01Purpose of Accounting & Users
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- Purpose: Record financial transactions, provide information for decision-making, measure profit/loss, show financial position
- Internal users: Owners, managers, employees — need info for planning and control
- External users: Banks/creditors (can we lend?), Government/HMRC (tax), Suppliers (can they pay?), Customers (are they stable?), Investors (should I invest?)
- Bookkeeping = recording transactions. Accounting = recording + analysing + reporting
Paper 1 often asks: "State TWO users of financial statements and explain why each needs the information."
02Source Documents & Books of Original Entry
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| Document | Used For | Book of Entry |
|---|---|---|
| Invoice (received) | Credit purchases | Purchases journal |
| Invoice (sent) | Credit sales | Sales journal |
| Credit note (received) | Purchases returns | Purchases returns journal |
| Credit note (sent) | Sales returns | Sales returns journal |
| Receipt / cheque counterfoil | Cash/bank transactions | Cash book |
| Petty cash voucher | Small cash expenses | Petty cash book |
| Journal voucher | Non-routine entries | General journal |
Books of original entry are NOT part of the double-entry system — they are prime entry records only.
03Double Entry & Ledger Accounts
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Accounting EquationAssets = Capital + Liabilities
| Account Type | Debit ↑ | Credit ↑ |
|---|---|---|
| Asset | Increase | Decrease |
| Liability | Decrease | Increase |
| Capital | Decrease | Increase |
| Expense | Increase | Decrease |
| Revenue/Income | Decrease | Increase |
Cash A/c — Example
DR (Debit)
Capital10 000
Sales3 000
Balance c/d8 500
CR (Credit)
Purchases2 500
Rent1 000
Wages1 000
- Every transaction = DR one account, CR another (same amount)
- Balance c/d closes account at period end; Balance b/d opens next period
- Contra entry: cash to bank or bank to cash — both in cash book
04Trial Balance & Types of Errors
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- Trial Balance: List of all ledger balances — DR total must equal CR total
- DR side: Assets, Expenses, Drawings, Purchases, Returns inwards
- CR side: Liabilities, Capital, Income, Sales, Returns outwards
| Error Type | Effect on TB | Example |
|---|---|---|
| Omission | No effect ✓ | Invoice not recorded at all |
| Commission | No effect ✓ | Posted to wrong person's account (same type) |
| Principle | No effect ✓ | Capital expenditure treated as revenue |
| Reversal | No effect ✓ | DR and CR swapped |
| Original Entry | No effect ✓ | Wrong amount on both sides |
| Compensating | No effect ✓ | Two equal but opposite errors cancel out |
| One-sided / Transposition | IMBALANCE ✗ | Only one side entered; wrong digit order |
A Suspense Account is opened when TB doesn't balance. Both sides of correction journal must be shown.
05Cash Book & Petty Cash (Imprest System)
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- Cash book = ledger account + book of original entry for cash & bank
- Has Discount Allowed (DR) and Discount Received (CR) columns — these are NOT part of double entry in cash book itself; posted separately to discount accounts
- Imprest system: Fixed float (e.g. $200); petty cashier spends, records vouchers; reimbursed to restore float at period end
- Advantage of imprest: Easy to control; simple to audit; limits risk of large losses
ReimbursementFloat − Remaining cash = Amount to reimburse
06Bank Reconciliation Statement
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BRS FormatBalance per cash book ± adjustments = Bank statement balance
| Reason for difference | Effect on BRS |
|---|---|
| Unpresented cheques (sent but not cleared) | Deduct from bank statement balance |
| Outstanding lodgements (deposited, not on statement) | Add to bank statement balance |
| Bank charges / interest (on statement, not in CB) | Deduct from cash book balance |
| Standing order / Direct debit on statement only | Update cash book (reduce balance) |
| Dishonoured cheque | Reverse original entry in cash book |
Bank overdraft shown as Credit balance in cash book and debit on bank statement — signs flip!
