Calculator Guides
201 in-depth guides · 14 categories · comparison tables & worked examples
Finance
27 guidesEqual Payment vs Equal Principal: Which Saves More?
Equal payment (annuity) keeps monthly payments constant; equal principal costs less interest overall but starts higher. Real test on a $1M loan: $67k interest difference over 30 years. Formulas, comparison tables, rate sensitivity analysis and a decision tree inside.
Take-Home Pay: Social Insurance, Housing Fund & Income Tax
Gross ¥10,000/month takes home roughly ¥7,500–8,000 after social insurance (~10.5%), housing fund (5–12%) and income tax. Includes the full deduction formula, 2025 social insurance base caps, special additional deductions and year-end bonus tax strategy.
Fund SIP Returns: Annualized Yield, Fees & Real Examples
SIP returns use XIRR, not simple ROI: ¥1k/month for 3 years (¥36k in) growing to ¥42k is ~10.4% annualized, not 16.7%. Covers averaging math, fee drag (1.5%/yr costs ~30% of gains over 30 years), and when to stop or switch a losing SIP.
Where to Park Cash: Deposits, CDs & Money Market
Demand deposits ~0.2%, 1-year fixed ~1.5%, 3-year ~2.6%, large CDs ~2.9%, money market ~2%. ¥100k over 3 years: demand earns ¥600 vs ¥7,800 in a 3-year fixed. Covers laddering, deposit insurance limits and early-withdrawal traps.
Mortgage Payment: Formula, Rates & Money-Saving Tips
Monthly payment = principal × r × (1+r)ⁿ ÷ [(1+r)ⁿ − 1]. $1M at 3.0% for 30 years → $4,216/mo, $518k total interest. Covers both repayment methods, rate tables, prepayment strategy and 6 proven ways to cut interest.
Auto Loan Payment: Down Payment, Rates & the 0% Trap
Payment follows the same annuity formula as mortgages. $150k over 3 years at 4% → $4,429/mo. Dealer "0% interest" deals often hide financing fees or a higher car price. Includes rate tables, total cost comparisons and the fee checklist.
Compound Interest: Formula, Rule of 72 & Doubling Times
A = P(1+r/n)^(nt). ¥100k at 8% becomes ¥216k in 10 years, ¥466k in 20 — growth accelerates. Rule of 72: 72÷rate = years to double. Includes compounding-frequency tables and why starting 10 years earlier beats saving twice as much.
Inflation: Purchasing Power, CPI & Real Returns
At 3% inflation, ¥100 today buys only ¥74.40 worth in 10 years, ¥55.40 in 20. Real return = (1+nominal) ÷ (1+inflation) − 1. Includes purchasing-power tables, CPI composition, and which assets historically outpace inflation.
ROI: Return on Investment Formula & Annualized Returns
ROI = (gain − cost) ÷ cost × 100%. $10k → $13k = 30% ROI. But ROI ignores time — annualize it: (1+ROI)^(1/years) − 1. 30% over 3 years = 9.14%/year. Includes worked examples across stocks, property and business decisions.
How to Compare Loans: Total Interest, APR & Terms
Never compare monthly payments — compare APR and total repayable. A 5-year loan at 4.5% beats a 7-year at 4.0% on total cost despite the higher payment. Includes the full comparison framework: rate type, fees, prepayment terms and total interest.
Early Mortgage Payoff: Shorten Term or Lower Payment?
Prepaying $1M/30yr/3.0% by $200k in year 3 saves ~$180k interest (term-shortening mode). Best timing is the first third of the loan when payments are interest-heavy. Covers both prepayment modes, penalty clauses and the invest-vs-prepay decision.
Savings Goal: How Much Per Month to Reach Your Target
Monthly saving = goal ÷ [((1+r)ⁿ − 1) ÷ r]. ¥500k in 5 years at 3% → ¥7,734/mo. Covers the FV-of-annuity formula, goal tables by timeline, where to park goal money, and how to adjust when returns disappoint.
Annuity: Future & Present Value Formulas with Examples
FV of annuity = PMT × [(1+r)ⁿ − 1] ÷ r; PV = PMT × [1 − (1+r)⁻ⁿ] ÷ r. ¥3k/month at 5% for 20 years grows to ¥1.23M. Covers ordinary vs due annuities, retirement drawdown math and how much a pension is really worth.
Currency Conversion: Direct Quotes, Cross Rates & Costs
CNY/USD 7.20 means ¥7.20 buys $1 — converting $1,000 costs ¥7,200 at mid-market, but banks add 0.3-2% spreads plus fees. Covers direct vs indirect quotes, cross-rate calculation, and how to minimize exchange costs.
Dividend Yield: Payouts, Ex-Dividend & Dividend Tax
Yield = annual dividend ÷ price. ¥1 dividend on a ¥25 stock = 4%. Buy before the ex-date to receive the payout, but the price drops by the dividend amount. Covers yield traps, tax tiers by holding period, and dividend reinvestment math.
Bond Prices: Yield, Interest Rates & Pricing Formula
Price = Σ coupon ÷ (1+y)ᵗ + face ÷ (1+y)ⁿ. When market yields rise, existing bond prices fall — a 30-year bond drops ~20% when yields rise 1pp. Covers coupon vs yield, duration, premium/discount logic and the rate-sensitivity table.
Stock Returns: Price Change, Dividends & Total Return
Total return = (sale price − cost + dividends) ÷ cost. Buy at ¥20, sell at ¥26 with ¥1 dividend → 35%. Annualize for fair comparison, subtract fees and tax, and never confuse a 50% loss with needing only 50% to recover (you need 100%).
Credit Card Minimum Payment Trap: The Real Cost of Paying the Minimum
Paying only the minimum (usually 5–10% of balance) on a ¥20,000 balance at 18% APR takes 10+ years and ~¥10,000 in interest to clear. How minimums are calculated, the interest spiral mechanics, and escape strategies.
FIRE Number Explained: The 4% Rule and Your 25× Target
FIRE number = annual expenses × 25 (the 4% rule). Spend ¥150k/year → need ¥3.75M invested. Covers the Trinity Study basis, safe withdrawal rates, savings-rate timelines, and China-specific adjustments.
How Much House Can I Afford? Income-Based Affordability Guide
Price ceiling ≈ 4–6× annual household income; monthly payment ≤ 40% of take-home pay. Household earning ¥20k/month supports roughly ¥7,000–8,000 payment → ¥1.6–1.9M loan at 3.0%/30yr. Full worksheet with down payment and DTI logic.
50/30/20 Budget Rule: The Simplest Way to Split Your Paycheck
After-tax income split: 50% needs, 30% wants, 20% savings. On ¥10,000 take-home: ¥5,000 essentials, ¥3,000 lifestyle, ¥2,000 savings. Includes category sorting table, income-tier templates, and how to fix a broken ratio.
NPV vs IRR: Net Present Value and Internal Rate of Return Guide
NPV > 0 means a project beats your cost of capital; IRR is the rate that makes NPV zero. ¥100k in, ¥25k/year for 5 years at 8% discount → NPV ≈ −¥182, skip it. Formulas, Excel functions, and what to do when NPV and IRR disagree.
Debt-to-Income Ratio (DTI): What Lenders Check Before Approving You
DTI = total monthly debt payments ÷ gross monthly income. ≤36% is healthy, 36–43% is tight, >50% means near-certain mortgage denial. Includes tier table, five ways to lower DTI fast, and a 6-month pre-application checklist.
Options Pricing & Grid Trading: Strategy Parameters
Black-Scholes prices options from five inputs; Delta hedges direction, Theta bleeds daily. Grid spacing must beat fees (0.1% → ≥0.3%). Tables plus examples.
Best Mortgage Calculator: 6 Features That Matter
A good mortgage calculator needs LPR tiers, both repayment modes, prepayment simulation and combo loans. Check: ¥1M, 30Y, 3.0% equal payment = ¥4,216/month.
