Aptitude & Exam Prep Cheatsheet

─── HCF (Highest Common Factor / GCD) ───
  Largest number that divides all given numbers

─── LCM (Least Common Multiple) ───
  Smallest number that is a multiple of all given numbers

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

─── Finding HCF — Prime Factorisation ───
  12 = 2^2 x 3
  18 = 2 x 3^2
  HCF = 2^1 x 3^1 = 6      (take minimum powers)

─── Finding LCM — Prime Factorisation ───
  12 = 2^2 x 3
  18 = 2 x 3^2
  LCM = 2^2 x 3^2 = 36     (take maximum powers)

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

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

─── Percentage Basics ───
  X% of Y  =  (X / 100) x Y
  X is what % of Y  =  (X / Y) x 100
  % Change  =  (Change / Original) x 100

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

─── Profit & Loss ───
  Profit = SP - CP            (when SP > CP)
  Loss   = CP - SP            (when CP > SP)
  Profit % = (Profit / CP) x 100
  Loss %   = (Loss / CP) x 100

  SP = CP x (1 + Profit%/100)      [when profit]
  SP = CP x (1 - Loss%/100)        [when loss]
  CP = SP x 100 / (100 + Profit%)
  CP = SP x 100 / (100 - Loss%)

─── Discount ───
  Discount = Marked Price (MP) - Selling Price (SP)
  Discount % = (Discount / MP) x 100
  SP = MP x (1 - Discount%/100)

─── Successive Discounts ───
  If two discounts a% and b%:
    Final Price = MP x (1 - a/100) x (1 - b/100)
    Equivalent single discount = a + b - (a x b / 100)

─── Important Percentage Equivalents ───
  1%   = 1/100          10%  = 1/10
  2%   = 1/50           11.11% = 1/9
  4%   = 1/25           12.5% = 1/8
  5%   = 1/20           14.28% = 1/7
  6.25% = 1/16          16.67% = 1/6
  8.33% = 1/12          20%  = 1/5
  25%  = 1/4            33.33% = 1/3
  50%  = 1/2            66.67% = 2/3

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

  P = Principal, R = Rate % per annum, T = Time in years

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

  When compounded half-yearly:
    A = P x (1 + R/200)^(2n)

  When compounded quarterly:
    A = P x (1 + R/400)^(4n)

─── Example: SI vs CI on Rs 10,000 at 10% for 2 years ───
  SI = (10000 x 10 x 2) / 100 = Rs 2,000
  A(CI) = 10000 x (1.10)^2 = Rs 12,100
  CI = 12100 - 10000 = Rs 2,100

  Difference in 2 years = P x (R/100)^2
  Difference in 3 years = P x (R/100)^2 x (3 + R/100)

─── Instalment Problems ───
  Total amount paid in instalments is always more than
  the principal. The extra amount is the interest charged.

─── Time & Work ───
  Work = Men x Days x Hours per day

  If A can do a work in n days:
    A's 1 day work = 1/n

  If A does 1/n work per day, A finishes in n days.

  A and B together (1 day work) = 1/a + 1/b = (a + b) / ab
  Time taken together = ab / (a + b)

─── Work & Wages ───
  Wages are distributed in ratio of work done
  Wages of A : B = Efficiency of A : Efficiency of B
    = (1/a) : (1/b) = b : a

─── Pipes & Cisterns ───
  Inlet pipe (fills): positive work rate
  Outlet pipe (empties): negative work rate

  Net rate = Sum of inlet rates - Sum of outlet rates
  Time to fill = 1 / Net rate

  Example: Pipe A fills in 20 min, Pipe B empties in 30 min
  Net = 1/20 - 1/30 = 1/60
  Tank fills in 60 minutes

─── Time & Distance ───
  Speed = Distance / Time
  Distance = Speed x Time
  Time = Distance / Speed

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

─── Average Speed ───
  When distances are equal:
    Avg Speed = 2xy / (x + y)
    (x, y = two speeds)

  When times are equal:
    Avg Speed = (x + y) / 2

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

─── Trains ───
  Time to cross a pole = Length / Speed
  Time to cross a platform = (Length + Platform) / Speed
  Time to cross another train = (L1 + L2) / Relative Speed

─── 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

  Time Downstream = Distance / Downstream Speed
  Time Upstream   = Distance / Upstream Speed

─── Races ───
  If A beats B by x metres in a race of d metres:
    When A finishes d metres, B has run (d - x) metres
    Ratio of speeds = d : (d - x)

  Dead Heat: Both finish at the same time
  Start/Head Start: A gives B a start of t seconds or x metres

─── Averages ───
  Average = Sum of observations / Number of observations
  Sum = Average x Number of observations

  When a value is added:
    New Avg = (Old Sum + New Value) / (n + 1)

  When a value is removed:
    New Avg = (Old Sum - Removed Value) / (n - 1)

  When a value is replaced:
    New Sum = Old Sum - Removed + Added
    New Avg = New Sum / n

  Weighted Average = (w1*x1 + w2*x2 + ...) / (w1 + w2 + ...)