07Control Accounts (SLCA & PLCA)
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| Sales Ledger Control A/c (SLCA) | Purchases Ledger Control A/c (PLCA) |
|---|---|
| Opening debtors (DR) | Opening creditors (CR) |
| Credit sales (DR) | Credit purchases (CR) |
| Cash received (CR) | Cash paid (DR) |
| Returns inwards (CR) | Returns outwards (DR) |
| Discount allowed (CR) | Discount received (DR) |
| Bad debts written off (CR) | — |
| Closing balance = Total debtors (DR) | Closing balance = Total creditors (CR) |
Control accounts verify the accuracy of the sales/purchases ledger totals — they don't replace individual debtor/creditor accounts.
08Depreciation & Disposal
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Straight-Line(Cost − Residual Value) ÷ Useful Life
Reducing BalanceNBV × Rate%
NBVCost − Accumulated Depreciation
- Straight-line: Equal charge every year. Best for assets that depreciate evenly (buildings, fixtures)
- Reducing balance: Higher charge early, lower later. Best for tech/vehicles that lose value quickly
- Disposal journal: DR Asset Disposal A/c (cost); CR Asset A/c (cost) → then DR Acc. Dep. A/c; CR Asset Disposal
- Profit on disposal = proceeds > NBV → CR Profit on disposal (Income)
- Loss on disposal = proceeds < NBV → DR Loss on disposal (Expense)
09Year-End Adjustments
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| Adjustment | P&L Effect | Balance Sheet |
|---|---|---|
| Accrued expense | ADD to expense | Current Liability |
| Prepaid expense | DEDUCT from expense | Current Asset |
| Accrued income | ADD to income | Current Asset |
| Deferred/Prepaid income | DEDUCT from income | Current Liability |
| Bad debt written off | Expense (debit bad debts) | Reduce debtors |
| Increase provision for bad debts | Expense (difference) | Deduct from debtors |
| Decrease provision for bad debts | Income (difference) | Deduct from debtors |
Remember: Provision for bad debts only the CHANGE goes to P&L, not the whole provision amount.
10Trading Account & P&L Account
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Gross ProfitRevenue − Cost of Sales
Cost of SalesOp. Stock + Net Purchases − Cl. Stock
Net PurchasesPurchases + Carriage In − Returns Out
Net ProfitGP + Other Income − All Expenses
- Carriage inwards → Cost of Sales (buying cost)
- Carriage outwards → Expense in P&L (delivery cost to customer)
- Returns inwards (from customers) deducted from Revenue
- Returns outwards (to suppliers) deducted from Purchases
- Discount received → Other income in P&L
- Discount allowed → Expense in P&L
11Statement of Financial Position (Balance Sheet)
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| Section | Examples |
|---|---|
| Non-Current Assets | Land, Buildings, Machinery, Equipment, Vehicles, Goodwill |
| Current Assets | Inventory (stock), Trade receivables, Prepayments, Cash, Bank |
| Current Liabilities | Trade payables, Accruals, Bank overdraft, Short-term loans |
| Non-Current Liabilities | Long-term bank loan, Mortgage, Debentures |
| Capital | Opening capital + Net profit − Drawings |
Working CapitalCurrent Assets − Current Liabilities
Net AssetsTotal Assets − Total Liabilities = Capital
12Accounting Ratios & Analysis
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Gross Profit MarginGP ÷ Revenue × 100
Net Profit MarginNP ÷ Revenue × 100
ROCENP ÷ Capital Employed × 100
Current RatioCA : CL (ideal ~2:1)
Liquid (Acid Test)(CA − Inventory) : CL (ideal ~1:1)
Trade Receivables DaysReceivables ÷ Revenue × 365
Trade Payables DaysPayables ÷ Purchases × 365
Inventory TurnoverCost of Sales ÷ Avg Inventory
Always explain what a ratio means and why it changed — don't just calculate it. State if it's better or worse and suggest a reason.
