Quantitative Aptitude (Quant) Mastery

• Sum of first n natural numbers: n(n+1)/2
• Sum of first n squares: n(n+1)(2n+1)/6
• Sum of first n cubes: [n(n+1)/2]²
• Divisibility Rules:
  • 3: Sum of digits divisible by 3.
  • 4: Last two digits divisible by 4.
  • 11: (Sum of odd place digits) - (Sum of even place digits) = 0 or multiple of 11.
• Dividend Formula: Dividend = (Divisor × Quotient) + Remainder

• Product Rule: 1st Number × 2nd Number = HCF × LCM
• Fractions:
  • HCF of Fractions = (HCF of Numerators) / (LCM of Denominators)
  • LCM of Fractions = (LCM of Numerators) / (HCF of Denominators)

• BODMAS: Bracket → Of → Division → Multiplication → Addition → Subtraction.
• Important Identities:
  • (a+b)² = a² + b² + 2ab
  • (a-b)² = a² + b² - 2ab
  • a² - b² = (a+b)(a-b)
  • (a+b)³ = a³ + b³ + 3ab(a+b)
  • a³ + b³ = (a+b)(a² - ab + b²)
  • a³ - b³ = (a-b)(a² + ab + b²)

• Percentage Change: [(Final - Initial) / Initial] × 100
• Fraction Table (Must Memorize):
  

  1/2 = 50%	1/3 = 33.33%	1/4 = 25%
  1/5 = 20%	1/6 = 16.66%	1/8 = 12.5%
  1/10 = 10%	1/20 = 5%

• Profit %: (Profit / CP) × 100
• Loss %: (Loss / CP) × 100
• Selling Price (SP): CP × [(100 ± P/L%)/100]
• Discount: Always calculated on Marked Price (MP).
• SP (with Discount): MP × [(100 - D%)/100]
• CP vs MP Relation: CP / MP = (100 - D%) / (100 + P%)

• Simple Interest (SI): (P × R × T) / 100
• Compound Interest Amount (A): P(1 + R/100)ᵀ
• Difference (CI - SI):
  • 2 Years: P(R/100)²
  • 3 Years: P(R/100)² × [(300 + R)/100]

• Mean Proportional (a, b): √(ab)
• Third Proportional (a, b): b²/a
• Partnership Rule: Profit Ratio = (Investment A × Time A) : (Investment B × Time B)

• Average: Sum of observations / Number of observations
• Average Speed: 2xy / (x+y) (For equal distances)
• Alligation Rule:
  (Qty Cheaper) / (Qty Dearer) = (Price Dearer - Mean) / (Mean - Price Cheaper)

• Work = Efficiency × Time
• MDH Formula: (M₁×D₁×H₁) / W₁ = (M₂×D₂×H₂) / W₂
• Shortcut: If A takes x days and B takes y days, Together = xy / (x+y) days.

• Distance = Speed × Time
• Conversions:
  km/hr → m/s: Multiply by 5/18
  m/s → km/hr: Multiply by 18/5
• Boats & Streams:
  • Downstream (D) = u + v
  • Upstream (U) = u - v
  • Boat Speed (u) = (D + U) / 2
  • Stream Speed (v) = (D - U) / 2

• If x + 1/x = a, then x² + 1/x² = a² - 2
• If x - 1/x = a, then x² + 1/x² = a² + 2
• If x + 1/x = a, then x³ + 1/x³ = a³ - 3a
• Quadratic Equation (ax² + bx + c = 0):
  Sum of roots (α+β) = -b/a
  Product of roots (αβ) = c/a

• Scalene Triangle: √[s(s-a)(s-b)(s-c)], where s=(a+b+c)/2
• Equilateral Δ: Area = (√3/4)a²; Height = (√3/2)a
• Circle: Area = πr²; Circumference = 2πr
• Rhombus: Area = ½ × d₁ × d₂
• Trapezium: Area = ½ × (Sum of parallel sides) × height

• Cuboid: Vol = lbh; TSA = 2(lb+bh+hl)
• Cube: Vol = a³; TSA = 6a²; Diagonal = a√3
• Cylinder: Vol = πr²h; CSA = 2πrh
• Cone: Vol = (1/3)πr²h; CSA = πrl; l = √(r²+h²)
• Sphere: Vol = (4/3)πr³; TSA = 4πr²
• Hemisphere: Vol = (2/3)πr³; TSA = 3πr²