Best Take-Home Pay Calculator: 4 Accuracy Checks
A good salary calculator needs city insurance rates, 7 special deductions, cumulative withholding tax, and a breakdown. Check: ¥20k Beijing nets ≈¥14,660.
Best Finance Calculators: Interest to FIRE
Five essentials: compound, DCA, deposit, NPV/IRR, FIRE. Verify: ¥100k at 5% for 10 years = ¥162,889 — a wrong result means a wrong formula.
Real Estate
3 guidesHousing Provident Fund Loans: Rates, Limits & How to Apply
HPF rates are ~0.6 percentage points below commercial loans (2.6% vs 3.3% for first homes). Max loan is typically 0.6–1.2M depending on city and account balance. Covers rate tables, limit formulas, the commercial-to-HPF conversion process and common pitfalls.
Home Buying Costs: Down Payment, Taxes & Fees Checklist
Beyond the down payment: deed tax 1–3%, agency fees 1–2.7%, maintenance fund ~2%, plus appraisal and mortgage fees. A ¥3M first home typically needs ¥450k down + ¥80–120k in taxes and fees. Full cost tables and city policy differences inside.
Resale Home Taxes: VAT, Income Tax & Deed Tax Explained
Selling a resale home: VAT 5.3% if under 2 years (exempt after 2 years in most cities), income tax 1% of price or 20% of gain (exempt if owned 5+ years and only home), deed tax 1–3% paid by buyer. Includes the "满二/满五唯一" exemption rules and real cost tables.
Health
23 guidesBMI Standard Ranges: Formula, Tables & Limitations
BMI = weight(kg) ÷ height(m)². Chinese standard: 18.5–23.9 normal, 24–27.9 overweight, ≥28 obese. WHO uses 25/30 cutoffs. Includes full classification tables, examples at common heights, and why muscular people and the elderly should read BMI with caution.
Daily Calorie Needs: BMR, TDEE & Deficit Planning
Daily burn (TDEE) = BMR × activity factor (1.2–1.9). A 30-year-old office worker at 70kg burns ~2,000–2,200 kcal/day. Fat loss: eat 300–500 kcal below TDEE for ~0.5kg/week. Includes Mifflin-St Jeor formula, activity tables and plateau troubleshooting.
Ideal Weight: Standard Formulas & Healthy Ranges
Common formulas: Broca (height−105 for men, −110 for women), BMI-based range (18.5–23.9 × height²), and Devine. A 175cm man's ideal range is ~57–73kg. Why a range beats a single number, and how frame size and muscle mass shift your personal target.
Waist-to-Hip Ratio: Standards, Measurement & Health Risk
WHR = waist ÷ hip. Risk thresholds: men >0.90, women >0.85 (WHO). Apple shape (high WHR) carries 2–3× the cardiovascular risk of pear shape at the same BMI. Measurement technique, percentile tables and reduction strategies inside.
Body Fat Percentage: Formulas, Standards & Measurement
Healthy ranges: men 10-20%, women 18-28% (varies by age). Navy formula uses waist/neck/height (±3-4%); DEXA is the lab gold standard. Covers all measurement methods, age-adjusted tables, and why BMI misses skinny-fat.
Waist-to-Hip Ratio: How to Measure, Standards & Health Risk
WHR = waist ÷ hip. Risk lines: men >0.90, women >0.85 (WHO). Measure waist at the navel, hips at the widest point, standing relaxed. Apple vs pear shape, the WHtR alternative (waist ÷ height < 0.5), and how to improve.
BMR: Mifflin Formula, Calculation & How to Raise It
Mifflin-St Jeor: men BMR = 10W + 6.25H − 5A + 5; women − 161. 70kg/175cm/30yo man → 1,649 kcal. BMR is 60-75% of daily burn. Covers the formula, factor tables, and why muscle is the only lasting way to raise it.
TDEE: Daily Calorie Needs by Activity Level
TDEE = BMR × activity factor (1.2-1.9). 1,650 BMR × 1.55 (moderate) = 2,558 kcal. Most people overestimate their activity by one full level — desk job + 3 gym sessions = "light", not "moderate". Cut 300-500 for fat loss, add 200-300 for lean gain.
How Much Water a Day: Calculation & Drinking Schedule
Baseline: 30-35 ml per kg body weight — 70kg → 2.1-2.5 L from all sources (food provides ~20%). Exercise adds 500-750 ml per hour. Urine color is the best gauge: pale straw = hydrated. Covers the formula, schedules, and the 8-glasses myth.
Body Surface Area (BSA): Mosteller Formula & Medical Uses
Mosteller: BSA(m²) = √(height(cm) × weight(kg) ÷ 3600). 175cm/70kg → 1.84 m². BSA drives chemo dosing, burn assessment (rule of nines) and cardiac indexing. Includes all five formulas compared and worked medical examples.
Sleep Cycles: Best Bedtime Calculator & 90-Minute Rule
Sleep runs in ~90-minute cycles; waking mid-cycle causes grogginess. For a 7:00 wake time, fall asleep at 23:30 (5 cycles) or 22:00 (6). Add 10-20 min to fall asleep. Covers cycle math, sleep hygiene, and why 6h20m of in-time sleep beats 8 hours of badly-timed sleep.
Running Pace: Calculation, Conversion & Race Prediction
Pace = time ÷ distance. 5km in 27:30 → 5:30/km. Speed = 60 ÷ pace (5:30/km = 10.9 km/h). Race prediction via Riegel: T2 = T1 × (D2/D1)^1.06 — a 25:00 5K projects to ~51:50 10K. Includes pace zones and treadmill conversions.
Counting Calories: Food Energy, Tracking & Burn Rates
Calorie math: protein/carbs 4 kcal/g, fat 9 kcal/g, alcohol 7 kcal/g. A 500 kcal deficit = ~0.5kg fat loss/week (7,700 kcal ≈ 1kg). Covers label reading, the tracking errors that erase deficits, and burn-per-exercise tables.
Due Date Calculator: Naegele's Rule & Trimester Guide
Due date = first day of last period + 280 days (40 weeks). Naegele: LMP + 1 year − 3 months + 7 days. Only ~4-5% deliver on the exact date; 37-42 weeks is the normal window. Covers IVF dating, ultrasound adjustment and week-by-week milestones.
Heart Rate Zones: Max HR, Training Intensity & Fat Burn
Max HR ≈ 220 − age (±10-12). Five zones: Z2 (60-70%) builds aerobic base and burns the most fat per total effort; Z4-Z5 (80-100%) for performance. 40yo: max ~180, Z2 = 108-126 bpm. Includes the Karvonen reserve method and resting-HR health table.
Pregnancy Weight Gain: IOM Ranges by BMI
IOM targets: underweight BMI<18.5 → 12.5-18kg; normal 18.5-24.9 → 11.5-16kg; overweight 25-29.9 → 7-11.5kg; obese ≥30 → 5-9kg. Twins: 17-25kg. Distribution: ~1-2kg in trimester 1, then ~0.4kg/week. Where the weight goes, trimester tables and safe management.
Blood Pressure Chart: Normal, Elevated and High BP Ranges
Normal is below 120/80 mmHg; hypertension starts at 140/90 (clinic) or 135/85 (home). Systolic vs diastolic explained, age-stratified targets, home measurement protocol, and when to see a doctor.
Ovulation Calculator Guide: Fertile Window, Timing and Signs
Ovulation happens ~14 days BEFORE the next period (not mid-cycle). The fertile window spans the 5 days before ovulation plus ovulation day. Covers cycle-day math for 21–35 day cycles, body signs, and conception odds.
The True Cost of Smoking: What a Pack a Day Really Costs You
One pack a day at ¥25 = ¥9,125/year; over 20 years that's ¥182k in cash — or ¥338k+ if invested at 6%. Full math on direct costs, compound opportunity cost, health economics, and the payback timeline after quitting.