13Partnerships
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- Appropriation Account: Net profit → Interest on Capital → Partners' Salaries → Remaining profit (shared in ratio)
- Capital Account: Fixed — only changes on admission/retirement/revaluation
- Current Account: Salary (CR), Interest on capital (CR), Share of profit (CR) vs Drawings (DR), Interest on drawings (DR)
- Goodwill on admission: Create goodwill in old ratio → Write off in new ratio
- Revaluation Account: Used when assets revalued on admission/retirement. Profit/loss shared in old ratio
Interest on CapitalCapital Balance × Rate%
Interest on DrawingsDrawings × Rate% × (months/12)
14Limited Companies
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| Term | Meaning |
|---|---|
| Ordinary shares | Voting rights; variable dividend based on profit |
| Preference shares | Fixed % dividend; paid before ordinary; usually no vote |
| Authorised share capital | Maximum shares company can issue |
| Issued share capital | Shares actually issued to shareholders |
| Share premium | Amount received above nominal value of shares |
| Retained profit (Revenue Reserve) | Accumulated profit kept in business |
| General reserve | Profit transferred for specific business purposes |
| Debentures | Long-term loans to company; fixed interest (expense not dividend) |
DividendShares issued × Nominal value × Rate%
EPSNet profit ÷ Number of ordinary shares
15Manufacturing Accounts
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- Prime Cost = Direct Materials + Direct Labour + Direct Expenses
- Factory Cost of Production = Prime Cost + Factory Overheads + Op. WIP − Cl. WIP
- Factory overheads: indirect labour, factory rent, machine depreciation, power for machines
- Factory cost transfers to Trading Account as "Cost of Finished Goods Produced"
- Market value may be used instead of factory cost if goods could be bought cheaper externally
16Non-Profit Organisations (Clubs & Societies)
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- Receipts & Payments Account: Cash summary — all cash received and paid (like a cash book)
- Income & Expenditure Account: Like P&L — accruals basis; surplus (profit) or deficit (loss)
- Accumulated Fund: Like Capital in a business — opening fund + surplus − deficit
- Subscriptions: Adjust for arrears (add) and in advance (deduct) to find income for year
- Trading sections (bar, cafe, tuck shop): calculate profit separately, transfer surplus to I&E
- Life membership: Received in full but spread over member's life; deferred portion = liability
17Inventory Valuation
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- FIFO (First In First Out): Oldest stock sold first. Closing stock at most recent prices. Higher profit in rising prices
- AVCO (Weighted Average Cost): Recalculate average after each purchase. Smooths out price fluctuations
- Lower of Cost and NRV: Prudence concept — never overstate inventory value
- NRV = Estimated selling price − Costs to complete and sell
- Different methods give different GP — FIFO gives higher closing stock (rising prices) → lower CoS → higher GP
IGCSE Paper 1 often tests: calculate closing inventory using FIFO or AVCO, and compare the effect on profit.
18Accounting Concepts & Qualitative Characteristics
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| Concept | Meaning | Example |
|---|---|---|
| Going Concern | Business will continue operating | Assets valued at cost, not liquidation value |
| Accruals / Matching | Income matched to expenses in same period | Prepayments and accruals |
| Prudence | Caution — never overstate assets/income | Provision for bad debts; NRV of stock |
| Consistency | Same methods period to period | Same depreciation method each year |
| Business Entity | Business separate from owner | Owner's personal expenses not in business books |
| Money Measurement | Only record what has a monetary value | Staff morale not recorded |
| Historical Cost | Assets recorded at original cost | Property at purchase price |
| Materiality | Only significant items need full disclosure | Small items expensed, not capitalised |
