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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.

AvionicsIntermediate

Effective Isotropic Radiated Power

EIRP_{dBW} = P_t + G_t - L_c

Apparent power a transmitter would need if it radiated equally in all directions; combines transmit power, antenna gain and cable loss.

AvionicsAdvanced

Radar Range Equation

P_r = \dfrac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4}

Received echo power for a monostatic radar, combining transmit power, antenna gain, target size and range.

AvionicsAdvanced

GPS Pseudorange

\rho = c\,(t_r - t_t) + c\,\delta t

Measured distance to a GPS satellite from signal travel time, biased by the receiver clock error — hence “pseudo” range.

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

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

Average Revenue Per User

ARPU = \dfrac{TR}{N}

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

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 & 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 & AcousticsBasic

Perceived Loudness Doubling

\Delta L = 10\,\text{dB}

A 10 dB increase is heard as roughly twice as loud.

Sports & Motion AnalyticsIntermediate

Elo Expected Score

E = \frac{1}{1 + 10^{(R_b - R_a)/400}}

Probability player A beats player B given their ratings.

Sports & Motion AnalyticsBasic

Baseball Batting Average

BA = \frac{H}{AB}

Hits divided by at-bats — baseball's classic hitting measure.

Sports & Motion AnalyticsIntermediate

Power-to-Weight Ratio

PWR = \frac{P}{m}

Cyclist's sustainable power divided by body mass — key to climbing.

Sports & Motion AnalyticsBasic

Reaction Distance

d = v \times t_r

How far you travel during reaction time before braking — vital in motorsport and driving.

Cryptography & SecurityAdvanced

RSA Decryption

m = c^d \bmod n

Recovers the message using the private exponent d.

Cryptography & SecurityAdvanced

Diffie–Hellman Shared Secret

s = B^a \bmod p

Two parties derive the same secret over a public channel.

Cryptography & SecurityIntermediate

Brute-Force Time

t = \frac{K}{r}

Time to try every key at a given guess rate.

Photography & ImagingBasic

F-Number

N = \frac{f}{D}

Aperture f-number is focal length divided by aperture diameter.

Photography & ImagingBasic

Equivalent Focal Length

f_{eq} = f \times C

Focal length on a crop sensor expressed in full-frame terms.

Photography & ImagingBasic

Print Resolution (PPI)

PPI = \frac{W_{px}}{W_{in}}

Pixels per inch when a photo is printed at a given width.

Automotive & EnginesIntermediate

Power from Torque & RPM

P = \frac{\tau \times N}{9549}

Engine power in kW from torque and rotational speed.

Automotive & EnginesBasic

Horsepower from Torque (Imperial)

HP = \frac{\tau \times N}{5252}

Horsepower from torque (lb-ft) and engine rpm.

Automotive & EnginesIntermediate

Compression Ratio

CR = \frac{V_d + V_c}{V_c}

Ratio of cylinder volume at bottom vs top of the stroke.

Automotive & EnginesBasic

Engine Displacement

V = \frac{\pi}{4} b^2 s\, n

Total swept volume of all cylinders.

Automotive & EnginesBasic

Gear Ratio

GR = \frac{N_{out}}{N_{in}}

Ratio of driven to driving gear teeth.

Automotive & EnginesIntermediate

Vehicle Speed from RPM

v = \frac{\pi d N}{GR \cdot G_f}

Road speed from engine rpm, gearing and tyre diameter.

Automotive & EnginesBasic

Fuel Economy (L/100km)

E = \frac{100\, F}{d}

Litres of fuel used per 100 km travelled.

Automotive & EnginesBasic

Miles Per Gallon

MPG = \frac{d}{F}

Distance in miles per gallon of fuel.

Automotive & EnginesIntermediate

Braking Distance

d = \frac{v^2}{2 \mu g}

Distance to stop from speed v with friction μ.

Automotive & EnginesIntermediate

Power-to-Weight Ratio

PWR = \frac{P}{m}

Engine power divided by vehicle mass — the key to acceleration.

Automotive & EnginesAdvanced

Aerodynamic Drag

F_d = \tfrac{1}{2}\rho v^2 C_d A

Air resistance a car must overcome, rising with speed squared.

Automotive & EnginesBasic

Kinetic Energy of Vehicle

KE = \tfrac{1}{2} m v^2

Energy that brakes must dissipate to stop a moving car.

Automotive & EnginesBasic

Torque from Force

\tau = F \times r

Turning effect of a force applied at radius r.

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.

Marine & NauticalBasic

Estimated Time of Arrival

t = \frac{d}{v}

Voyage time from distance and speed over ground.