Marathon Pace Chart: Target Finish Times and Per-Km Splits
Sub-4 marathon = 5:41/km, 3:30 = 5:00/km, sub-3 = 4:16/km. Pace = target minutes ÷ 42.195. Full split tables for marathon and half, five training zones, and a first-marathon race-day plan.
Baby Care Numbers: Milk, Vaccines, Fever, Height
Daily milk ≈150ml×kg; 11 free vaccines on schedule; ibuprofen 5-10mg/kg; target height=(dad+mom±13)÷2 ±7.5cm.
Kidney & Body Composition: eGFR, FFMI, Blood Type, Fasting
eGFR <60 for 3 months = CKD 3; creatinine μmol/L÷88.4=mg/dL; FFMI caps ~25 naturally; O×O parents cannot make AB; fat-burning past 12h.
Best BMI Calculator: Which Standard Does It Use?
China cutoffs 24/28 vs WHO 25/30 flip the verdict at BMI 24.5. A good tool shows both standards, a healthy weight range, and waist/body-fat links.
Math
26 guidesPercentage: How to Calculate Shares, Change & Discounts
Three core questions: X% of Y = Y×X÷100; what % is A of B = A÷B×100; percent change = (new−old)÷old×100. Covers discounts, growth rates, percentage points vs percent, and the classic traps that fool most people.
Mean, Median & Mode: When to Use Each
Mean is skewed by outliers (average salary myth); median is the middle value and robust; mode is the most frequent. House prices and salaries should use median. Includes calculation steps, comparison table, and the "average trap" in real-world statistics.
Standard Deviation: Formula, Meaning & Use Cases
Standard deviation measures spread: ~68% of data falls within ±1σ, 95% within ±2σ (normal distribution). Sample version divides by n−1, population by n. Includes step-by-step calculation, CV for comparing volatility, and real uses from quality control to fund risk.
Growth Rate & CAGR: Formulas and Annualized Returns
CAGR = (end ÷ start)^(1/years) − 1. 100k → 200k in 5 years = 14.87% CAGR (not 20%). Covers YoY vs CAGR vs average growth, the arithmetic-mean trap, and how to compare investments with different horizons.
Scientific Calculator: Functions, Buttons & Order of Operations
Master the core keys: sin/cos/tan need the right angle mode (DEG vs RAD), x²/√/xʸ/10ˣ/ln/log, and parentheses beat memory. 2+3×4 = 14 (multiplication first). Covers angle modes, scientific notation, and the errors everyone makes.
Fractions: Add, Subtract, Multiply, Divide & Simplify
Same denominators: add numerators (1/4 + 2/4 = 3/4). Different: common denominator first (1/2 + 1/3 = 5/6). Multiply straight across; divide by flipping the second. Always simplify with GCD. Includes mixed numbers and decimal conversion.
Number Base Conversion: Binary, Octal, Decimal & Hex
Binary 1011 = 1×8+0×4+1×2+1 = 11 in decimal. Decimal→binary: divide by 2, read remainders upward (13 → 1101). Hex uses 0-9,A-F: 255 = FF. Octal groups bits by 3, hex by 4. Includes conversion tables and programmer tricks.
Permutations & Combinations: P and C Formulas Explained
Order matters → permutation: P(n,r) = n!/(n−r)!. Order irrelevant → combination: C(n,r) = n!/(r!(n−r)!). P(5,3) = 60, C(5,3) = 10. Lottery odds, committee counting, and the lock-vs-combination naming irony explained.
Random Numbers: True vs Pseudo, Uses & Generation
Computers generate pseudo-random numbers from seeds — deterministic but statistically random. True randomness needs physical entropy (atmospheric noise, hardware events). For games pseudo is fine; for cryptography only CSPRNG counts. Includes ranges, seeds, and fairness testing.
Prime Numbers: Definition, Tests & the Table to 100
A prime has exactly two factors: 1 and itself. 2 is the only even prime; 1 is NOT prime. Test n by trial division up to √n. The 25 primes under 100: 2, 3, 5, 7, 11, ..., 97. Includes the sieve method, prime factorization and why crypto runs on primes.
GCD & LCM: Euclidean Algorithm with Examples
GCD by Euclid: gcd(48,18) = gcd(18,12) = gcd(12,6) = 6. LCM = a×b ÷ gcd — lcm(12,18) = 36. Covers the algorithm, prime-factorization method, and real uses: fraction simplification, gear cycles, and scheduling.
Arithmetic & Geometric Sequences: Formulas & Sums
Arithmetic: aₙ = a₁ + (n−1)d, sum = n(a₁+aₙ)/2. Geometric: aₙ = a₁ × rⁿ⁻¹, sum = a₁(rⁿ−1)/(r−1). 1+2+...+100 = 5,050 (Gauss trick). Infinite geometric sum = a₁/(1−r) when |r|<1. Worked examples inside.
Solving Equations: Linear, Systems & the Quadratic Formula
Linear: isolate x by inverse operations — 3x + 5 = 20 → x = 5. Quadratic: x = (−b ± √(b²−4ac)) ÷ 2a. Systems: substitution or elimination. Discriminant tells you the answer count before solving. Step-by-step examples for all three types.
Square Roots: Calculation, Estimation & Simplifying
√a × √a = a. √144 = 12, √2 ≈ 1.414. Estimate non-perfects by bracketing: √50 sits between 7 and 8, closer to 7.07. Simplify radicals by factoring out squares: √72 = 6√2. Includes the digit-pair longhand method and key values table.
Cube Roots: Calculation, Estimation & Quick Values
∛a × ∛a × ∛a = a. ∛27 = 3, ∛8 = 2. Unlike square roots, negatives work: ∛−8 = −2. Estimate by bracketing perfect cubes: ∛100 is between 4 (64) and 5 (125) ≈ 4.64. Includes the perfect-cube table and volume-to-side applications.
Absolute Value: Rules, Equations & Inequalities
|x| = distance from zero: |5| = 5, |−5| = 5. Equations split in two: |x−3| = 7 → x = 10 or x = −4. Inequalities flip direction: |x| < 3 means −3 < x < 3; |x| > 3 means x > 3 or x < −3. Rules, graphs and error patterns inside.
Logarithms: Definition, Laws & Common Log Tables
log₁₀100 = 2 because 10² = 100 — a log asks "what exponent?". Three laws: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a. ln uses base e ≈ 2.718. Includes solving exponential equations and decibel/pH applications.
Exponent Rules: Powers, Negative & Fractional Exponents
Six laws: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a. 2¹⁰ = 1,024; 2⁻³ = 1/8; 8^(2/3) = 4. All rules with worked examples and the zero/negative traps.
Factorials: n! Definition, Properties & the Table to 20
n! = n × (n−1) × ... × 1. 5! = 120, 10! = 3,628,800. Special case: 0! = 1. Factorials count arrangements and feed permutations/combinations. Includes the growth table, trailing-zeros trick, and Stirling approximation for huge n.
Modular Arithmetic: Remainders, Rules & Applications
a mod n = remainder of a ÷ n: 17 mod 5 = 2. Rules: (a+b) mod n = ((a mod n)+(b mod n)) mod n — same for ×. Powers cycle: 3ⁿ mod 7 repeats 3,2,6,4,5,1. Clock math, week cycles, ISBN check digits and hashing all run on mod.
Quadratic Formula Explained: Solving ax²+bx+c=0 Step by Step
x = [−b ± √(b²−4ac)] ÷ 2a. The discriminant b²−4ac decides everything: >0 two real roots, =0 one repeated root, <0 two complex roots. Worked examples, common sign errors, and when to factor instead.
Statistical Inference: CI, Margin of Error, Bayes
95% CI=x̄±1.96σ/√n; ±3% margin needs n≈1067; at 1% prevalence a 99% accurate positive is truly sick only ~50%.
Descriptive Stats: Quartiles, Regression, Probability
Quartiles split ranked data in four; beyond 1.5×IQR flags outliers. Regression ŷ=a+bx; independent probabilities multiply; sci notation a×10ⁿ.