| Dual Aspect | Every transaction has two effects | Foundation of double entry |
🧠 Accounting Quick Fire
1. An error where DR and CR entries are swapped is called an error of:
2. Carriage inwards is included in:
3. Under the imprest system, petty cash balance is $200. Total vouchers = $148. Reimbursement is:
4. A provision for bad debts DECREASES. The effect on P&L is:
5. Which concept explains why a business records assets at purchase price, not current market value?
6. Partner A has capital of $30,000. Interest on capital is 5% p.a. What is the annual interest on capital?
7. FIFO is used. Stock: 10 units @ $4, then 10 units @ $6 bought. 12 units sold. What is the value of closing inventory?
8. Which of these is a NON-CURRENT liability?
01Measurement & Units
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| Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Temperature | kelvin | K (°C in practice) |
| Electric current | ampere | A |
| Force | newton | N |
| Energy / Work | joule | J |
| Power | watt | W |
| Pressure | pascal | Pa |
| Frequency | hertz | Hz |
| Charge | coulomb | C |
| Voltage | volt | V |
| Resistance | ohm | Ω |
- Scalar: magnitude only (speed, distance, mass, energy, temperature)
- Vector: magnitude AND direction (velocity, displacement, force, acceleration, momentum)
- Significant figures: match to least precise measurement given
02Motion — Distance, Speed, Velocity, Acceleration
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Speedv = d / t
Accelerationa = (v − u) / t
v² = u² + 2asSUVAT (extended)
s = ut + ½at²SUVAT (extended)
| Graph | Gradient = | Area = |
|---|---|---|
| Distance–time | Speed | — |
| Velocity–time | Acceleration | Distance (displacement) |
- Horizontal line on v-t graph = constant velocity (zero acceleration)
- Curve on d-t graph = changing speed (accelerating or decelerating)
- Negative gradient on v-t = deceleration
- Free fall: acceleration due to gravity g = 10 m/s² (9.8 in extended)
03Forces & Newton's Laws
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Newton's 2nd LawF = ma
WeightW = mg (g = 10 N/kg)
MomentM = F × d (perpendicular)
- Newton's 1st: Object at rest or constant velocity unless resultant force acts
- Newton's 2nd: F = ma — larger force or smaller mass → greater acceleration
- Newton's 3rd: Equal and opposite reaction forces between two objects
- Terminal velocity: Drag = Weight → zero resultant → constant velocity
- Principle of moments: Clockwise moments = Anticlockwise moments (equilibrium)
- Centre of gravity: Point where weight appears to act. Lower CoG = more stable
- Friction: always opposes motion; increases with normal force and rougher surfaces
04Energy, Work & Power
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Work DoneW = F × d (cos θ for angle)
KE½mv²
GPEmgh
PowerP = W/t = Fv
Efficiency(Useful output ÷ Total input) × 100%
- Conservation of energy: Energy cannot be created or destroyed, only transferred
- At max height: KE = 0, GPE = maximum; at bottom: GPE = 0, KE = maximum
- Energy types: kinetic, gravitational potential, elastic potential, thermal, chemical, nuclear, electrical, sound, light
- Wasted energy usually becomes thermal energy (heat) due to friction or resistance
05Pressure in Solids, Liquids & Gases
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Solid PressureP = F / A (Pa = N/m²)
Fluid PressureP = ρgh
Boyle's LawP₁V₁ = P₂V₂ (constant T)
- Pressure increases with depth in a fluid (more weight above)
- Pressure acts equally in all directions in a fluid (Pascal's principle)
- Atmospheric pressure = 101,000 Pa (≈101 kPa); decreases with altitude
- Manometer: measures pressure difference using liquid column height
- Hydraulic systems: pressure transmitted equally — small piston → large force at large piston
06Thermal Physics
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SHCQ = mcΔT
Latent HeatQ = mL
Charles' LawV/T = constant (const. P)
Pressure LawP/T = constant (const. V)
| Heat Transfer | Medium | Key Point |
|---|---|---|
| Conduction | Solids (best: metals) | Free electrons transfer energy; no bulk movement |
| Convection | Liquids & gases only | Density difference; hot rises, cool falls |
| Radiation | No medium needed | IR waves; black matt = best emitter/absorber; shiny = best reflector |