─── Ratios ───
  Ratio a:b is same as a/b. It has no units.
  Compound Ratio: a:b and c:d = ac : bd
  Duplicate Ratio of a:b = a^2 : b^2
  Sub-duplicate Ratio = sqrt(a) : sqrt(b)
  Triplicate Ratio = a^3 : b^3

  If a:b = c:d then (a+b):(a-b) = (c+d):(c-d)   [Componendo & Dividendo]

─── Proportions ───
  a:b = c:d  implies  a x d = b x c      (Product of extremes = Product of means)
  a:b :: c:d  =>  a, b, c, d are in proportion

  Fourth Proportional: a:b = c:x  =>  x = bc/a
  Third Proportional:  a:b = b:x  =>  x = b^2/a
  Mean Proportional:   a:x = x:b  =>  x = sqrt(ab)

─── Fundamental Rules ───
  P(E) = Favourable Outcomes / Total Outcomes

  0 <= P(E) <= 1
  P(Sure Event) = 1
  P(Impossible Event) = 0
  P(E) + P(not E) = 1

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

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

─── Important Results ───
  Coins:  n coins tossed → 2^n outcomes
  Dice:   n dice rolled → 6^n outcomes
  Cards:  52 cards → 4 suits x 13 ranks
    - 4 Aces, 4 Kings, 13 Hearts, 13 Spades
    - 26 Red, 26 Black, 12 Face cards

─── Types of Questions ───
  At least one: P(at least 1) = 1 - P(none)
  Exactly one:  P(exactly 1) = P(A) x P(not B) + P(not A) x P(B)
  Neither:      P(neither A nor B) = P(not A) x P(not B)

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

─── Permutation (Order Matters) ───
  nPr = n! / (n-r)!       [Arrangement]

  nPn = n!                [All n items arranged]
  0Pr = 0 (for r > 0)

  Repetition allowed: n^r

─── Combination (Order Does NOT Matter) ───
  nCr = n! / [r! x (n-r)!]     [Selection]

  nCr = nC(n-r)           [Symmetry property]
  nC0 = nCn = 1
  nC1 = nC(n-1) = n

─── Key Relationship ───
  nPr = r! x nCr

─── Circular Permutation ───
  (n-1)!  arrangements of n distinct objects in a circle
  (n-1)! / 2  if clockwise and anti-clockwise are same

─── Division into Groups ───
  n! / (n1! x n2! x ...)   [if all groups distinct]
  n! / [(n1! x n2! x ...) x k!]  [if k groups are identical]

─── Logarithm Rules ───
  If a^m = N then log_a(N) = m    (a > 0, a != 1, N > 0)

  Product:      log(M x N)    = log M + log N
  Quotient:     log(M / N)    = log M - log N
  Power:        log M^n       = n x log M
  Base Change:  log_a N       = log N / log a
  Reciprocal:   log(1/N)      = -log N
  Self:         log_a a       = 1
  Unity:        log_a 1       = 0

─── Surds & Indices Rules ───
  a^m x a^n = a^(m+n)         a^m / a^n = a^(m-n)
  (a^m)^n = a^(mn)            (ab)^n = a^n x b^n
  (a/b)^n = a^n / b^n         a^0 = 1  (a != 0)
  a^(-n) = 1 / a^n

  a^(m/n) = n-th root of (a^m) = (n-th root of a)^m

─── Simplification of Surds ───
  sqrt(50) = sqrt(25 x 2) = 5 sqrt(2)
  sqrt(75) = sqrt(25 x 3) = 5 sqrt(3)
  sqrt(98) = sqrt(49 x 2) = 7 sqrt(2)

  Like surds: a*sqrt(m) + b*sqrt(m) = (a+b)*sqrt(m)
  Multiply: a*sqrt(m) x b*sqrt(n) = ab*sqrt(mn)

─── Blood Relation Notation (Drawing a Tree) ───
  Use + for Male, - for Female
  ----  = Married couple (horizontal line)
  |     = Parent-child (vertical line)
  ---   = Siblings (horizontal line with dots)

  Generation levels (top to bottom):
    Grandparents → Parents → Self → Children → Grandchildren

