Quant Formula Sheet

Percentages & Profit-Loss14

• a% of b = b% of a
• x% = x/100; to convert decimal→%, ×100
• Successive change a% then b%: net% = a + b + ab/100 (use signs)
• Increase x% then decrease x%: net = −x²/100 (always a loss)
• CP→SP at profit p%: SP = CP(1 + p/100) · at loss: SP = CP(1 − l/100)
• Profit% = (SP−CP)/CP ×100 · Loss% = (CP−SP)/CP ×100
• Marked price M, discount d%: SP = M(1 − d/100)
• Two successive discounts a%, b%: net = a + b − ab/100
• Dishonest seller using false weight: gain% = (error/(true−error))×100
• If A is x% more than B, B is [x/(100+x)]×100% less than A
• Markup on CP = m%, discount d%: profit% = m − d − md/100
• Sell two items at same SP, +p% & −p%: net loss = p²/100 %
• Margin% (on SP) = markup% / (1 + markup%)
• % → fraction: 25%=¼, 20%=⅕, 12.5%=⅛, 16⅔%=⅙, 33⅓%=⅓, 11⅑%=1/9

Time, Speed & Distance14

• Speed = Distance/Time · 1 km/h = 5/18 m/s, 1 m/s = 18/5 km/h
• Average speed = total distance / total time
• Average speed for equal distances at x, y = 2xy/(x+y) (harmonic mean)
• Fixed distance: speed ∝ 1/time; speeds a:b ⇒ times b:a
• Relative speed — same direction: |x−y|; opposite: x+y
• Train crossing pole: t = L/v · crossing platform of length P: t = (L+P)/v
• Two trains lengths L₁,L₂ crossing: t = (L₁+L₂)/relative speed
• Boats: downstream d = b+s, upstream u = b−s; b = (d+u)/2, s = (d−u)/2
• Races: 'A beats B by t s / d m'; head start = distance given
• Circular track (start same pt): meet time = LCM of individual lap-meeting times
• Two bodies towards each other: meet time = d/(u+v); distances ∝ speeds
• After crossing, t = √(t₁·t₂); speed ratio = √(t₂/t₁)
• Escalator: visible steps = (man's rate ± escalator rate) × time taken
• Early/late by t: speed s₁ vs s₂ ⇒ distance from (d/s₁ − d/s₂) = total time gap

Time & Work7

• Work rate = 1/days; combined rate = sum of individual rates
• A in a days, B in b days, together = ab/(a+b) days
• M₁D₁H₁/W₁ = M₂D₂H₂/W₂ (men·days·hours per work)
• Efficiency ∝ 1/time; if A:B efficiency = 2:1, A takes half the time
• Work done = rate × time; total work = LCM(individual days) units
• Pipes & cisterns: filling rate +, emptying rate − ; net = sum
• If a leak empties a full tank in x h, it reduces fill rate by 1/x

Averages, Mixtures & Alligation11

• Average = (sum of terms)/(number of terms)
• Average of first n naturals = (n+1)/2
• Average of n consecutive numbers = (first+last)/2
• New average after adding a value = (n·old + new)/(n+1)
• Alligation: (cheaper qty)/(dearer qty) = (d − mean)/(mean − c)
• Mean price = (c·q₁ + d·q₂)/(q₁+q₂)
• Repeated replacement: final/initial = (1 − r/V)ⁿ after n draws of r from V
• Weighted average = Σ(value×weight)/Σweight
• Average of an AP = middle term (odd count) = (first+last)/2
• Adding a term shifts avg by (new − old avg)/(n+1)
• Two groups merged: avg = (n₁a₁ + n₂a₂)/(n₁+n₂)

Ratio, Proportion & Variation6

• a:b = c:d ⇒ ad = bc (product of means = product of extremes)
• If a/b = c/d = e/f = k, then (a+c+e)/(b+d+f) = k
• Componendo-dividendo: (a+b)/(a−b) = (c+d)/(c−d)
• Mean proportional of a & b = √(ab) · third proportional = b²/a
• Direct variation: a ∝ b ⇒ a = kb · Inverse: a ∝ 1/b ⇒ ab = k
• If a:b = m:n and b:c = p:q ⇒ a:b:c = mp:np:nq