- Latent heat of fusion: solid → liquid (no temperature change)
- Latent heat of vaporisation: liquid → gas (much larger than fusion)
- Thermometers: liquid-in-glass uses expansion of liquid; thermocouple uses voltage
07Waves — Properties & Types
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Wave Equationv = fλ
PeriodT = 1/f
- Transverse: oscillation ⊥ direction of travel — light, EM waves, water, strings
- Longitudinal: oscillation ∥ direction of travel — sound, ultrasound (compressions & rarefactions)
- Amplitude: max displacement from rest; relates to energy/loudness
- Frequency: waves per second (Hz)
- Wavelength (λ): distance between two identical points on adjacent waves
- All EM waves: same speed in vacuum (3×10⁸ m/s), transverse, travel without medium
| EM Spectrum (low f → high f) | Use |
|---|---|
| Radio waves | Broadcasting, communications |
| Microwaves | Cooking, satellite communication |
| Infrared (IR) | Remote controls, thermal imaging, heating |
| Visible light | Sight, photography |
| Ultraviolet (UV) | Sterilisation, fluorescence, suntan |
| X-rays | Medical imaging, airport security |
| Gamma rays | Cancer treatment, sterilisation, tracers |
08Light — Reflection, Refraction & TIR
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Snell's Lawn = sin i / sin r = v₁/v₂
Critical Anglesin c = 1/n
- Law of reflection: Angle of incidence = Angle of reflection (both from normal)
- Refraction: Light bends when entering a different density medium — towards normal when slowing down
- TIR (Total Internal Reflection): Occurs when angle > critical angle AND light travels from dense to less dense medium
- TIR applications: optical fibres (endoscopes, internet cables), prisms in binoculars
- Converging (convex) lens: Brings parallel rays to a focal point. Used in cameras, eyes, magnifying glasses
- Diverging (concave) lens: Spreads rays. Used in spectacles for short-sight
- Real image: formed by actual rays converging; can be projected on screen
- Virtual image: rays appear to come from; cannot be projected (e.g. mirror image)
09Sound
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- Sound = longitudinal mechanical wave; needs a medium (cannot travel in vacuum)
- Speed of sound: ~340 m/s in air; faster in liquids; fastest in solids
- Pitch = frequency. Higher frequency → higher pitch
- Loudness = amplitude. Larger amplitude → louder sound
- Human hearing range: 20 Hz to 20,000 Hz (20 kHz)
- Ultrasound (>20 kHz): used in medical scanning, sonar, quality control
- Echo = reflection of sound. Distance = speed × (time/2)
- Oscilloscope: displays sound waves. X-axis = time; Y-axis = amplitude
10Electricity & Circuits
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Ohm's LawV = IR
PowerP = IV = I²R = V²/R
ChargeQ = It
EnergyE = VIt = Pt = QV
Series RR_total = R₁ + R₂ + ...
Parallel R1/R_t = 1/R₁ + 1/R₂
| Property | Series | Parallel |
|---|---|---|
| Current | Same everywhere | Splits (adds up) |
| Voltage | Splits (adds up) | Same across each branch |
| Total Resistance | Increases (sum) | Decreases (< smallest R) |
- Ammeter: connected in series; very LOW resistance
- Voltmeter: connected in parallel; very HIGH resistance
- LDR: resistance decreases as light increases
- NTC Thermistor: resistance decreases as temperature increases
- Diode: allows current in one direction only; forward bias only
- Fuse: thin wire that melts if current too high; always connected in LIVE wire
11Magnetism & Electromagnetism
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- Magnetic field lines: N→S outside; never cross; closer lines = stronger field
- Electromagnet: coil of wire + iron core. Increase strength: more turns, more current, soft iron core
- Motor effect: Current-carrying wire in magnetic field experiences a force. F = BIL
- Fleming's Left Hand Rule: thuMb = Motion/Force, First finger = Field, seCond = Current
- Electromagnetic induction: Moving a conductor in a field induces an EMF. Increase: faster movement, stronger magnet, more turns
- Fleming's Right Hand Rule: Generator — induced current direction
- AC generator: slip rings; DC generator: split-ring commutator
TransformerVp/Vs = Np/Ns
Ideal TransformerVp × Ip = Vs × Is
Step-up transformer: more turns on secondary (Ns > Np) → higher voltage, lower current. Used for power transmission to reduce energy loss.