─── Key Concepts ───
  Right Turn  = 90 degrees clockwise
  Left Turn   = 90 degrees anti-clockwise
  About Turn  = 180 degrees (reverses direction)

  Shadow direction:
    Morning (sunrise):  Shadow falls to the WEST
    Evening (sunset):   Shadow falls to the EAST
    Noon:               No shadow (sun directly overhead)

  Finding shortest distance between start and end:
    Use Pythagoras theorem after finding net N-S and E-W displacement

  Example:
    Walk 4 km North, turn right, walk 3 km East
    Distance from start = sqrt(4^2 + 3^2) = 5 km (right angle triangle)

─── Linear Arrangement ───
  People sit in a single straight line
  Left and Right are based on the person's perspective
  When facing North: Left = West, Right = East
  When facing South: Left = East, Right = West

─── Circular Arrangement ───
  People sit around a circle (usually 5-8 people)
  Facing Centre:
    - Left = clockwise
    - Right = anti-clockwise
  Facing Outward:
    - Left = anti-clockwise
    - Right = clockwise

─── Rectangular / Square Arrangement ───
  People sit at corners and/or middle of sides
  Combination of linear and circular logic

─── Double Row Arrangement ───
  Two rows facing each other (A facing B and vice versa)
  - A's left = B's right and vice versa

─── Key Tips ───
  1. Always note who faces which direction first
  2. Start with fixed/clues with definite positions
  3. Look for "sits next to" (adjacent) and "sits between" clues
  4. Draw diagrams for every arrangement problem
  5. Verify final arrangement with all given clues

─── Steps to Solve ───
  1. Read the question and identify what is asked
  2. Check Statement I alone
  3. Check Statement II alone
  4. Check both together

─── Answer Options (Standard) ───
  A: Statement I alone is sufficient
  B: Statement II alone is sufficient
  C: Both I and II together are sufficient
  D: Either I or II alone is sufficient
  E: Neither I nor II is sufficient (need more data)

─── Tips ───
  - Do NOT assume anything beyond given data
  - Do NOT solve completely; just check if answer CAN be found
  - "What is the value" → need a unique numerical answer
  - "Is X greater than Y?" → need comparison, not exact values
  - If I alone gives answer, answer is A (do not bother checking II)

─── Common Input-Output Patterns ───
  1. Shifting: Elements rearranged by position (left/right shift)
  2. Arrangement: Words sorted alphabetically, numbers sorted
  3. Mathematical: Each number operated upon (+, -, x, /)
  4. Replacement: Letters replaced by next/previous letter
  5. Combination: Two or more of the above

─── Solving Steps ───
  1. Write the given input carefully
  2. Compare Input → Step 1 → Step 2 to find the pattern
  3. Check if the pattern applies consistently
  4. Apply the pattern to answer the question

─── Example Pattern ───
  Input:  96  48  72  24  36
  Step 1: 6   12  18  24  30      (divide by 8, then 4, then 4, ...)
  Step 2: 18  12  24  6   30      (rearranged ascending by first digit)
  Always verify the pattern with at least 2-3 steps before answering.

─── Rule 1: Subject-Verb Agreement ───
  - Singular subject → Singular verb
  - Plural subject → Plural verb
  - Two singular subjects joined by "and" → Plural verb
  - "Each", "Every", "Either", "Neither", "Everyone" → Singular verb
  - "Either...or", "Neither...nor" → Verb matches nearer subject

─── Rule 2: Tense Consistency ───
  - Keep the same tense throughout a sentence
  - Past tense for past events
  - Present perfect for recently completed + effect

─── Rule 3: Pronoun Errors ───
  - Subject position → I, he, she, they, we (nominative)
  - Object position → me, him, her, them, us (objective)
  - Possessive → my, his, her, their, our

─── Rule 4: Common Prepositions ───
  - Accused OF, Acquitted OF, Afraid OF
  - Absorbed IN, Abundant IN
  - Blind IN/TO, Deprived OF
  - Comply WITH, Deal WITH, Conversant WITH
  - Fond OF, Deprived OF, Devoid OF

─── Rule 5: Article Usage ───
  - "The" for unique things (sun, earth, longest)
  - "A/An" before consonant/vowel SOUND (not letter)
  - No article before: proper nouns, abstract nouns (general), meals

─── Strategies ───
  1. Read the original sentence carefully
  2. Identify the error (grammar/usage/meaning)
  3. Check all options — choose the best one
  4. Look for: tense, preposition, parallelism, idioms

  Common improvement areas:
  - Replace wrong preposition
  - Fix tense consistency
  - Use correct article
  - Correct subject-verb agreement
  - Use appropriate idiom
  - Remove redundancy (e.g., "revert back" → "revert")
  - Use active voice over passive for conciseness