Marine & NauticalAdvanced

Metacentric Height

GM = KM - KG

Stability measure; positive GM means the ship rights itself.

Renewable EnergyIntermediate

Battery Runtime

t = \frac{E \times DoD}{P}

How long a battery powers a load given depth of discharge.

Actuarial & InsuranceBasic

Loss Ratio

LR = \frac{C}{P}

Claims paid divided by premiums earned.

Actuarial & InsuranceAdvanced

Curtate Life Expectancy

e_x = \sum_{k=1}^{\infty} {}_k p_x

Expected remaining whole years of life from survival probabilities.

Actuarial & InsuranceIntermediate

Solvency Ratio

SR = \frac{A - L}{L}

Insurer's capital buffer relative to its liabilities.

Robotics & ControlAdvanced

PID Controller

u(t) = K_p e + K_i \!\int e\,dt + K_d \frac{de}{dt}

Control output from proportional, integral and derivative of the error.

Agriculture & SoilBasic

Nutrient from Fertiliser

n = F \times c

Actual nutrient delivered by a mass of fertiliser.

Agriculture & SoilBasic

Irrigation Water Requirement

V = A \times d

Water volume to apply a given depth over a field.

Agriculture & SoilBasic

Livestock Stocking Rate

SR = \frac{N}{A}

Number of animals per unit of grazing land.

Agriculture & SoilIntermediate

Feed Conversion Ratio

FCR = \frac{F}{G}

Feed eaten per unit of body-weight gain.

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

Shapley Value (2-player)

\phi_i = \tfrac{1}{2}(v(i) + [v(N) - v(j)])

Fair share of the total from a player's marginal contributions.

Real Estate & MortgagesBasic

Price per Square Metre

p = \frac{V}{A}

Property price divided by floor area.

Real Estate & MortgagesIntermediate

Cash-on-Cash Return

CoC = \frac{ACF}{I_{cash}} \times 100

Annual cash flow relative to actual cash invested.

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

Salt Percentage

S = \frac{m_{salt}}{m_{flour}} \times 100

Salt relative to flour — too little makes bland bread, too much slows the yeast.

Culinary & Baking MathBasic

Fresh to Dry Yeast

m_{dry} = \frac{m_{fresh}}{3}

Converts fresh (cake) yeast to active dry yeast by weight.

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.

HVAC & RefrigerationIntermediate

Coefficient of Performance (Cooling)

COP = \frac{Q_c}{W}

Cooling delivered per unit of work input — higher is more efficient.

HVAC & RefrigerationBasic

Relative Humidity

RH = \frac{p_v}{p_{sat}} \times 100

Vapour pressure as a percentage of the saturation pressure.

HVAC & RefrigerationBasic

Thermal Resistance (R-value)

R = \frac{L}{k}

Insulating power of a material layer of given thickness.

HVAC & RefrigerationAdvanced

Log Mean Temperature Difference

LMTD = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}

Effective average temperature gap driving heat across an exchanger.

HVAC & RefrigerationIntermediate

Seasonal Energy Efficiency (SEER)

SEER = \frac{\text{annual BTU}}{\text{annual Wh}}

Cooling output over a season divided by electrical energy used.

HVAC & RefrigerationIntermediate

Heating Degree Days

HDD = \sum (T_{base} - T_{avg})

Cumulative measure of how much and how long outdoor temperature sits below a base.

Logistics & Supply ChainIntermediate

Little's Law

L = \lambda W

Average items in a system equals arrival rate times average time in the system.

Logistics & Supply ChainBasic

Cycle Stock

CS = \frac{Q}{2}

Average working inventory between deliveries, half the order quantity.

Logistics & Supply ChainBasic

Capacity Utilisation

U = \frac{output}{capacity} \times 100

How fully productive capacity is being used.

Logistics & Supply ChainBasic

Cost Per Unit Shipped

C_u = \frac{C_{total}}{n}

Total shipment cost spread across the units delivered.

Logistics & Supply ChainBasic

On-Time Delivery Rate

OTD = \frac{n_{on\ time}}{n_{total}} \times 100

Percentage of orders delivered by the promised date.

Logistics & Supply ChainBasic

Dwell Time

D = t_{depart} - t_{arrive}

How long a shipment or container sits idle at a facility.

Logistics & Supply ChainBasic

Order Cycle Time

OCT = t_{delivered} - t_{ordered}

Elapsed time from a customer placing an order to receiving it.

Petroleum & DrillingBasic

API Gravity

API = \frac{141.5}{SG} - 131.5

Standard measure of how light or heavy a crude oil is relative to water.