Simple & Compound Interest10

• SI = P·R·T/100 · Amount = P + SI
• CI: Amount = P(1 + R/100)ᵀ · CI = Amount − P
• Half-yearly: rate R/2, time 2T · Quarterly: R/4, 4T
• CI for 2 yrs at r%: CI = P[2r/100 + (r/100)²]
• Difference (CI − SI) for 2 yrs = P(r/100)²
• Difference for 3 yrs = P(r/100)²·(3 + r/100)
• Population/depreciation: future = present(1 ± r/100)ⁿ
• Sum doubles in SI in 100/R years; triples in 200/R years
• If a sum becomes x times in t yrs (CI), in 2t yrs it becomes x² times
• Effective annual rate (compounded k times) = (1 + r/k)ᵏ − 1

Number Systems16

• For N = pᵃ·qᵇ·… : number of divisors = (a+1)(b+1)…
• Sum of divisors = Π (pᵃ⁺¹ − 1)/(p − 1)
• Product of divisors = N^(d/2), d = number of divisors
• HCF × LCM = product of the two numbers
• HCF of fractions = HCF(num)/LCM(den) · LCM = LCM(num)/HCF(den)
• Unit digit cycles repeat with period 4 (e.g. 2,4,8,6…)
• Divisibility: 3/9 → digit sum; 11 → alt-sum; 8 → last 3 digits
• Sum 1..n = n(n+1)/2 · squares = n(n+1)(2n+1)/6 · cubes = [n(n+1)/2]²
• Highest power of prime p in n! = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …
• Remainders: (a·b) mod m = [(a mod m)(b mod m)] mod m
• Fermat: if p prime, a^(p−1) ≡ 1 (mod p) for gcd(a,p)=1
• Number of trailing zeros in n! = power of 5 in n!
• Number of even divisors of N = total divisors − odd divisors
• Pairs (a,b) with a×b = N (ordered) = number of divisors of N
• N is a perfect square ⇔ it has an odd number of divisors
• Co-prime pairs ≤ N counted via Euler's totient φ(N)

Surds, Indices & Algebra15

• aᵐ·aⁿ = aᵐ⁺ⁿ · aᵐ/aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ · a⁰ = 1
• a⁻ⁿ = 1/aⁿ · a^(1/n) = ⁿ√a · a^(m/n) = ⁿ√(aᵐ)
• (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)
• 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 then a³+b³+c³ = 3abc
• Quadratic ax²+bx+c=0: roots = (−b ± √(b²−4ac))/2a
• Sum of roots = −b/a · product = c/a
• Discriminant b²−4ac: >0 real & distinct, =0 equal, <0 imaginary
• Sum of squares of roots = (sum)² − 2·product
• Rationalise: 1/(√a+√b) = (√a−√b)/(a−b)
• Irrational/surd roots occur in conjugate pairs (p ± √q)
• x²−Sx+P: roots real & positive ⇒ S>0, P>0, S² ≥ 4P

Logarithms10

• log_a(xy) = log_a x + log_a y
• log_a(x/y) = log_a x − log_a y
• log_a(xⁿ) = n·log_a x
• Change of base: log_a b = log b / log a = 1/log_b a
• a^(log_a x) = x · log_a a = 1 · log_a 1 = 0
• log_a b · log_b c = log_a c (chain)
• Number of digits in N = ⌊log₁₀N⌋ + 1
• log₁₀2 ≈ 0.3010, log₁₀3 ≈ 0.4771, log₁₀7 ≈ 0.8451
• a^(log_b c) = c^(log_b a)
• log is increasing: if x>y>0 then log_a x > log_a y (a>1)