12Atomic Structure & Radioactivity
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- Atom: nucleus (protons + neutrons) surrounded by electrons. Nucleus = tiny, dense, positive
- Proton number (Z) = number of protons; Nucleon number (A) = protons + neutrons
- Isotopes: same proton number, different neutron number (e.g. ¹²C and ¹⁴C)
| Radiation | Nature | Stopped by | Ionising | Charge |
|---|---|---|---|---|
| Alpha (α) | 2p + 2n (⁴₂He) | Paper / few cm air | Strongest | +2 |
| Beta (β) | Fast electron | 3 mm aluminium | Medium | −1 |
| Gamma (γ) | EM wave (high E) | Several cm lead | Weakest | 0 |
- Half-life: time for activity (or number of undecayed nuclei) to halve
- Background radiation: cosmic rays, radon gas, medical sources, nuclear power
- α decay: A−4, Z−2 | β decay: A unchanged, Z+1 | γ: no change in A or Z
- Uses: α = smoke detectors; β = paper thickness; γ = cancer treatment, sterilisation
13Space PhysicsExtended
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- Solar system: Sun (star) → 8 planets → moons, asteroids, comets
- Galaxy: billions of stars. Observable universe: billions of galaxies
- Gravitational force keeps planets in orbit; orbital speed must equal centripetal requirement
- Stars: nebula → protostar → main sequence (fusion H→He) → red giant → white dwarf (small star) OR supernova → neutron star/black hole (large star)
- Red-shift: galaxies moving away — light wavelength increases. Evidence for expanding universe and Big Bang
- CMB (cosmic microwave background radiation): remnant heat from Big Bang
🧠 Physics Quick Fire
1. A 4 kg object accelerates at 3 m/s². What is the resultant force?
2. A wave has frequency 50 Hz and wavelength 2 m. What is its speed?
3. Which radiation type has NO charge and the highest penetrating power?
4. Two resistors of 4Ω and 12Ω are connected in parallel. Total resistance?
5. A transformer has 200 primary turns and 1000 secondary turns. If input voltage is 12V, output is?
6. What happens to pressure in a fluid as depth increases?
7. A moving object has mass 2 kg and velocity 6 m/s. Its kinetic energy is?
8. Total internal reflection occurs when light moves from:
Paper 1 is NON-CALCULATOR. Focus on mental methods, exact values, estimation, and showing clear working.
01Number — Types, Operations & Rules
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- Integers: whole numbers (positive, negative, zero)
- Rational: can be written as p/q (includes fractions, terminating and recurring decimals)
- Irrational: cannot be written as fraction — π, √2, √3
- Prime: only divisible by 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19, 23...
- HCF: highest common factor — use prime factorisation or Venn diagram
- LCM: lowest common multiple — product of highest powers in prime factorisation
- BIDMAS/BODMAS: Brackets, Indices, Division/Multiplication (left→right), Addition/Subtraction
- Reciprocal of x = 1/x. Reciprocal of fraction a/b = b/a
For prime factorisation: use factor trees. e.g. 60 = 2² × 3 × 5. HCF = product of common factors; LCM = product of all factors at highest power.
02Fractions, Decimals & Percentages
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% Change(Change ÷ Original) × 100
Reverse %Amount ÷ (1 ± r/100)
Multiplier method15% increase → × 1.15
- Adding/subtracting fractions: common denominator first
- Multiplying fractions: top × top, bottom × bottom. Cancel first!
- Dividing fractions: flip the second fraction then multiply (KFC: Keep, Flip, Change)
- Recurring decimal → fraction: multiply by 10ⁿ to eliminate recurring part, then subtract
- Key conversions: 1/2=0.5, 1/4=0.25, 3/4=0.75, 1/3=0.333..., 1/5=0.2, 1/8=0.125
03Ratio, Proportion & Rate
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- Simplify ratio: divide both sides by HCF (e.g. 12:8 = 3:2)
- Share in ratio: total ÷ (sum of parts) × each part
- Direct proportion: y = kx — if x doubles, y doubles
- Inverse proportion: y = k/x — if x doubles, y halves
- Map scale: ratio e.g. 1:50,000 means 1 cm on map = 50,000 cm = 500 m in reality
- Speed = distance ÷ time. Units matter: km/h, m/s — convert carefully
Compound InterestA = P(1 + r/100)ⁿ
DepreciationA = P(1 − r/100)ⁿ
04Indices & Standard Form
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Product ruleaᵐ × aⁿ = aᵐ⁺ⁿ
Quotient ruleaᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power rule(aᵐ)ⁿ = aᵐⁿ
Zero indexa⁰ = 1
Negative indexa⁻ⁿ = 1/aⁿ
Fractional indexa^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ
Standard FormA × 10ⁿ (1 ≤ A < 10)
To multiply standard form: multiply A values, add powers. To add/subtract: first make powers equal.
