The Ultimate Quantitative Aptitude Cheatsheet

─── Squares (1 to 30) ───
  1²=1   2²=4   3²=9   4²=16  5²=25  6²=36  7²=49
  8²=64  9²=81  10²=100  11²=121  12²=144  13²=169  14²=196
  15²=225  16²=256  17²=289  18²=324  19²=361  20²=400
  21²=441  22²=484  23²=529  24²=576  25²=625  26²=676
  27²=729  28²=784  29²=841  30²=900

─── Cubes (1 to 15) ───
  1³=1   2³=8   3³=27  4³=64  5³=125  6³=216  7³=343
  8³=512  9³=729  10³=1000  11³=1331  12³=1728  13³=2197
  14³=2744  15³=3375

─── Powers of 2 ───
  2¹=2   2²=4   2³=8   2⁴=16  2⁵=32  2⁶=64  2⁷=128
  2⁸=256  2⁹=512  2¹⁰=1024  2¹¹=2048  2¹²=4096

─── Powers of 3 ───
  3¹=3   3²=9   3³=27  3⁴=81  3⁵=243  3⁶=729

─── Square Root Tricks ───
  Last digit of perfect square tells last digit of root:
  Last digit: 0→0, 1→1/9, 4→2/8, 5→5, 6→4/6, 9→3/7
  Example: √529 → ends in 9, so root ends in 3 or 7
  Since 22²=484 and 23²=529, answer = 23

─── HCF (Highest Common Factor / GCD) ───
  Largest number that divides all given numbers
  Method: Prime factorization, take minimum powers

─── LCM (Least Common Multiple) ───
  Smallest number divisible by all given numbers
  Method: Prime factorization, take maximum powers

─── Key Relationships ───
  HCF(a, b) × LCM(a, b) = a × b           [For two numbers]
  HCF always divides LCM
  HCF of co-prime numbers = 1
  LCM of co-prime numbers = a × b

─── HCF and LCM of Fractions ───
  HCF(a/b, c/d) = HCF(a,c) / LCM(b,d)
  LCM(a/b, c/d) = LCM(a,c) / HCF(b,d)

─── Example: HCF and LCM of 12 and 18 ───
  12 = 2² × 3¹
  18 = 2¹ × 3²
  
  HCF = 2^min(2,1) × 3^min(1,2) = 2¹ × 3¹ = 6
  LCM = 2^max(2,1) × 3^max(1,2) = 2² × 3² = 36
  
  Verification: HCF × LCM = 6 × 36 = 216 = 12 × 18 ✓

─── BODMAS / PEMDAS Order of Operations ───
  B - Brackets (first: (), then {}, then [])
  O - Orders (Powers, Roots)
  D - Division (left to right)
  M - Multiplication (left to right)
  A - Addition (left to right)
  S - Subtraction (left to right)

  Note: Division and Multiplication have equal priority → solve left to right
  Addition and Subtraction have equal priority → solve left to right

─── Example ───
  18 - [6 - {4 - (8 - 6 + 3)}]
  = 18 - [6 - {4 - (8 - 6 + 3)}]
  = 18 - [6 - {4 - (5)}]
  = 18 - [6 - {-1}]
  = 18 - [7]
  = 11

─── Surds Simplification ───
  √50 = √(25 × 2) = 5√2
  √75 = √(25 × 3) = 5√3
  √98 = √(49 × 2) = 7√2
  √128 = √(64 × 2) = 8√2
  √180 = √(36 × 5) = 6√5

─── Percentage Formulas ───
  X% of Y = (X/100) × Y
  X is what % of Y = (X/Y) × 100
  Percentage change = (Change/Original) × 100
  New value after % increase = Original × (1 + R/100)
  New value after % decrease = Original × (1 - R/100)

─── Important Percentage-Fraction Equivalents ───
  1% = 1/100       2% = 1/50        3% = 3/100
  4% = 1/25        5% = 1/20        6% = 3/50
  8% = 2/25        10% = 1/10       12.5% = 1/8
  15% = 3/20       16.67% = 1/6    20% = 1/5
  25% = 1/4        33.33% = 1/3    37.5% = 3/8
  40% = 2/5        50% = 1/2        60% = 3/5
  62.5% = 5/8      66.67% = 2/3    75% = 3/4
  80% = 4/5        87.5% = 7/8     100% = 1

