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Percentages
Percentage = (Value / Total) × 100New Value = Original × (100 ± P) / 100For increase: multiply by (100 + P)/100. For decrease: multiply by (100 - P)/100Discount = Marked Price - Selling PriceSuccessive change: Net% = a + b + (a × b)/100Fraction equivalents: 50% = 1/2, 25% = 1/4, 20% = 1/5, 10% = 1/10
Profit & Loss
Profit = Selling Price - Cost PriceProfit % = (Profit / CP) × 100Loss = Cost Price - Selling PriceLoss % = (Loss / CP) × 100SP = CP × (100 + Profit%) / 100SP = CP × (100 - Loss%) / 100False weight profit% = (Error / Actual) × 100
Ratio & Proportion
a:b = a/b — ratio compares two quantitiesIf a:b = x:y, then a = kx, b = ky (multiplier method)Proportion: a:b :: c:d means a/b = c/d, cross multiply: ad = bcSum of parts = (a+b)x, Difference = |a-b|x
Time & Work
Work = Time × Rate (Rate = efficiency)If person takes T days: Rate = 1/T work per dayCombined Rate = Rate₁ + Rate₂ + ... (add rates, NOT times)LCM method: Total Work = LCM of individual timesPipe problems: Inlet = +ve rate, Outlet = -ve rate
Time-Speed-Distance
Speed = Distance / Time, Distance = Speed × Time, Time = Distance / Speedkm/hr to m/s: multiply by 5/18. m/s to km/hr: multiply by 18/5Avg Speed = Total Distance / Total Time (NOT average of speeds)Equal distances: Avg = (2 × S₁ × S₂) / (S₁ + S₂)Relative: Towards → add speeds. Same direction → subtract speedsTrain + Platform: Distance = Train length + Platform length
Simple & Compound Interest
SI = (P × R × T) / 100Amount = P + SICI: Amount = P × (1 + R/100)^TCI = Amount - PDifference CI - SI for 2 years = P × (R/100)²Half-yearly compounding: Use R/200 and 2T periods
Averages
Average = Sum of values / Number of valuesSum = Average × Number of valuesMissing value = (Average × N) - Sum of known valuesWeighted Avg = (W₁V₁ + W₂V₂ + ...) / (W₁ + W₂ + ...)
Number Series
Arithmetic progression: Tn = a + (n-1)d, Sum = n/2[2a + (n-1)d]Geometric progression: Tn = a × r^(n-1)Check differences first, then ratios, then squares/cubesAlternating series: check even/odd position patterns separately
Mixtures & Alligations
Concentration = (Solute / Solution) × 100Alligation rule: (Q₁/Q₂) = (C₂ - Cm) / (Cm - C₁)Replacement: Final = Original × (1 - x/V)^nMean price method for mixing two items
Permutations & Combinations
n! = n × (n-1) × (n-2) × ... × 1Permutations: nPr = n! / (n-r)!Combinations: nCr = n! / [r!(n-r)!]nC₀ = nCn = 1, nC₁ = n, nCr = nC(n-r)Arrangement with repetitions: n^r
Probability
P(Event) = Favorable outcomes / Total possible outcomesFor coins: Total outcomes = 2^n. For dice: Total outcomes = 6^nIndependent events: P(A and B) = P(A) × P(B)Mutually exclusive: P(A or B) = P(A) + P(B)Not mutually exclusive: P(A or B) = P(A) + P(B) - P(A and B)P(not A) = 1 - P(A)
Algebraic Equations
(a+b)² = a² + 2ab + b²(a-b)² = a² - 2ab + b²a² - b² = (a+b)(a-b)Linear: ax + b = c → x = (c-b)/aTwo variables: solve by substitution or elimination
Quadratic Problems
ax² + bx + c = 0Roots: x = [-b ± sqrt(b²-4ac)] / 2aSum of roots = -b/a, Product of roots = c/aDiscriminant: D = b² - 4ac (D>0 real, D=0 equal, D<0 complex)
Geometry Basics
Circle: Area = πr², Circumference = 2πrTriangle: Area = 1/2 × base × heightRight triangle: a² + b² = c² (Pythagoras)Equilateral triangle: Area = (√3/4) × side²Rectangle: Area = l × w, Perimeter = 2(l + w)Square: Area = side², Perimeter = 4 × side
Data Interpretation
Percentage share = (Part / Whole) × 100Growth rate = [(New - Old) / Old] × 100Average of values = Total / Number of itemsInferred total from a percentage share = Value / (Percentage/100)
Logical Puzzles
Draw constraints as a board or table firstApply each clue one at a time, eliminating impossible casesRanking: draw horizontal positions (left to right)Syllogisms: use Venn diagrams or rules of inference
Multi-Step Word Problems
Break into sub-problems: Part 1 → Result 1 → Part 2 → Result 2Define all variables before writing equationsCheck units match across all stepsValidate final answer against original question
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