Progressions (AP / GP / HP)12

• AP nth term: Tₙ = a + (n−1)d
• AP sum: Sₙ = n/2 [2a + (n−1)d] = n × (first+last)/2
• Three terms in AP: a−d, a, a+d
• GP nth term: Tₙ = a·rⁿ⁻¹
• GP sum: Sₙ = a(rⁿ−1)/(r−1), r≠1
• Infinite GP (|r|<1): S∞ = a/(1−r)
• GM of a & b = √(ab) · HP: reciprocals form an AP
• Relation: GM² = AM × HM
• n arithmetic means between a & b: common diff d = (b−a)/(n+1)
• Terms equidistant from ends of an AP have a constant sum
• GP: product of terms equidistant from ends is constant
• Sum of squares of an AP / GP — use Σn² and GP-sum identities

Permutations & Combinations11

• ⁿPr = n!/(n−r)! · ⁿCr = n!/(r!(n−r)!)
• ⁿCr = ⁿCₙ₋ᵣ · ⁿCr + ⁿCr₋₁ = ⁿ⁺¹Cr
• ⁿC₀ + ⁿC₁ + … + ⁿCₙ = 2ⁿ
• Arrangements of n with p,q alike = n!/(p!·q!)
• Circular arrangement of n = (n−1)! · with reflection = (n−1)!/2
• Selecting ≥1 from n distinct = 2ⁿ − 1
• Distributing n identical into r groups = ⁿ⁺ʳ⁻¹Cᵣ₋₁
• Identical into r groups, each ≥1 = ⁿ⁻¹Cᵣ₋₁
• Selecting r of n in a row, no two adjacent = ⁿ⁻ʳ⁺¹Cr
• Sum of all r-digit numbers from n distinct digits uses (n−1)! repetition
• Number of rectangles in m×n grid = ⁽ᵐ⁺¹⁾C₂ × ⁽ⁿ⁺¹⁾C₂

Probability11

• P(event) = favourable outcomes / total outcomes (0 ≤ P ≤ 1)
• P(not A) = 1 − P(A)
• P(A∪B) = P(A) + P(B) − P(A∩B)
• Independent events: P(A∩B) = P(A)·P(B)
• Mutually exclusive: P(A∩B) = 0
• Conditional: P(A|B) = P(A∩B)/P(B)
• Odds in favour = favourable : unfavourable
• At least one of independent events = 1 − Π(1 − pᵢ)
• Bayes: P(A|B) = P(B|A)·P(A) / P(B)
• Expected value E = Σ(outcome × probability)
• Binomial: P(r successes) = ⁿCr·pʳ·(1−p)ⁿ⁻ʳ

Set Theory & Venn6

• |A∪B| = |A| + |B| − |A∩B|
• |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |B∩C| − |C∩A| + |A∩B∩C|
• Exactly two = Σ(pairwise) − 3×(all three)
• Exactly one = total in single regions only
• Neither (none) = Total − |A∪B∪C|
• De Morgan: (A∪B)' = A'∩B' · (A∩B)' = A'∪B'

Geometry — Triangles & Circles16

• Angle sum of triangle = 180° · exterior angle = sum of remote interiors
• Pythagoras: a² + b² = c² (right triangle)
• Area = ½·base·height = ½ab·sinC = √(s(s−a)(s−b)(s−c)), s=(a+b+c)/2
• Equilateral side a: area = (√3/4)a², height = (√3/2)a
• Similar triangles: sides proportional, areas ∝ side²
• Mid-point theorem: segment joining midpoints ∥ base & half its length
• Circle: circumference = 2πr, area = πr²
• Arc length = (θ/360)·2πr · Sector area = (θ/360)·πr²
• Angle at centre = 2 × angle at circumference (same arc)
• Tangent ⟂ radius at point of contact; tangents from a point are equal
• Cyclic quadrilateral: opposite angles sum to 180°
• Intersecting chords: AP·PB = CP·PD
• Tangent-secant from external point: PT² = PA·PB
• Area of triangle = r·s (inradius × semi-perimeter)
• Exterior angle = sum of the two remote interior angles
• Direct common tangent length = √(d² − (r₁−r₂)²) · transverse = √(d² − (r₁+r₂)²)