─── Successive Percentage Change ───
  If two changes a% and b%:
  Net % = a + b + (a × b / 100)
  Example: 10% increase then 20% decrease:
  Net = 10 + (-20) + (10 × -20/100) = -10 - 2 = -12% (net decrease)

─── Population & Depreciation ───
  After n years (increase): P × (1 + R/100)^n
  After n years (decrease): P × (1 - R/100)^n

─── Basic Formulas ───
  Profit = SP - CP (when SP > CP)
  Loss = CP - SP (when CP > SP)
  Profit % = (Profit/CP) × 100
  Loss % = (Loss/CP) × 100

  SP = CP × (1 + Profit%/100)
  SP = CP × (1 - Loss%/100)
  CP = SP × 100 / (100 + Profit%)
  CP = SP × 100 / (100 - Loss%)

─── Discount ───
  Discount = MP - SP
  Discount % = (Discount/MP) × 100
  SP = MP × (1 - Discount%/100)

  Successive Discounts (a% and b%):
  Net discount = a + b - (a × b / 100)
  SP = MP × (1-a/100) × (1-b/100)

─── Example: Multiple Transactions ───
  A trader buys at 20% below MP and sells at MP.
  Profit % = ?
  Let MP = 100, CP = 80, SP = 100
  Profit = 20, Profit% = (20/80) × 100 = 25%

─── Example: False Weights ───
  Shopkeeper sells at cost price but gives 900g instead of 1kg.
  Gain % = (Error/Actual given) × 100
  = (100/900) × 100 = 11.11%

─── Simple Interest (SI) ───
  SI = (P × R × T) / 100
  Amount = P + SI = P(1 + RT/100)

─── Compound Interest (CI) ───
  Amount = P(1 + R/100)^n
  CI = Amount - P

  Half-yearly: A = P(1 + R/200)^(2n)
  Quarterly: A = P(1 + R/400)^(4n)

─── Difference: SI vs CI ───
  For 2 years: CI - SI = P(R/100)²
  For 3 years: CI - SI = P(R/100)² × (3 + R/100)

─── Example ───
  P = ₹10,000, R = 10%, T = 2 years
  SI = (10000 × 10 × 2)/100 = ₹2,000
  CI Amount = 10000 × (1.10)² = ₹12,100
  CI = 12100 - 10000 = ₹2,100
  Difference = ₹100 = 10000 × (10/100)² = 10000 × 0.01 = 100 ✓

─── Ratio Basics ───
  Ratio a:b has no units; same as fraction a/b
  Compound Ratio: a:b and c:d → ac:bd
  Duplicate Ratio: a:b → a²:b²
  Sub-duplicate: a:b → √a:√b
  Triplicate: a:b → a³:b³

─── Componendo & Dividendo ───
  If a/b = c/d, then:
  (a+b)/(a-b) = (c+d)/(c-d)

─── Proportion ───
  a:b = c:d → a×d = b×c (Product of extremes = Product of means)
  
  Fourth Proportional: a:b = c:x → x = bc/a
  Third Proportional: a:b = b:x → x = b²/a
  Mean Proportional: a:x = x:b → x = √(ab)

─── Partnership (Profit Sharing) ───
  Profit is shared in ratio of (Investment × Time)
  A invests ₹X for T1 months, B invests ₹Y for T2 months
  A's share : B's share = X×T1 : Y×T2

  Example: A invests ₹5000 for 6 months, B invests ₹8000 for 4 months
  A:B = 5000×6 : 8000×4 = 30000 : 32000 = 15:16

─── Alligation Rule ───
  When two ingredients at different prices are mixed:
  
  (Cheaper)          (Dearer)
    Quantity     :     Quantity
  (Dear - Mean)  :   (Mean - Cheap)
  
  Diagram:
      Cheaper ─────────┐
                       ├─ Mean Price
      Dearer ──────────┘
  
  Ratio = |Dear - Mean| : |Mean - Cheap|

─── Example ───
  Tea at ₹60/kg mixed with tea at ₹75/kg to get mixture at ₹68/kg
  