Linear Algebra Basics: Matrices, Vectors & Complex Numbers
Dot product tests angles; matrix products are rows×columns and not commutative; (a+bi)(c+di)=(ac−bd)+(ad+bc)i. Tables plus worked examples.
Coordinate Geometry: Slope, Distance & Ratios
Slope k=Δy/Δx, perpendicular when k₁k₂=−1; distance d=√(Δx²+Δy²); ratios via cross-products. Grade table plus ramp and map examples.
Best Online Scientific Calculator: 5 Tests
Five tests: precedence (2+3×4=14), DEG/RAD switch, full functions, keyboard input, history tape. Failing precedence disqualifies a tool outright.
Business
15 guidesPrice-Tax Split: Net Price, VAT & Gross Calculation
Net = gross ÷ (1 + tax rate); tax = gross − net. ¥113 with 13% VAT → net ¥100, tax ¥13. Covers the 13%/9%/6%/3%/1% rate table, why "gross × rate" is wrong, and invoicing rules for general vs small taxpayers.
Small-Scale Taxpayer VAT: 1% Rate, Filing & Thresholds
Small-scale taxpayers pay 3% collection rate, temporarily 1% through 2027; monthly sales ≤¥100k (quarterly ≤¥300k) are fully exempt. Covers the exemption rules, when to issue special invoices, the general-taxpayer decision, and common filing traps.
Corporate Income Tax: 25% Rate, Preferences & Filing
Standard CIT rate is 25%; small low-profit enterprises pay an effective 5% on the first ¥3M of profit; high-tech enterprises 15%. Covers taxable income adjustments, the small-enterprise thresholds, R&D super-deduction, and quarterly prepayment rules.
Invoice Amounts: Tax-Inclusive vs Tax-Exclusive Pricing
Tax-inclusive price = exclusive × (1 + rate); exclusive = inclusive ÷ (1 + rate). ¥113 with 13% VAT → ¥100 + ¥13 tax. Invoicing quotes totals inclusive; bookkeeping records them exclusive. Includes rate tiers (13/9/6%), small-scale 1%/3% rules and mixed-price traps.
Gross Margin: Formula, Benchmarks & vs Markup
Gross margin = (revenue − cost) ÷ revenue. Sell ¥100, cost ¥60 → 40% margin. Markup divides by COST (40 ÷ 60 = 66.7%) — margin and markup are never equal. Includes conversion table, industry benchmarks and pricing from target margin (price = cost ÷ (1 − margin)).
Depreciation: Straight-Line, Double-Declining & Sum-of-Years
Straight-line: (cost − salvage) ÷ years. ¥100k equipment, 5 years, 5% salvage → ¥19k/year. Double-declining: 2/years × book value (front-loaded). Sum-of-years-digits: weighted by remaining life. Includes tax lives (3/4/5/10/20 years) and monthly proration.
Break-Even Analysis: Formula, Margin of Safety & Pricing
Break-even units = fixed costs ÷ (price − variable cost). Fixed ¥300k, price ¥100, variable ¥60 → 7,500 units = ¥750k revenue. Contribution margin ratio 40% → BEV = fixed ÷ 0.4. Includes target-profit units, margin of safety and the multi-product version.
Markup Pricing: Rates, Formulas & Industry Standards
Markup = (price − cost) ÷ cost. Cost ¥60, markup 50% → price ¥90 (margin 33.3%). Keystone = 100% markup (50% margin). Includes the markup↔margin conversion table, industry rates (restaurants 200-300% on food) and cost-plus pricing limits.
Commission: Rates, Tiers & Sales Compensation Math
Commission = sales × rate; tiered plans accelerate: 3% to ¥100k, 5% beyond. ¥150k sales → 100k×3% + 50k×5% = ¥5,500. Includes flat vs tiered vs margin-based plans, clawback rules, and designing quotas that do not distort behavior.
Annualized Return (ROI): CAGR, XIRR & Comparison Rules
Annualized return = (final/initial)^(1/years) − 1. ¥100k → ¥150k in 3 years: 14.47%/yr (not 50%÷3). CAGR for lump sums, XIRR for irregular cash flows (SIP). Includes the return-vs-time trap, volatility drag and comparing investments fairly.
China Individual Income Tax: 7-Bracket Table & Annual Bonus
Comprehensive income: 5,000/month threshold + social insurance + special deductions, then 3%-45% seven brackets with quick deduction. ¥20k/month gross, standard setup → ~¥9.5k/year tax. Annual bonus can be taxed separately (often better). Full table and optimization inside.
Net Pay: Gross-to-Net Breakdown (Social Insurance + Tax)
Net = gross − social insurance (pension 8%, medical 2%, unemployment ~0.5%) − housing fund (5-12%) − income tax. ¥15k gross with 12% fund → ~¥11.5-11.8k net. Includes the contribution-base cap/floor rules and why the same gross nets differently by city.
Profit Margins: Gross, Operating & Net Margin Explained
Three levels: gross (revenue−COGS)/revenue, operating (after opex), net (after everything). Revenue ¥1M: gross 40%, operating 12%, net 8% — each answers a different question. Includes the waterfall, industry comparison rules and why net margin alone misleads.
True Cost of an Employee: What a ¥10k Salary Really Costs a Company
A ¥10,000/month salary costs the employer ~¥13,300–13,400 after social insurance and housing fund contributions (~27–34% on top) — about 1.6× the employee's ¥8,135 take-home. Full rate tables, two worked examples, and compliance red lines.
Startup Metrics: CAC, LTV, Inventory Turns & Dilution
LTV/CAC > 3 is healthy with payback under 12 months; inventory turns 6–12 suit e-commerce; each 15–20% round compounds dilution. Tables plus examples.
Science
16 guidesOhm's Law: V=IR Formulas, Wheel & Power Calculations
V = I × R links voltage, current and resistance; P = V × I gives power. 12V across 10Ω → 1.2A, 14.4W. Includes the 12-formula wheel, series vs parallel examples, wire sizing intuition and why LEDs need resistors.
Equivalent Resistance: Simplify Series-Parallel & Bridge Circuits
Series resistances add (R = R1 + R2); parallel reciprocals add (1/R = 1/R1 + 1/R2). Two 10Ω: 20Ω series, 5Ω parallel. Includes the product-over-sum shortcut, n-resistor formulas, mixed-network reduction steps and voltage divider design.
Ohm's Law: U=IR Formulas & Circuit Examples
U = I × R — voltage = current × resistance. Any two give the third: I = U/R, R = U/I. 12V across 4Ω → 3A; power P = UI = I²R = U²/R → 36W. Includes the power triangle, series/parallel voltage-current rules and household safety math.
Wavelength & Frequency: λ = v/f and the EM Spectrum
λ = v/f: wavelength = wave speed ÷ frequency. Sound (343 m/s) at 1 kHz → 34.3 cm; light (3×10⁸ m/s) at 100 MHz → 3 m. The EM spectrum from radio to gamma is one formula wide. Includes music acoustics, antenna sizing and color wavelengths.
Force: F = ma, Weight, Friction & Pressure Examples
F = ma: 1 N accelerates 1 kg at 1 m/s². Weight = mg (g ≈ 9.8; 70 kg → 686 N). Friction = μN. Pressure = F/A — why snowshoes work. Includes kgf vs N, braking-force intuition and the elevator scale illusion.
Density: ρ = m/V, Unit Conversion & Identification
ρ = m/V: mass ÷ volume. Water = 1,000 kg/m³ = 1 g/cm³ — the reference. Iron 7.87, aluminum 2.70, gold 19.32 g/cm³. Floats if ρ < water. Includes unit bridges (g/cm³ × 1,000 = kg/m³), the gold-fake test and oil-on-water logic.