Petroleum & DrillingAdvanced

Equivalent Circulating Density

ECD = MW + \frac{\Delta P}{0.052 \times TVD}

Effective mud density while circulating, including friction pressure.

Petroleum & DrillingAdvanced

Cuttings Transport Ratio

CTR = 1 - \frac{V_{slip}}{V_a}

Fraction of annular velocity effectively lifting cuttings out of the hole.

Surveying & GeomaticsIntermediate

Relative Precision

RP = \frac{E}{P}

Closure error expressed as a fraction of the traverse perimeter.

Surveying & GeomaticsBasic

Map Scale (RF)

RF = \frac{d_{map}}{d_{ground}}

Representative fraction linking map distance to real-world distance.

Telecom & NetworkingAdvanced

Shannon Channel Capacity

C = B \log_2(1 + SNR)

Maximum error-free data rate of a channel given bandwidth and signal-to-noise ratio.

Telecom & NetworkingBasic

Throughput

R = \frac{D}{t}

Actual data delivered per unit time.

Telecom & NetworkingBasic

Wavelength

\lambda = \frac{c}{f}

Physical length of one wave cycle for a given frequency.

Telecom & NetworkingAdvanced

Queue Utilisation

\rho = \frac{\lambda}{\mu}

Fraction of a server's capacity being used by arriving traffic.

Telecom & NetworkingIntermediate

Bit Error Rate

BER = \frac{E}{N}

Fraction of transmitted bits received in error.

Telecom & NetworkingIntermediate

Antenna Gain (dBi)

G_{dBi} = 10\log_{10} G

Directional gain of an antenna relative to an isotropic radiator, in decibels.

Digital Marketing & AnalyticsBasic

Engagement Rate

ER = \frac{\text{Likes} + \text{Comments} + \text{Shares}}{\text{Followers}} \times 100

How actively an audience interacts with content on social media.

Digital Marketing & AnalyticsBasic

Churn Rate

\text{Churn} = \frac{\text{Customers Lost}}{\text{Customers at Start}} \times 100

Percentage of customers who stop using a product in a given period.

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

Incidence Rate

IR = \frac{\text{New Cases}}{\text{Population at Risk}} \times k

Rate at which new cases of a disease appear in a population over a given period.

Epidemiology & Public HealthIntermediate

Basic Reproduction Number

R_0 = \beta \times D

Average number of secondary infections produced by one infected individual in a fully susceptible population.

Epidemiology & Public HealthIntermediate

Effective Reproduction Number

R_e = R_0 \times S

Actual average infections per case when part of the population is already immune.

Epidemiology & Public HealthAdvanced

SIR Model — Recovered

\frac{dR}{dt} = \gamma I

Rate at which infected individuals recover and gain immunity.

Epidemiology & Public HealthBasic

Case Fatality Rate

CFR = \frac{\text{Deaths}}{\text{Confirmed Cases}} \times 100

Proportion of diagnosed individuals who die from the disease.

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.

Epidemiology & Public HealthIntermediate

Number Needed to Treat

NNT = \frac{1}{ARR}

Number of patients who must receive a treatment for one additional patient to benefit.

Quantum ComputingBasic

Pauli-X Gate (Quantum NOT)

X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Flips |0⟩ to |1⟩ and vice versa — the quantum equivalent of a classical NOT gate.

Pharmacology & DosageIntermediate

Bioavailability

F = \frac{AUC_{oral}}{AUC_{IV}} \times 100

Fraction of an oral dose that reaches systemic circulation compared with an IV dose.

Pharmacology & DosageBasic

IV Drip Rate

\text{Rate} = \frac{V \times gtt}{T}

Drops per minute to deliver a given volume over a set time via IV drip.

Pharmacology & DosageBasic

Therapeutic Index

TI = \frac{TD_{50}}{ED_{50}}

Ratio of the dose causing toxicity to the effective dose — a safety measure for drugs.

Scientific Laws & PrinciplesBasic

Law of Definite Proportions

\frac{m_A}{m_B} = \text{constant}

A given chemical compound always contains its component elements in the same fixed ratio by mass, regardless of source.

Scientific Laws & PrinciplesAdvanced

Bragg's Law

n\lambda = 2d \sin\theta

X-rays reflected from parallel crystal planes interfere constructively when the path difference is a whole number of wavelengths.

Scientific Laws & PrinciplesAdvanced

Biot–Savart Law

dB = \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2}

Gives the magnetic field produced at a point by a small element of current-carrying wire.

Scientific Laws & PrinciplesAdvanced

Wiedemann–Franz Law

\frac{\kappa}{\sigma} = L \, T

For metals, the ratio of thermal conductivity to electrical conductivity is proportional to absolute temperature, with a nearly universal constant.