       60 ──── 68 ──── 75
  Ratio = |75-68| : |68-60| = 7:8
  
  So mix 7 parts of ₹60 tea with 8 parts of ₹75 tea

─── Replacement / Removal Problems ───
  A vessel contains x litres of liquid. y litres removed and replaced.
  After n operations: Remaining = x × (1 - y/x)^n
  
  Example: 40L milk, 4L removed and replaced with water, done 3 times
  Remaining milk = 40 × (1 - 4/40)³ = 40 × (9/10)³ = 40 × 0.729 = 29.16L

─── Basic Work Formulas ───
  If A can do work in n days: A's 1 day work = 1/n
  A and B together: 1/a + 1/b = (a+b)/ab
  Time together = ab/(a+b)

─── Work Equivalence ───
  Total Work = Men × Days × Hours per day
  If M1 men do W1 work in D1 days, and M2 men do W2 work in D2 days:
  M1 × D1 / W1 = M2 × D2 / W2
  (with hours: M1 × D1 × H1 / W1 = M2 × D2 × H2 / W2)

─── Pipes & Cisterns ───
  Inlet (fills): Positive rate → +1/n
  Outlet (empties): Negative rate → -1/n
  Net rate = Σ inlet rates - Σ outlet rates
  Time = 1 / Net rate

  Example: A fills in 20 min, B empties in 30 min
  Net = 1/20 - 1/30 = (3-2)/60 = 1/60
  Tank fills in 60 minutes

─── Alternate Days Work ───
  A works Day 1, B works Day 2, alternating:
  Work done in 2 days = 1/a + 1/b
  Total days = ceil(Total work / Work per cycle) × 2

─── Example ───
  A can do work in 12 days, B in 18 days. They work together for 4 days,
  then A leaves. How long for B to finish?
  A + B per day = 1/12 + 1/18 = 5/36
  In 4 days: 4 × 5/36 = 20/36 = 5/9 done
  Remaining: 4/9
  B alone: (4/9) / (1/18) = 4/9 × 18 = 8 days

─── Basic Formulas ───
  Speed = Distance / Time
  Distance = Speed × Time
  Time = Distance / Speed

  Conversions:
  1 km/hr = 5/18 m/s
  1 m/s = 18/5 km/hr

─── Average Speed ───
  Equal distances: Avg = 2xy/(x+y) where x, y are two speeds
  Equal times: Avg = (x+y)/2
  Three equal distances: Avg = 3xyz/(xy+yz+zx)

─── Relative Speed ───
  Same direction: |Speed1 - Speed2|
  Opposite direction: Speed1 + Speed2

─── Trains ───
  Time to cross pole/post = Length of train / Speed
  Time to cross platform = (Length of train + Length of platform) / Speed
  Time to cross another train = (L1 + L2) / Relative Speed
  (moving in opposite direction: relative speed = S1 + S2)
  (moving in same direction: relative speed = |S1 - S2|)

─── Boats & Streams ───
  Downstream Speed = Speed of Boat + Speed of Stream
  Upstream Speed = Speed of Boat - Speed of Stream
  Speed of Boat = (Downstream + Upstream) / 2
  Speed of Stream = (Downstream - Upstream) / 2

─── Circular Track ───
  Time to meet first time = Track Length / Relative Speed
  Time to meet at starting point = LCM of individual times

─── Example ───
  Train 200m long passes a pole in 20 seconds. Speed = ?
  Speed = 200/20 = 10 m/s = 10 × 18/5 = 36 km/hr

─── Square Identities ───
  (a + b)² = a² + 2ab + b²
  (a - b)² = a² - 2ab + b²
  a² - b² = (a+b)(a-b)
  (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

─── Cube Identities ───
  (a + b)³ = a³ + 3a²b + 3ab² + b³
  (a - b)³ = a³ - 3a²b + 3ab² - b³
  a³ + b³ = (a+b)(a² - ab + b²)
  a³ - b³ = (a-b)(a² + ab + b²)
  a³ + b³ + c³ - 3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)
  If a+b+c = 0: a³ + b³ + c³ = 3abc