05Algebra — Expressions, Equations & Inequalities
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- Expanding: multiply every term in bracket by outside term; FOIL for double brackets
- Factorising: common factor → HCF out; trinomials → two brackets; difference of squares: a²−b² = (a+b)(a−b)
- Solving linear: isolate variable — same operation both sides
- Simultaneous (elimination): make coefficients equal, add/subtract equations
- Simultaneous (substitution): rearrange one equation, substitute into other
- Inequalities: solve like equations but FLIP sign when × or ÷ by negative
- Number line: open circle = strict (< >); closed circle = inclusive (≤ ≥)
- Change of subject: rearrange formula step by step (inverse operations)
06Quadratics
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Quadratic Formulax = (−b ± √(b²−4ac)) / 2a
Discriminantb²−4ac: >0 two roots, =0 one, <0 none
- Factorising method: find two numbers that multiply to c and add to b (for x²+bx+c)
- Completing the square: x²+bx+c → (x + b/2)² − (b/2)² + c
- Vertex of parabola: x = −b/2a, substitute to find y
- Positive a → U-shape (minimum); Negative a → ∩-shape (maximum)
- Roots = where graph crosses x-axis (y = 0)
07Sequences
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Arithmetic nth terma + (n−1)d
Geometric nth termarⁿ⁻¹
Sum of arithmeticSₙ = n/2(2a + (n−1)d)
- Linear sequence: constant first difference → nth term = an + b. Find a (common difference), then b
- Quadratic sequence: constant second difference → nth term has n² term
- Geometric: constant ratio r between terms (multiply each time)
- Fibonacci-type: each term = sum of two previous terms
- To find if a value is in a sequence: set nth term = value and solve for n. If n is a positive integer, it's in the sequence
08Coordinate Geometry & Straight Lines
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Gradientm = (y₂−y₁)/(x₂−x₁)
Equation of liney = mx + c
Midpoint((x₁+x₂)/2, (y₁+y₂)/2)
Distance√((x₂−x₁)² + (y₂−y₁)²)
- Parallel lines: same gradient (m₁ = m₂)
- Perpendicular lines: gradients multiply to −1 (m₁ × m₂ = −1; so m₂ = −1/m₁)
- To find equation: find m, then substitute a point into y−y₁ = m(x−x₁)
- y-intercept: set x = 0; x-intercept: set y = 0
09Functions & Graph Transformations
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- f(x): function notation — substitute x value to evaluate
- fg(x): composite — apply g first, then f to result
- f⁻¹(x): inverse — swap x and y, then rearrange for y
- Domain = set of allowed input values; Range = set of output values
| Transformation | Effect |
|---|---|
| f(x) + a | Shift UP by a units |
| f(x) − a | Shift DOWN by a units |
| f(x + a) | Shift LEFT by a units |
| f(x − a) | Shift RIGHT by a units |
| af(x) | Stretch vertically by factor a |
| f(ax) | Stretch horizontally by factor 1/a |
| −f(x) | Reflect in x-axis |
| f(−x) | Reflect in y-axis |
10Geometry — Angles & Polygons
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Interior angle (regular polygon)(n−2)×180 / n
Exterior angle360 / n
Sum of interior angles(n−2) × 180°
- Angles on straight line = 180°; at a point = 360°; vertically opposite are equal
- Parallel lines: alternate angles equal (Z); co-interior = 180° (C); corresponding equal (F)
- Triangle angles sum = 180°; exterior angle = sum of two opposite interior angles
- Congruent triangles: SSS, SAS, ASA, RHS
- Similar triangles: same shape, different size. Corresponding sides in same ratio
11Circle Theorems
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| # | Theorem |
|---|---|
| 1 | Angle at centre = 2 × angle at circumference (same arc) |
| 2 | Angles in the same segment are equal |
| 3 | Angle in a semicircle = 90° (angle subtended by diameter) |
| 4 | Opposite angles of cyclic quadrilateral add up to 180° |
| 5 | Tangent is perpendicular to radius at the point of contact |
| 6 | Two tangents from an external point are equal in length |
| 7 | Alternate segment theorem: angle between tangent & chord = angle in alternate segment |
| 8 | Perpendicular from centre to chord bisects the chord |
Always give the theorem name as your reason — e.g. "Angle in alternate segment" or "Angle at centre is twice angle at circumference."
