Rate

Divide the total by the quantity to get the cost (or amount) per single unit.

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Turn a Discount into a Fraction

Convert common discounts to easy fractions: 25% = 1/4 off, 20% = 1/5 off, 50% = half off.

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Quick 20% Tip

For a 20% tip: find 10% (move the decimal) and double it.

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Running Pace per Kilometre

Pace = total minutes ÷ distance in km, giving minutes per kilometre.

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Estimate Calories Burned Walking

Rough rule: calories ≈ your weight in kg × kilometres walked.

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