Math Tricks (57)

Quick Multiply by 5

Multiply by 10 and divide by 2 (or vice versa) for instant results.

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Divisibility Quick Checks

Quickly check if a number is divisible by 2 through 12 without dividing.

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Breaking Down Large Multiplications

Split large multiplications into simpler parts using the distributive property.

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Quick Multiply by 25

Since 25 = 100 ÷ 4, multiply by 100 and divide by 4 for instant results.

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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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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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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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Divide by 5 Instantly

Dividing by 5 is awkward; doubling then moving the decimal one place is far faster.

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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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Quick Multiply by 125

Since 125 = 1000 ÷ 8, multiplying by 125 is just adding three zeros and dividing by 8.

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Paravartya Yojayet (Flag Division)

A Vedic method for dividing by numbers close to a power of 10. Use the complement as a "flag" and perform synthetic-style division mentally.

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Dhvajanka (Flag Division)

Divide large numbers by multi-digit divisors using a flag (the non-leading digits of the divisor). Process one digit at a time, subtracting flag × quotient digit as you go.

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Ballpark Division by 9

Divide by 9 using repeated subtraction of digit sums. Each step: bring down the next digit, the quotient digit equals the running remainder.

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Multiply by 37

Since 37 × 3 = 111, multiply by 37 by first dividing the other number by 3, then multiplying by 111. Works perfectly when the number is divisible by 3.

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Divide by 25

Dividing by 25 = multiplying by 4 then dividing by 100. Much simpler for mental math since multiplying by 4 (double twice) is easy.

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Divide by 8 (Halve Three Times)

Divide by 8 by halving the number three times in a row. Since 8 = 2³, three successive halvings give the exact answer.

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Chained Percentage Discounts

Multiple successive discounts are NOT additive. Multiply the remaining fractions: 20% off then 10% off = 0.8 × 0.9 = 0.72, meaning 28% total discount (not 30%).

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Pascal Triangle Expansion

Expand (a + b)ⁿ using Pascal triangle coefficients. Row n gives the coefficients: e.g. row 3 = 1, 3, 3, 1 for (a+b)³.

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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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Build Quadratic from Roots

If you know the roots r₁ and r₂, the quadratic is x² − (r₁+r₂)x + r₁×r₂ = 0. Sum gives the middle term, product gives the constant.

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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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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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Multiply by 95

Use 95 = 100 − 5: multiply by 100, then subtract five times the number.

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Multiply by 105

Use 105 = 100 + 5: multiply by 100, then add five times the number.

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Multiply by 250

Use 250 = 1000 ÷ 4: multiply by 1000, then divide by 4.

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Multiply by 2.5

Use 2.5 = 10 ÷ 4: multiply by 10, then divide by 4.

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Multiply by 7.5

Use 7.5 = 15 ÷ 2 = 30 ÷ 4: multiply by 30, then divide by 4.

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Multiply by 0.25 (Quarter)

Multiplying by 0.25 is the same as dividing by 4 (halve twice).

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Square Numbers Near 200

For n near 200: n² = 200² + 400×d + d², where d = n − 200 (d can be negative).

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Divide by 4 (Halve Twice)

Divide by 4 by halving the number twice.

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Divide by 6

Divide by 6 by dividing by 2, then by 3.

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Divide by 12

Divide by 12 by dividing by 4 (halve twice), then by 3.

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Divide by 15

Use 15 = 30 ÷ 2: multiply by 2, then divide by 30 — or divide by 3 then by 5.

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Divide by 50

Use 50 = 100 ÷ 2: multiply by 2, then divide by 100.

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Divide by 125

Use 125 = 1000 ÷ 8: multiply by 8, then divide by 1000.

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Divide by 250

Use 250 = 1000 ÷ 4: multiply by 4, then divide by 1000.

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Break Apart Multiplication

Use the distributive law: split one factor into easy parts, multiply each, then add.

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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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Five Percent is Half of Ten Percent

Get 5% instantly: find 10% and halve it.

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Take a Fraction of a Number

To find a fraction of a number, divide by the denominator, then multiply by the numerator.

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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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Arithmetic Mean ≥ Geometric Mean

For two positive numbers, (a + b)/2 ≥ √(ab). Equality only when a = b.

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Remainder Theorem

The remainder when polynomial P(x) is divided by (x − a) is simply P(a).

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Pounds to Kilograms

Divide pounds by 2.2 to get kilograms.

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Centimetres to Inches

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

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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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Everyday Percentage Change

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

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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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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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Decimal to Binary

Repeatedly divide by 2 and record remainders bottom-up.

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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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Running Pace per Kilometre

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

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