─── Quadratic Equation ───
  ax² + bx + c = 0
  Sum of roots = -b/a
  Product of roots = c/a
  Discriminant D = b² - 4ac
  D > 0: Two distinct real roots
  D = 0: Two equal real roots
  D < 0: No real roots (complex)
  
  Roots = (-b ± √D) / (2a)

─── Sum of Series ───
  Sum of first n natural numbers = n(n+1)/2
  Sum of first n even numbers = n(n+1)
  Sum of first n odd numbers = n²
  Sum of squares of first n natural numbers = n(n+1)(2n+1)/6
  Sum of cubes of first n natural numbers = [n(n+1)/2]²

─── Factorial ───
  n! = n × (n-1) × (n-2) × ... × 2 × 1
  0! = 1, 1! = 1

─── Permutation (Order Matters) ───
  nPr = n! / (n-r)!     [Arrange r from n items]
  nPn = n!              [Arrange all n items]
  Repetition allowed: n^r

─── Combination (Order Does NOT Matter) ───
  nCr = n! / [r! × (n-r)!]    [Select r from n items]
  nCr = nC(n-r)           [Symmetry property]
  nC0 = nCn = 1
  Key: nPr = r! × nCr

─── Circular Permutation ───
  (n-1)! arrangements in a circle
  (n-1)!/2 if clockwise = anti-clockwise

─── Grouping ───
  n distinct items into r groups: Complex formula
  n identical items into r distinct groups: C(n+r-1, r-1)

─── Basic Rules ───
  P(E) = Favourable Outcomes / Total Outcomes
  0 ≤ P(E) ≤ 1
  P(Sure Event) = 1
  P(Impossible) = 0
  P(E) + P(not E) = 1

─── Addition Rule ───
  P(A or B) = P(A) + P(B) - P(A and B)
  [If mutually exclusive: P(A or B) = P(A) + P(B)]

─── Multiplication Rule ───
  P(A and B) = P(A) × P(B)  [Independent events]

─── Common Scenarios ───
  Coins: n coins → 2^n outcomes
  Dice: n dice → 6^n outcomes
  Cards: 52 cards = 4 suits × 13 ranks
  4 Aces, 4 Kings, 26 Red, 26 Black, 12 Face cards

─── Useful Formulas ───
  P(at least 1) = 1 - P(none)
  P(exactly 1) = P(A and not B) + P(not A and B)
  P(neither A nor B) = P(not A) × P(not B)

─── Average (Mean) ───
  Average = Sum of values / Number of values
  Sum = Average × Count
  New average when value added: (Old Sum + New) / (n+1)
  New average when value removed: (Old Sum - Removed) / (n-1)

─── Weighted Average ───
  W.Avg = (w1×x1 + w2×x2 + ...) / (w1 + w2 + ...)

─── Median ───
  Middle value when sorted
  Odd count: value at position (n+1)/2
  Even count: average of values at n/2 and (n/2)+1

─── Mode ───
  Most frequently occurring value
  A data set can have 0, 1, or multiple modes

─── Quick Multiplication ───
  × 5: Add 0 and divide by 2
  × 25: Add 00 and divide by 4
  × 50: Add 00 and divide by 2
  × 125: Add 000 and divide by 8
  × 9: × 10 - 1 (e.g., 47 × 9 = 470 - 47 = 423)
  × 11: Add adjacent digits (47 × 11 = 4, 4+7=11, 7 → 517)
  × 99: × 100 - 1 (e.g., 47 × 99 = 4700 - 47 = 4653)

─── Percentage Quick Math ───
  10% of X = X/10
  5% of X = X/20
  20% of X = X/5
  25% of X = X/4
  50% of X = X/2
  75% of X = 3X/4

─── Square Quick Tricks ───
  Number ending in 5: (n)(n+1) and append 25
  e.g., 35² = 3×4=12, append 25 → 1225
  e.g., 85² = 8×9=72, append 25 → 7225
  
  Near base of 100: Use (100±a)² = 10000 ± 200a + a²
  e.g., 97² = 10000 - 600 + 9 = 9409
  e.g., 104² = 10000 + 800 + 16 = 10816