Resistors in Series & Parallel: Equivalent Resistance
Series: R = R₁ + R₂ (add). Parallel: 1/R = 1/R₁ + 1/R₂; two resistors: R = R₁R₂/(R₁+R₂) ("product over sum"). 4Ω + 12Ω series = 16Ω; parallel = 3Ω. Includes voltage dividers, current splitting and equal-resistor shortcuts.
Electric Power & Bills: kWh, Tiered Pricing & Savings
P = UI; energy (kWh) = kW × hours; bill = Σ kWh by tier. 1.5匹 AC ≈ 1.1 kW → 8 h night ≈ 6-9 kWh. A 2 kW water heater 2 h/day = 120 kWh/month. Appliance tables, standby drains and the highest-ROI savings inside.
Speed of Sound: 340 m/s, Echoes & Distance Ranging
Sound in air: 331 m/s at 0°C, 343 m/s at 20°C (+0.6 per °C). Thunder distance: seconds ÷ 3 ≈ km. Water 1,480 m/s, steel ~5,100 m/s. Includes echo ranging, supersonic Mach and why you see the flash before the bang.
Half-Life Formula: Decay Math, Tables, Drugs and Carbon Dating
After n half-lives, remaining = initial × (1/2)ⁿ. Seven half-lives leave <1%; ten leave <0.1% — effectively gone. Decay table 0–10 half-lives, drug-dosing logic, and carbon-14 dating worked examples.
Kinematics: Free Fall, Projectiles, Momentum, Energy
Free fall h=½gt², impact v=√(2gh); projectiles split into horizontal+vertical motions; momentum p=mv always conserves, KE ½mv² only in elastic collisions.
Chemistry Math: Molar Mass, Ideal Gas, Dilution, pH
n=m/M; PV=nRT with R=8.314; C₁V₁=C₂V₂ for dilution; pH=−lg[H⁺], each unit a 10× jump. Molar mass sums atomic weights — expand parentheses first. Tables inside.
Energy & Periodic Motion: PE, Pendulum, Centripetal
PE=mgh turns into kinetic energy on falling; pendulum T=2π√(L/g) ignores bob mass; centripetal F=mv²/r quadruples when speed doubles.
Heat, Wind Chill & Decibels Explained
Water’s specific heat 4186 tops common substances — boiling 1L costs 336kJ. Wind chill makes -5℃ feel like -12℃. +10dB roughly doubles loudness.
Everyday Physics: Battery, Lightning, Distance, DoF
Battery Wh=mAh×V÷1000; thunder gap ÷3 ≈ km; Haversine gives great-circle distance; f/8 yields ~4× the depth of field of f/2.
Reading Spec Sheets: Wire Gauge, Hardness & Energy Units
AWG runs backward: every −6 doubles diameter, 12AWG≈2.05mm carries 20A; HRC/HB/HV hardness scales compared; kWh↔MJ↔kcal tables plus wiring examples.
Education
2 guidesGPA: How to Calculate Grade Point Average
GPA = Σ(grade point × credits) ÷ Σ credits. Standard 4.0 scale: A=4, B=3, C=2, D=1, F=0. A 3-credit A and 4-credit B → (12+12)/7 = 3.43. Covers weighted vs unweighted, Chinese 5.0/4.0 conversions, and what top universities expect.
Target Score: What Do I Need on the Final Exam?
Needed score = (target − current × current weight) ÷ final weight. Current 85 (60% weight), target 90, final worth 40% → need 97.5. Covers weighted grade math, feasibility checks, and how to plan across multiple remaining assessments.
Auto
4 guidesCar On-Road Price: The 6 Fees Beyond the Sticker
On-road price = sticker + purchase tax (10% of net price) + insurance + registration + extras. A ¥150k car lands at ~¥168k. Covers each fee's formula, new-energy exemptions, and the dealer add-ons you can refuse.
Gas vs Electric Car: 5-Year Real Cost Comparison
15,000 km/year: gas car at 8L/100km × ¥7.5/L = ¥9,000/yr fuel; EV at 15kWh/100km × ¥0.6/kWh = ¥1,350/yr home charging. Five-year gap ~¥45k — but depreciation and charging access decide the real winner.
Vehicle Purchase Tax: The 10% Rule & NEV Exemption
Tax = sticker ÷ 1.13 × 10%. ¥200k car → ¥17,699. New-energy vehicles are exempt (with price caps), imported cars use a composite assessable price. Covers the formula, exemption rules, and why the "minimum taxable price" exists.
Annual Car Ownership Cost: Fuel, Insurance, Tax & More
A ¥150k gas car costs ¥22-30k/year all-in: fuel ¥9k + insurance ¥5.5k + parking ¥6k + maintenance ¥2k + tax ~¥400 — before depreciation (¥10-15k/yr). The full breakdown plus per-category tables and cost-cutting tactics.
Construction
2 guidesRenovation Budget: Price Tiers & Where the Money Goes
Full renovation runs ¥1,000-3,000/sqm: basic ¥1-1.5k, mid ¥1.5-2.5k, high-end ¥3k+. A 100sqm mid-tier = ¥150-250k split roughly: labor 30%, materials 40%, furniture/appliances 20%, contingency 10%. Full allocation tables inside.
Tile Quantity Calculator: Coverage, Waste & Box Math
Tiles needed = area ÷ tile area × (1 + waste 5-15%). 20sqm floor with 600×600mm tiles: 20 ÷ 0.36 × 1.08 ≈ 60 tiles. Covers waste rates by pattern, box rounding, and the extra-box rule for future repairs.
Life
19 guidesTiered Electricity Pricing: How Your Bill Is Calculated
Tiered pricing charges blocks progressively: 0-260 kWh at ¥0.55, 261-600 at ¥0.60, 600+ at ¥0.85 (varies by province). 800 kWh = 260×0.55 + 340×0.60 + 200×0.85 = ¥517. Includes the block method, annual vs monthly tiers, and peak-valley arbitrage.
Overtime Pay: The 1.5× / 2× / 3× Rules Explained
Weekday overtime = 1.5×, rest days (no comp leave) = 2×, statutory holidays = 3×. Hourly base = monthly salary ÷ 21.75 ÷ 8. ¥10k/month → ¥57.5/h base; one holiday day = ¥1,724 extra. Full rules, base definition traps and calculation examples inside.
Monthly Utility Bills: Water, Power & Gas for a Family of 3
Typical family-of-3 monthly bill: electricity 300-450 kWh → ¥170-300; water 12-18 tons → ¥40-70; gas 25-40 m³ → ¥70-120. Total ¥280-490, doubling in AC/heating months. Full breakdown, usage benchmarks and where the waste hides.
Mobile Data: How Long Does 1GB Last? Usage Guide
1GB ≈ 1 hour of HD video, 4 hours of SD video, 17 hours of music, or 200+ hours of messaging. Video quality is the master switch: dropping 1080p→480p cuts usage 75%. Per-app consumption table and data-saving settings inside.
Phone Plans: How to Choose Data, Minutes & Save Money
Match the plan to your real usage: check 3 months of actual data/minutes first. Most users overpay — a ¥129 unlimited plan for someone using 25GB/month wastes ¥50+/month vs a matched plan. Includes the audit method, plan types, and negotiation tactics.
Discount Math: Stacked Sales, Thresholds & Real Savings
Sequential discounts multiply, not add: 30% off then 20% off = 44% off, not 50%. "¥100 off ¥300" = max 33% and only at exact thresholds. Real effective rate = final ÷ original. Includes stacking tables and the fake-discount detection checklist.
Tipping Guide: Percentages, Scenarios & Country Rules
US baseline: restaurants 15-20%, delivery 10-15% or $3-5 minimum, taxis 10-15%. Tip = bill × percentage, split per person. Japan: tipping can offend; Europe: service often included. Covers calculation shortcuts, split-bill math and country-by-country norms.
Size Conversion: Shoe & Clothing Sizes (CN/US/EU/UK)
Shoes: EU = CN old system + 30ish; foot length (cm) is the universal anchor — 25cm ≈ EU 40 ≈ US 7.5 (men). Clothing: CN 170/92A ≈ M ≈ EU 48. Foot-measuring method, brand variance warnings and full conversion tables inside.
