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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.

Math Tricks (8)

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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