12Mensuration — Area, Volume & Surface Area
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Circle areaπr²
Circumference2πr
Sector areaθ/360 × πr²
Arc lengthθ/360 × 2πr
Triangle area½bh or ½ab sinC
Trapezium½(a+b)h
Cylinder Vπr²h
Cone V⅓πr²h
Sphere V⁴⁄₃πr³
Sphere SA4πr²
Cone SAπrl + πr² (l = slant)
Prism Vcross-section area × length
- For similar shapes: length ratio k → area ratio k², volume ratio k³
- Leave answers in terms of π on non-calculator paper unless asked to evaluate
13Trigonometry
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SOH-CAH-TOAsinθ=O/H · cosθ=A/H · tanθ=O/A
Sine Rulea/sinA = b/sinB = c/sinC
Cosine Rulea² = b²+c²−2bc cosA
Area of triangle½ab sinC
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
- Sine rule: use when given angle-opposite side pairs
- Cosine rule: use for SAS or SSS triangles
- Bearings: always measured clockwise from North, always 3 digits (045°, not 45°)
- Angle of elevation: measured up from horizontal; angle of depression: measured down
14Vectors
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Magnitude|v| = √(x² + y²)
ResultantAdd column vectors component-wise
- Vectors have magnitude AND direction. Scalar multiple changes length, negative reverses direction
- Parallel vectors: one is a scalar multiple of the other
- To prove collinear: show vectors are parallel AND share a common point
- Translate position: OB = OA + AB (add vectors along a path)
15Probability
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P(A and B) independentP(A) × P(B)
P(A or B) mutually exclusiveP(A) + P(B)
P(A or B) generalP(A)+P(B)−P(A∩B)
ComplementaryP(A') = 1 − P(A)
- Probability is always between 0 (impossible) and 1 (certain)
- Tree diagram: multiply along branches (AND); add branches for OR outcomes
- Without replacement: denominator reduces by 1 after each pick
- Venn diagrams: ∩ = intersection (AND); ∪ = union (OR); ξ = universal set
- Expected frequency = P(event) × number of trials
- Conditional probability: P(A|B) = P(A∩B) / P(B)
16Statistics
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Mean (grouped)Σfx / Σf
IQRQ3 − Q1
| Average | How to find | When to use |
|---|---|---|
| Mean | Sum ÷ count | All values, no extreme outliers |
| Median | Middle value when ordered | Skewed data / outliers present |
| Mode | Most frequent value | Categorical/non-numeric data |
- Cumulative frequency curve: plot at upper class boundaries; read off median at n/2
- Box plot: minimum, Q1, median, Q3, maximum
- Histogram: area = frequency (not height!). Frequency density = frequency ÷ class width
- Scatter graph: positive correlation → both increase; negative → one increases as other decreases
- Line of best fit: drawn through the mean point; used for interpolation (within data) not extrapolation (outside)
17Transformations
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| Transformation | Must State |
|---|---|
| Translation | Vector (column) |
| Reflection | Line of reflection (equation) |
| Rotation | Centre, angle, direction (CW or CCW) |
| Enlargement | Centre of enlargement, scale factor |
- Scale factor < 0: enlargement on opposite side of centre (negative enlargement)
- Enlargement with SF = 1/k is the inverse of enlargement with SF = k
- Area scale factor = (length scale factor)²
🧠 Maths Quick Fire (No Calculator!)
1. Factorise completely: 6x² − 15x
2. What is the gradient of a line perpendicular to y = 3x + 2?
3. Solve: 2x² − 8 = 0
4. The nth term of a sequence is 4n − 3. What is the first term that exceeds 50?
5. Express 72 as a product of its prime factors.
6. A bag has 4 red, 3 blue balls. Two drawn without replacement. P(both red)?
7. f(x) = 2x + 1. What is f⁻¹(x)?
8. Interior angle of a regular hexagon?