Color Systems: RGB, CMYK, Pantone & Print Conversion
RGB (screen, additive) vs CMYK (print, subtractive) — gamuts differ: vivid screen blues/greens cannot print exactly. Pantone = premixed spot inks for brand consistency. Convert RGB→CMYK at export, expect dulling on saturated colors, and always proof brand colors physically.
Password Security: Strength Rules & Management
Length beats complexity: 12+ chars of random words ("correct-horse-battery-staple") > 8 chars of symbols. 8-char passwords fall in hours; 14-char random ones take centuries. Unique per site (a breach dominoes otherwise), manager-generated, 2FA on. Full math inside.
How to Choose Clothing Size: Measuring & Fit Guide
Size charts beat size labels: measure chest, waist, hips, shoulder and inseam, then match the brand's cm table. Between sizes — check fabric stretch and fit style. Includes the measuring method, fit types (slim/regular/oversize) and return-rate-saving tactics.
Splitting the Bill Fairly: AA Rules for Groups & Trips
Simplest fair split: total ÷ headcount, then adjust for non-drinkers and no-shows. For trips, track every shared expense and settle once at the end with the "who paid vs who owes" net method. Includes weighted splits, couple counting and settlement examples.
Rent vs Buy: The Long-Term Cost Comparison Framework
Price-to-rent ratio under 15 favors buying, over 20 favors renting (annual: price ÷ yearly rent). ¥2M home renting at ¥4k/month → ratio 41 → rent. True comparison includes mortgage interest, maintenance (1%/yr), opportunity cost of down payment, and transaction costs (~8-10% round trip).
Fuel Consumption: L/100km, Real-World vs Official & Costs
Real consumption = fuel added ÷ km driven × 100. Fill-to-fill method: 40L over 520km = 7.69 L/100km. Official figures run 15-30% below reality. Includes the measurement method, mpg conversion, and which driving habits actually cut fuel use.
Renovation Materials: Concrete, Rebar, Staircase
Concrete volume ×1.02 waste; C30 ≈380kg cement/m³. Rebar kg/m=d²×0.00617. Stairs: 2h+b≈600-640mm, risers ≤175mm.
Digital Gear Math: Power Bank, RAID, CCTV, Projector
Power bank Wh=mAh×3.7÷1000, ≤100Wh flies; RAID 5 yields (n−1) disks; CCTV GB/day=bitrate×10.55; throw=ratio×width.
Travel & Documents: Tolls, Luggage, ID Photos, Shipping
Tolls ≈0.4-0.5 CNY/km with 5% ETC off; carry-on sides ≤115cm; 1-inch photo 25×35mm=295×413px; couriers bill max(volumetric, real).
Home Theater Setup: Lumens & Viewing Distance
Living rooms need 2000+ ANSI lumens (1500 with blackout); 4K distance ≈ screen height ×1.5; TV size ≈ distance ÷1.2–1.6. Tables plus a 100-inch example.
Best Online Calculator Sites: Honest Comparison
Four tests for any calculator site: stated formulas, matching standards (China BMI, LPR), free with no signup, mobile-friendly. Comparison table inside.
Date
13 guidesDate Difference: Days Between Dates & Date Arithmetic
Days between dates: count end − start; 2025-01-15 to 2025-03-20 = 64 days. Inclusive counting adds 1. Month lengths vary (28-31) and Feb jumps on leap years — the two traps. Includes the 30/360 banking convention and week/day breakdowns.
Age Calculation: Full Age (周岁) vs Nominal Age (虚岁)
Full age (周岁): completed years since birth — birthday passes, +1. Nominal age (虚岁): 1 at birth, +1 every Lunar New Year — typically 1-2 years higher than full age. Includes exact age-to-the-day math, legal age rules and zodiac-year pitfalls.
Retirement Age: The Delayed Retirement Schedule Explained
China's delayed retirement (from 2025): men 60 → 63 over 15 years (+1 month per 4 months); women cadres 55 → 58 (+1 per 4 months); women workers 50 → 55 (+1 per 2 months). Born later = retire later. Includes the lookup method, flexible early/late options and pension math.
Countdown Calculation: Days Until Any Date
Days until = target date − today. Countdowns exclude today, include the target (or not — define it). 100-day milestones, workday-only countdowns, and the psychology of deadline math. Includes formulas for weeks+days display and repeated-event countdowns.
Workday Calculation: Business Days Excluding Weekends & Holidays
Workdays = total days − weekend days − holidays (+ makeup workdays). A 30-day span has ~22 workdays. China's twist: 调休 makeup weekends where Saturdays become workdays. Includes the counting algorithm, deadline math and project-schedule pitfalls.
Time Zones: UTC, Beijing Time & World Clock Conversion
Beijing = UTC+8, all of China in one zone. Conversion: target = local + (target offset − local offset). 9:00 Beijing = 20:00 New York (previous day, EST) = 01:00 London. Includes the DST trap, the date line, and meeting-planning windows.
Day of the Week: Zeller's Formula & Mental Calculation
Zeller's congruence computes any date's weekday; the doomsday method does it in your head (2025 doomsday = Friday; 4/4, 6/6, 8/8, 10/10, 12/12 all fall on it). Includes anchor dates, leap-year handling and the century codes.
Chinese Public Holidays: Schedule Rules & Countdown
China has 7 statutory holidays totaling 13 days: New Year 1, Spring Festival 4 (from Eve), Qingming 1, Labor Day 2, Dragon Boat 1, Mid-Autumn 1, National Day 3. Actual breaks extend via 调休 weekend swaps. Rules, countdown math and pay rules inside.
Chinese Lunar Calendar: Conversion Rules Explained
The lunar calendar is lunisolar: months follow the moon (29.53 days, alternating 29/30), years track the sun via 7 leap months in 19 years. Lunar 1/1 = Spring Festival. Conversion needs astronomical tables — no simple formula. Rules, leap months and birthday questions inside.
The 24 Solar Terms: Dates, Rules & Meaning
24 solar terms divide the sun's annual path into 15° steps: 立春 (Feb 3-5), 春分 (Mar 20-22), 夏至 (Jun 21-22), 秋分 (Sep 22-24), 冬至 (Dec 21-23). Dates drift ±1-2 days by year. Includes the full table, the 中气/节气 split, and why they predict weather proverbs.
Chinese Zodiac: Year Calculation, Lichun vs New Year Debate
Zodiac = birth year mod 12 mapped to 12 animals: 2024 = Dragon, 2025 = Snake, 2026 = Horse. Boundary debate: folk tradition switches at Spring Festival (lunar 1/1); 八字 astrology switches at 立春 (Feb 3-5). Jan-Feb births must check both. Full table and 本命年 rules inside.
Date Arithmetic: Add & Subtract Days, Weeks, Months
Date + N days: count forward handling month lengths (Mar 10 + 45 days = Apr 24). "+1 month" from Jan 31 → Feb 28/29 (clamping rule). Weekday-preserving jumps use weeks. Includes the 30/60/90-day business rules and the add-vs-set distinction.
Paid Annual Leave in China: Days by Tenure, Proration and Payouts
Statutory annual leave: 5 days at 1–10 years of cumulative work, 10 days at 10–20 years, 15 days at 20+ years. Tenure counts ACROSS employers. Pro-rata rules for joining/leaving, and the 300% payout for unused days.
Convert
17 guidesJin to Kilograms: Chinese Weight Units Explained
1 jin = 500 g = 0.5 kg; 1 kg = 2 jin. 1 liang = 50 g (10 liang = 1 jin). Grocery prices in jin: ¥8/jin apples = ¥16/kg. Includes the old 16-liang jin, regional variants (Hong Kong jin = 604.8 g), and quick price conversion.