• Basic: H² = P² + B² (Pythagoras)
• sinθ=P/H, cosθ=B/H, tanθ=P/B
• Values Table:
  

  θ	0°	30°	45°	60°	90°
  sin	0	1/2	1/√2	√3/2	1
  cos	1	√3/2	1/√2	1/2	0
  tan	0	1/√3	1	√3	∞

• Identities:
  sin²θ + cos²θ = 1
  1 + tan²θ = sec²θ

• Sum of angles in a triangle = 180°
• Circle Properties:
  • Angle in a semi-circle is 90°.
  • Tangents from an external point are equal.
  • Radius is perpendicular to Tangent.

• Distance Formula: √[(x₂-x₁)² + (y₂-y₁)²]
• Section Formula: x = (mx₂ + nx₁) / (m+n)
• Slope (m): (y₂ - y₁) / (x₂ - x₁)

• Permutation (Arrangement): ⁿPᵣ = n! / (n-r)!
• Combination (Selection): ⁿCᵣ = n! / [r!(n-r)!]
• Probability: Favorable Outcomes / Total Outcomes
• Coin Toss: Total outcomes = 2ⁿ

• Percentage change: [(New − Old) / Old] × 100
• Share of total: (Part / Whole) × 100
• Average of two groups combined: (n₁×A₁ + n₂×A₂) / (n₁+n₂)
• Pie chart angle: (Value / Total) × 360°
• Bar/line charts: Read axis labels, units (lakhs vs crores), and whether values are absolute or percentage of something.

• If current ages are in ratio a : b, after t years: (a + kt) : (b + kt) only when the same multiple k of years passes for both—usually just add t to each part conceptually.
• n years ago: subtract n from each present age before forming ratio equations.
• Equation method: Let present age of A = x; translate “after 5 years A is twice B” into algebra and solve.

• Treat filling as positive work rate, emptying (leak) as negative.
• If pipe A fills in a hours: part per hour = 1/a. Together: 1/a + 1/b (if both fill).
• Alternate pipes: Find work done in 2-hour (or 1-hour) cycles, then count full cycles toward 1 tank.

• If A beats B by x metres in r m race: while A runs r, B runs r − x. Speed ratio A : B = r : (r − x).
• Head start: “A gives B a start of t seconds” means B begins earlier; distances covered in same running time still follow speed ratio.

1. First-level difference (subtract consecutive terms)
2. Second-level difference (difference of differences)
3. n², n³, n²±n, 2ⁿ mixed patterns
4. Alternating series (two interleaved sequences)
5. Prime numbers, factorials (less common in clerical level)

Always verify the pattern on at least three terms before marking the next term.

• BODMAS strictly; round only when question says “approximate” or options are wide apart.
• Learn squares (1–30), cubes (1–15), and tables up to 25 for faster factoring.
• For ?² near a perfect square: bracket between known squares (e.g. 729 < 800 < 900).

When two quadratics are given, find roots sign (sum = −b/a, product = c/a) or factorize quickly. Compare roots of I and II for relationship (x > y, x < y, x = y, or no clear relation). Practice 10 questions daily—pattern recognition beats deriving from scratch in the hall.

• Minute hand moves 6° per minute; hour hand moves 0.5° per minute → relative speed 5.5° per minute.
• Angle between hands at H:M ≈ |30H − 5.5M|° (take smaller angle ≤ 180° if needed).
• In t minutes, minute hand gains 5.5t degrees on hour hand.

• Leap year: divisible by 4; century years must be divisible by 400.
• Odd days: ordinary year = 1 odd day; leap year = 2 odd days.
• Count net odd days from a reference (e.g. 1 Jan 1 AD Monday) or use code for month + year formula you prefer—consistency in one method matters more than memorizing five systems.

1. Percentage, profit/loss, SI/CI — highest ROI for DI and arithmetic.
2. Ratio, partnership, ages, mixture & alligation
3. Time & work, pipes, time-speed-distance, boats
4. Number system, HCF/LCM, simplification
5. Mensuration + basic trigonometry (SSC-heavy)
6. Permutation, combination, probability (moderate frequency)
7. DI sets daily once arithmetic is 60%+ comfortable