─── Approach ───
  1. Read the questions FIRST (to know what to look for)
  2. Skim the passage for overall idea (2-3 minutes)
  3. Read in detail, marking key points
  4. Answer questions based on the passage ONLY

─── Question Types ───
  - Main Idea / Central Theme
  - Specific Detail (directly from passage)
  - Inference (implied, not stated directly)
  - Vocabulary in Context (meaning of a word as used in passage)
  - Tone / Attitude of the author
  - Title of the passage

─── Tips ───
  - The answer is ALWAYS in the passage — do not use outside knowledge
  - "Inference" questions: look for what the author implies
  - "NOT" questions: eliminate options that ARE in the passage
  - Watch for extreme words: "always", "never", "all", "none"
  - First and last sentences of paragraphs are crucial
  - Read at least 2 passages daily for practice

─── Conversion Formula ───
  Degree = (Percentage / 100) x 360
  Percentage = (Degree / 360) x 100

  Quick conversions to memorize:
    1%  = 3.6 degrees
    10% = 36 degrees
    25% = 90 degrees
    50% = 180 degrees

─── Percentage Approximation ───
  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

  Successive % change:
    Net % = a + b + (a x b / 100)
    Example: 10% increase then 20% decrease:
      Net = 10 - 20 + (10 x -20 / 100) = -10 - 2 = -12%

─── Quick Multiplication ───
  x 50  → add two zeros and divide by 2
  x 25  → add two zeros and divide by 4
  x 125 → add three zeros and divide by 8
  x 5   → add one zero and divide by 2

─── Ratio Comparison ───
  a/b vs c/d  → Cross multiply: compare ad vs bc

─── Averages in DI ───
  If data is in ranges, approximate by midpoints
  Weighted average for grouped data is common in DI

─── Approach ───
  1. Identify exactly what is being asked
  2. Check each statement independently
  3. Determine if data is sufficient to get a UNIQUE answer

─── Common Traps ───
  - Statement gives ratio but question asks value → NOT sufficient
  - Statement gives total but needs individual breakup → check
  - Graph without scale/legend → data may be insufficient
  - Percentage without base value → cannot find absolute numbers

─── Practice Strategy ───
  - Solve at least 3 DI sets daily
  - Time yourself: target 8-10 minutes per set
  - Focus on accuracy over speed initially
  - Learn to skip difficult sets and come back later

─── Preparation Strategy ───
  1. Daily Sources:
     - The Hindu / Indian Express (editorials most important)
     - PIB (Press Information Bureau) — pib.gov.in
     - DD News / Rajya Sabha TV

  2. Monthly Compilation:
     - Make notes of important events by category
     - Focus on: Appointments, Awards, Summits, Indexes, Defence

  3. Key Areas to Cover:
     - National & International events
     - Government schemes & policies
     - Appointments & resignations
     - Sports (tournaments, winners, records)
     - Awards & honours (national and international)
     - Science & Technology (ISRO, DRDO)
     - Defence exercises (bilateral, multilateral)
     - Important indexes/rankings (HDI, Ease of Business, GHI)

  4. Timeline:
     - For most exams: last 6-12 months
     - For UPSC: last 12-18 months
     - For Bank exams: last 4-6 months

  5. Revision:
     - Revise weekly notes every Sunday
     - Take monthly quizzes
     - Make flashcards for important dates and names

─── What to Note in Every Union Budget ───
  1. Fiscal Deficit (% of GDP) — Target vs Actual
  2. Revenue Deficit and Capital Expenditure
  3. Tax Revenue (Direct and Indirect) estimates
  4. New Tax Slabs (if any changes to income tax)
  5. Key Allocations:
     - Education sector
     - Health sector
     - Defence sector
     - Agriculture and farmer welfare
     - Infrastructure (roads, railways, ports)
     - Rural development
     - Social welfare schemes

  6. New Schemes / Schemes discontinued
  7. Subsidy allocations (food, fertilizer, fuel)
  8. Disinvestment targets
  9. Borrowing estimates (market borrowing, external)
  10. GST collections trend

─── Key Budget Terms ───
  Fiscal Deficit  = Total Expenditure - Total Receipts (except borrowing)
  Revenue Deficit = Revenue Expenditure - Revenue Receipts
  Primary Deficit = Fiscal Deficit - Interest Payments
  Capital Expenditure (Capex) = Spending on assets (infrastructure, equipment)
  Revenue Expenditure = Day-to-day spending (salaries, subsidies, interest)

─── Section-Wise Time Allocation ───
  Rule of thumb: Spend time proportional to marks