Mu to Square Meters: Land Area Conversion Guide
1 mu = 666.67 m²; 1 hectare = 15 mu; 1 km² = 1,500 mu; 1 acre ≈ 6.07 mu. A 100 m² apartment plot = 0.15 mu. Includes the historical definition (60 square zhang), farmland context and fast mental conversion.
Cooking Measures: Tsp, Tbsp, Cup & Gram Conversions
1 tbsp = 3 tsp = 15 mL; 1 cup = 16 tbsp = 240-250 mL. Water 1 cup ≈ 240 g, flour 1 cup ≈ 125 g, sugar 1 cup ≈ 200 g — volume-to-weight depends on density. Full ingredient table and the scoop-vs-spoon accuracy debate inside.
Length Conversion: km, m, cm, Mile, Foot & Inch Tables
1 mile = 1.609 km; 1 inch = 2.54 cm exactly; 1 foot = 30.48 cm; 1 yard = 0.914 m. Metric moves by 10s: 1 km = 1,000 m = 100,000 cm. Quick anchors, precise factors, and the marathon (42.195 km = 26.2 mi) inside.
Area Conversion: m², mu, Hectare, Acre & Square Feet
1 hectare = 10,000 m² = 15 mu; 1 mu = 666.7 m²; 1 acre = 4,047 m² ≈ 6.07 mu; 1 m² = 10.76 sq ft. Key trap: linear conversions SQUARE — 1 m² = 10,000 cm², not 100. Full land-size tables and apartment-size intuition inside.
Volume Conversion: Liter, mL, m³, Gallon & Cup Tables
1 L = 1,000 mL = 1,000 cm³; 1 m³ = 1,000 L; 1 US gallon = 3.785 L; 1 UK gallon = 4.546 L (different!). 1 cup = 240-250 mL depending on standard. Includes kitchen measures, fuel tank math, and the gallon trap.
Temperature Conversion: °C, °F & K Formulas + Tables
°F = °C × 9/5 + 32; °C = (°F − 32) × 5/9; K = °C + 273.15. Key points: 0°C = 32°F, 100°C = 212°F, 37°C = 98.6°F, −40°C = −40°F (the crossover). Quick mental math and oven/body/climate tables inside.
Speed Conversion: km/h, m/s, mph & Knots Explained
1 m/s = 3.6 km/h exactly; 1 mph = 1.609 km/h; 1 knot = 1.852 km/h. 100 km/h = 27.8 m/s = 62.1 mph. Includes reaction-distance math (at 100 km/h you travel 27.8 m every second) and the running-pace connection.
Time Conversion: Hours, Minutes, Seconds, Days & Weeks
1 day = 24 h = 1,440 min = 86,400 s; 1 week = 168 h; 1 year ≈ 8,760 h (8,784 leap). Decimal hours: 1.5 h = 1 h 30 min — 0.1 h = 6 min, not 10. Includes work-hour math, timesheet decimals and the million/billion-seconds milestones.
Data Storage: KB, MB, GB, TB & Internet Speed Explained
Storage is decimal (1 GB = 1,000 MB, manufacturers) but RAM/OS is binary (1 GiB = 1,024 MiB) — why a "1 TB" drive shows 931 GB. Speed: Mbps is megaBITS, MB/s is megaBYTES — a 100 Mbps line downloads at 12.5 MB/s max. Full tables inside.
Pressure Units: Pa, kPa, MPa, bar, psi & atm Conversion
1 atm = 101.325 kPa ≈ 1.013 bar ≈ 14.7 psi. 1 bar = 100 kPa; 1 MPa = 10 bar. Tire pressure: 2.3-2.5 bar = 33-36 psi = 230-250 kPa. Includes the Pa definition (N/m²), mmHg for blood pressure and weather, and scuba depth math.
Power Units: kW, Horsepower, PS & Watts Explained
1 kW = 1,000 W; 1 hp (mechanical) = 745.7 W; 1 PS (metric) = 735.5 W — the two horsepowers differ 1.4%. EV 150 kW ≈ 204 PS. Car AC ≈ 2-3 kW. Covers engine ratings, the kW vs kWh confusion, and appliance electricity costs.
Angles: Degrees, DMS, Radians & π Explained
180° = π radians; 1 rad ≈ 57.3°. Degrees split into minutes and seconds: 1° = 60′ = 3600″. Radians measure arc/radius — the natural unit of calculus and physics. Includes conversion formulas, the special-angle table, and DMS arithmetic.
Frequency Units: Hz, kHz, MHz, GHz & RPM Explained
1 kHz = 1,000 Hz; 1 GHz = 10⁹ Hz. Period T = 1/f: 50 Hz mains = 20 ms cycle. RPM ÷ 60 = Hz (3,000 rpm = 50 rps). Includes audio range (20 Hz-20 kHz), radio bands, CPU clock speeds and the wavelength link (λ = v/f).
Shoe Size Conversion Chart: EU / US / UK / CN and Foot Length
Foot length 25.0 cm = CN 250 = EU 40 = US men's 7.5 / women's 9 = UK 6.5. Full men's, women's and kids' tables, how to measure your foot correctly, and 5 rules for buying shoes online.
Paper Sizes Explained: A4, A3, A5, B5, Photo Sizes and Book Formats
A4 = 210×297 mm, A3 = 297×420, A5 = 148×210. Every A-size folds in half into the next, keeping the √2:1 ratio — so A4 scaled 141% becomes A3. Full A/B/C tables, photo print sizes, and print-scaling cheat codes.
Best Unit Converter: Coverage and Accuracy
Judge by coverage, unit systems (US gallon 3.785L vs Imperial 4.546L), exact basis shown, and bidirectional solving. Gallon and ton traps inside.
Dev
22 guidesNumber Bases for Developers: Binary, Hex & Octal
Binary (0b), octal (0o), hex (0x) — hex is binary compressed 4 bits per digit: 0b1010 = 0xA = 10. 0xFF = 255, 0x100 = 256. Colors #RRGGBB are three hex bytes. Includes fast conversion, bit operations and the 0x7FFFFFFF landmarks.
Color Codes: HEX, RGB, HSL & Conversion Guide
HEX #3B82F6 = RGB(59,130,246) — each hex pair is one channel ÷ 255. HSL separates hue (0-360° wheel), saturation, lightness — best for building palettes: same hue, vary L for shades. Includes conversion math, alpha channels and contrast ratios.
JSON Formatting: Syntax Rules, Validation & Fixes
JSON: double-quoted strings only, no trailing commas, no comments, no undefined. Top causes of parse errors: single quotes, trailing comma, unquoted keys. Includes the type system, pretty-print conventions, JSONC/JSON5 escapes and debugging workflow.
Base64 Encoding: How It Works & When to Use It
Base64 packs 3 bytes into 4 ASCII chars (A-Z, a-z, 0-9, +, /) — +33% size, universally safe through text channels. Padding "=" marks the remainder. NOT encryption — trivially reversible. Uses: data URIs, email attachments, API tokens. URL-safe variant swaps +/ to −_.
URL Encoding: Percent-Encoding Rules & Examples
URL encoding replaces unsafe bytes with %XX hex: space → %20 (or + in query strings), 中 → %E4%B8%AD (UTF-8 bytes). Reserved chars: / ? & = # %. encodeURIComponent for parameters, encodeURI for whole URLs — the difference matters. Full table inside.
Unix Timestamps: Seconds, Milliseconds & Conversion
Unix timestamp = seconds since 1970-01-01 00:00:00 UTC. 1735689600 = 2025-01-01 00:00 UTC. Milliseconds version ×1,000 (JS default). Includes the 13-vs-10 digit check, timezone independence, the 2038 problem and language conversion snippets.
Regex Essentials: Patterns, Examples & Pitfalls
Core tokens: . any, \d digit, \w word, \s space, * 0+, + 1+, ? 0-1, {n,m} range, [] class, ^$ anchors, () groups, | or. Email: ^[\w.+-]+@[\w-]+\.[\w.]+$. Includes the 10 must-know patterns, greediness control (?), and catastrophic-backtracking warnings.
Hash Functions: MD5, SHA-256 & Password Hashing
Hash = one-way fingerprint: same input → same output; infeasible to reverse. MD5 (128-bit) and SHA-1 are BROKEN for security; SHA-256 for integrity; passwords need SLOW salted hashes (bcrypt/argon2) — never fast hashes. Includes collision math and file-verification workflow.
Character Counting: Length, Bytes & Platform Limits
Length depends on the ruler: "中文" = 2 chars, 6 UTF-8 bytes; JS .length counts UTF-16 units (emoji = 2!). Word vs char counts differ for CJK. Includes platform limits (titles 60 chars, meta 160), the emoji trap and byte math for APIs.
Lorem Ipsum: Placeholder Text Explained & Alternatives
Lorem ipsum = scrambled 1st-century Latin (Cicero's de Finibus) used as neutral filler since the 1500s — tests layout without content bias. Generate by paragraphs/sentences/words. Includes when to use it, when NOT to (real-content-first), and CJK alternatives.
Markdown Syntax: Complete Cheat Sheet & Preview Tips
Markdown = plain-text formatting: # headings, **bold**, *italic*, [links](url), , `code`, ``` fences, > quotes, - lists, 1. ordered, | tables |, --- rules. Renders to HTML everywhere (GitHub, docs, blogs). Includes flavor differences (GFM tables/task lists) and nesting rules.
HTML Entities: Escape Reference & Encoding Guide
Entities escape reserved/special chars: < (<), > (>), & (&), " ("), (non-breaking space), © (©). Numeric forms: < or <. Essential in HTML text/attributes to prevent breakage and XSS. Includes the must-know table and copy-paste symbols.
Text Diff: How Comparison Works & Reading Output
Diff finds line-level changes between texts: unified format shows −removed +added with @@ hunks of context. Myers algorithm minimizes the edit script. Uses: code review, config drift, contract versions. Includes word-level diff, patch application and merge-conflict anatomy.
Cron Expressions Explained: 5-Field Syntax, 20 Examples, Pitfalls
Five fields: minute hour day month weekday. Every day at 3am = 0 3 * * *; every 5 minutes = */5 * * * *; weekdays 9:30 = 30 9 * * 1-5. Syntax tables, 20 ready-to-paste expressions, and the 6 classic mistakes.
Subnet Mask Cheat Sheet: CIDR Table, Usable Hosts and Examples
/24 = 255.255.255.0 with 254 usable hosts; each +1 on the prefix halves the range. Usable hosts = 2^(32−prefix) − 2. Full /8–/30 table, splitting 192.168.1.0/24 into four, and how to check if two IPs share a subnet.
ASCII vs Unicode vs UTF-8 Explained
ASCII holds 128 chars; Unicode gives every character a code point (U+4E2D = 中); UTF-8 stores them in 1-4 bytes. Mojibake = wrong encoding declaration.
API Debugging: Status Codes, JWT, SLA Budgets
401 not authenticated, 403 no permission, 429 too fast, 502 upstream down. Decode JWT locally and check exp. 99.9% SLA = 43.8 min downtime per month.
Frontend Cheatsheet: px/rem, Aspect Ratio, Contrast
rem = px ÷ root font size; the 62.5% trick makes 1rem=10px. 16:9=1.778. WCAG AA body text needs 4.5:1 contrast. One viewport line fixes mobile.
CSV↔JSON, Slugs and Naming-Case Conversion Rules
CSV fields with commas/quotes/newlines need double quotes; slugs are lowercase and hyphenated; case conversion splits words then rebuilds. 9 styles compared.
Launch Checklist: Ports, Licenses, UUID, Download Time
Open only 80/443, never expose DB ports; MIT permissive, GPL copyleft; UUID v4 collisions negligible; download seconds = MB ÷ (Mbps÷8). Tables inside.
Why 0.1+0.2 Is Not 0.3: IEEE754 and Bitwise
IEEE754 doubles cannot hold 0.1 exactly — that is why 0.1+0.2≠0.3. Store money as integer cents, compare with ε. Bitwise ops and two's complement inside.
Best Online Developer Tools: Privacy-First List
Tokens and keys must stay in the browser. Recommended set: JSON formatter, Base64, regex, timestamp, hash, JWT decoder — with sanity checks.
Geometry
12 guidesCircle: Circumference & Area Formulas with Examples
C = 2πr = πd; A = πr². r=5 → C = 31.4, A = 78.5. Remember: area uses the radius SQUARED — doubling r quadruples the area. Includes arc/sector formulas, π approximations and worked pizza-size math.
Triangle Area: The 4 Formulas & When to Use Each
Base-height: A = bh/2. Three sides: Heron's formula √[s(s−a)(s−b)(s−c)]. Two sides + included angle: A = ½ab·sinC. Right triangle: A = ½ × legs. Includes the triangle inequality check and worked examples for each.
Rectangle & Square: Area, Perimeter & Diagonal
Rectangle: A = l×w, P = 2(l+w), diagonal = √(l²+w²). Square: A = s², P = 4s, diagonal = s√2. 5×3 rectangle → A = 15, P = 16, d = 5.83. Includes unit conversion traps and flooring/fencing worked examples.
Sphere: Surface Area & Volume Formulas Explained
Surface area = 4πr²; volume = (4/3)πr³. r = 3: SA = 113.1, V = 113.1 (numerically equal only at r=3). Doubling radius → 8× volume. Includes hemisphere formulas, the Archimedes cylinder relation, and ball-tank math.
Trapezoid Area: Formula, Midline & Examples
A = (a + b) × h ÷ 2 — average of the parallel sides times height. a=8, b=5, h=4 → A = 26. The midline (median) = (a+b)/2, so area = midline × height. Includes height-finding, isosceles diagonals, and the parallelogram connection.
Cone: Volume & Surface Area Formulas with Examples
V = (1/3)πr²h — exactly one third of the matching cylinder. Lateral area = πrl where slant l = √(r²+h²); total = πr(r+l). r=3, h=4 → l=5, V = 37.7, total SA = 75.4. Includes the 3-4-5 classic and ice-cream-cone math.
Cylinder: Volume & Surface Area Formulas with Examples
V = πr²h; lateral area = 2πrh; total = 2πr(r+h). r=2, h=5 → V = 62.8, total SA = 88.0. Tank capacity in liters (1 m³ = 1,000L), pipe volume, and why a can's height-to-diameter ratio affects material cost.
Ellipse: Area & Perimeter (Circumference) Formulas
Area = πab (a, b = semi-axes) — clean and exact. Perimeter has NO elementary formula; Ramanujan's approximation: C ≈ π[3(a+b) − √((3a+b)(a+3b))], accurate to ~0.01%. a=5, b=3 → A = 47.12, C ≈ 25.53.
Parallelogram Area: Formula & Relations to Rhombus & Rectangle
A = base × height (perpendicular!). a=8, h=5 → 40. The slant side is NOT the height. Rhombus bonus formula: A = d₁d₂/2 from diagonals. Includes the rectangle/rhombus/square family tree and diagonal properties.
Pythagorean Theorem: a²+b²=c² Formula, Triples and Uses
In a right triangle, legs a and b satisfy a²+b²=c² (c = hypotenuse). Classic triples 3-4-5 and 5-12-13. Covers finding legs vs hypotenuse, distance formula connection, and real-world measuring examples.
Regular Polygon Area Formula: Quick Coefficients from Triangle to Dodecagon
Area = n·a² ÷ [4·tan(π/n)] with side a. Hexagon = 2.598a², octagon = 4.828a². Coefficient table for n=3–12, apothem and circumradius formulas, plus flooring and pavilion worked examples.
Circle Geometry: Annulus, Segment, Spherical Cap, Torus
Annulus S=π(R²−r²); segment = sector − triangle; cap area S=2πRh uses only radius and height; torus V=2π²Rr²; ellipsoid V=(4/3)πabc. Tables and examples.