  For a 60-minute exam with equal sections:
  - English/VARC: 12-15 minutes
  - Quantitative Aptitude: 15-18 minutes
  - Reasoning/DILR: 15-18 minutes
  - GK/Computer/Current Affairs: 5-10 minutes (fastest)

─── Attempt Strategy ───
  1. FIRST: Attempt GK, Computer, English Vocab (quick, high accuracy)
  2. SECOND: Attempt Reasoning/Logical (moderate time, moderate-high accuracy)
  3. THIRD: Attempt Quantitative (most time-consuming)
  4. LAST: Attempt difficult/review marked questions

─── Question Selection ───
  - Round 1: Attempt all easy questions (30-40 seconds each)
  - Round 2: Attempt moderate questions (60-90 seconds each)
  - Round 3: Attempt difficult only if time remains

  NEVER waste more than 2 minutes on a single question in Round 1.
  Mark and skip — come back later with fresh perspective.

─── Speed Building ───
  - Learn tables up to 30, squares up to 30, cubes up to 15
  - Practice mental calculation daily (10 minutes)
  - Learn shortcut methods for common problem types
  - Take timed mock tests at least 3 times a week

─── Why Mock Tests Matter ───
  - Build exam temperament and time management
  - Identify weak and strong areas
  - Practice question selection strategy
  - Build stamina for long exams
  - Familiarize with computer-based test interface

─── Mock Test Schedule (3 months before exam) ───
  Month 3 (first month):
    - 1 mock test per week
    - Focus on understanding question types
    - Spend 2x time on analysis

  Month 2 (second month):
    - 2-3 mock tests per week
    - Focus on speed improvement
    - Track sectional accuracy

  Month 1 (last month before exam):
    - 4-5 mock tests per week
    - Simulate actual exam conditions (same time, no breaks)
    - Focus on maximizing score and minimizing mistakes

─── Post-Test Analysis (IMPORTANT) ───
  Spend 60-90 minutes analyzing EVERY mock test:
  1. Wrong questions: Why did I get it wrong? Concept gap or silly mistake?
  2. Unattempted: Could I have solved them? Should I attempt next time?
  3. Correct but slow: How can I solve faster? Learn shortcuts.
  4. Maintain an error log: Note repeated mistake patterns.
  5. Track progress: Section-wise scores, total score, accuracy percentage

─── Last Week Before Exam ───
  - Stop learning new topics; revise what you know
  - Go through your error log and formula sheet
  - Take 2-3 full-length mock tests
  - Revise important GK notes (current affairs)
  - Sleep well (7-8 hours); avoid late-night study
  - Prepare admit card, ID proof, photos in advance
  - Know your exam centre route; reach 1 hour early

─── Exam Day Strategy ───
  - Read instructions carefully before starting
  - Glance through the paper once (if allowed)
  - Start with your strongest section to build confidence
  - Do NOT discuss with others before the exam
  - Stay calm; skip questions that confuse you
  - Keep track of time; wear a wristwatch if allowed
  - Attempt all easy questions first; do NOT leave easy marks
  - Guess only if negative marking is low and you can eliminate 2+ options

─── During the Exam ───
  - Mark questions for review (if CBT allows)
  - Check that all attempted questions are submitted
  - Do NOT change answers in last 5 minutes (unless obvious error)
  - Use rough paper effectively for calculations

─── Must-Memorize Formulas ───

PERCENTAGES & PROFIT/LOSS
  SP = CP(1 + P/100)    |   SP = CP(1 - L/100)
  Equivalent discount: a + b - ab/100

INTEREST
  SI = PRT/100          |   A(CI) = P(1 + R/100)^n
  CI - SI (2yr) = P(R/100)^2

TIME & WORK
  Together: 1/a + 1/b = (a+b)/ab   |   Time = ab/(a+b)

SPEED & DISTANCE
  km/hr to m/s: x 5/18  |  m/s to km/hr: x 18/5
  Avg Speed (equal dist): 2xy/(x+y)
  Train: Time = (L1+L2) / Relative Speed

GEOMETRY
  Pythagoras: c^2 = a^2 + b^2
  Area of Triangle: 1/2 x base x height
  Circle: Area = pi*r^2  |  Circumference = 2*pi*r

PROBABILITY
  P(E) = Favourable / Total   |   P(E) + P(not E) = 1

PERMUTATIONS & COMBINATIONS
  nPr = n!/(n-r)!   |   nCr = n!/[r!(n-r)!]
  nCr = nC(n-r)     |   nPr = r! x nCr

PROPORTIONS
  a:b = c:d  =>  ad = bc
  Mean proportion: x = sqrt(ab)