Mensuration (Solids)6

• Cuboid: V = lbh · TSA = 2(lb+bh+hl) · diagonal = √(l²+b²+h²)
• Cube: V = a³ · TSA = 6a² · diagonal = a√3
• Cylinder: V = πr²h · CSA = 2πrh · TSA = 2πr(r+h)
• Cone: V = ⅓πr²h · CSA = πrl · l = √(r²+h²) · TSA = πr(r+l)
• Sphere: V = 4/3πr³ · SA = 4πr²
• Hemisphere: V = 2/3πr³ · TSA = 3πr²

Coordinate Geometry11

• Distance = √((x₂−x₁)² + (y₂−y₁)²)
• Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
• Section (ratio m:n) = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
• Slope m = (y₂−y₁)/(x₂−x₁) · line: y = mx + c
• Parallel: equal slopes · Perpendicular: m₁·m₂ = −1
• Area of triangle = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
• Distance of point (x₁,y₁) from ax+by+c=0 = |ax₁+by₁+c|/√(a²+b²)
• Centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
• Three points collinear ⇔ area of triangle = 0
• Circle: (x−h)² + (y−k)² = r² · general x²+y²+2gx+2fy+c=0
• Angle between lines: tanθ = |(m₁−m₂)/(1+m₁m₂)|

Trigonometry (basics)10

• sinθ = opp/hyp · cosθ = adj/hyp · tanθ = sinθ/cosθ
• sin²θ + cos²θ = 1 · 1 + tan²θ = sec²θ · 1 + cot²θ = cosec²θ
• Values 0°/30°/45°/60°/90° sin: 0, ½, 1/√2, √3/2, 1
• Heights & distances: tanθ = height / horizontal distance
• sin(90−θ) = cosθ · tan(90−θ) = cotθ
• sin(A±B) = sinA cosB ± cosA sinB
• cos(A±B) = cosA cosB ∓ sinA sinB
• sin2θ = 2 sinθ cosθ · cos2θ = 1 − 2sin²θ = 2cos²θ − 1
• Sine rule: a/sinA = b/sinB = c/sinC = 2R
• Cosine rule: c² = a² + b² − 2ab·cosC

Clocks & Calendars6

• Minute hand gains 5½° per minute over hour hand
• Hands coincide 11 times in 12 h (every 65 5/11 min)
• Angle between hands = |30H − 5.5M| degrees
• Odd days decide the day; 1 ordinary year = 1 odd day, leap = 2
• 100 yrs = 5 odd days, 400 yrs = 0 odd days
• Leap year: divisible by 4 (century only if divisible by 400)

Functions & Inequalities11

• Even function: f(−x) = f(x) · Odd: f(−x) = −f(x)
• |x| < a ⇒ −a < x < a · |x| > a ⇒ x < −a or x > a
• Parabola ax²+bx+c: vertex at x = −b/2a
• Min of ax²+bx+c (a>0) = c − b²/4a (max if a<0)
• For positives, x + 1/x ≥ 2 (equality at x = 1); for x<0, ≤ −2
• AM ≥ GM ≥ HM (all equal ⇒ equality)
• Product fixed ⇒ sum min when terms equal; sum fixed ⇒ product max when equal
• Triangle inequality: |a| + |b| ≥ |a + b|
• For positives: (a+b+c)(1/a+1/b+1/c) ≥ 9
• Min of |x−a| + |x−b| = |a − b| (between a and b)
• fog(x) = f(g(x)); inverse: f(f⁻¹(x)) = x

Polygons & Quadrilaterals7

• Sum of interior angles of n-gon = (n−2)×180°
• Regular n-gon: each interior = (n−2)180/n · exterior = 360/n
• Number of diagonals = n(n−3)/2
• Parallelogram area = base × height
• Rhombus area = ½·d₁·d₂ · all sides equal
• Trapezium area = ½(sum of parallel sides) × height
• Square: diagonal = a√2 · area from diagonal = d²/2

Approximations & shortcuts6

• √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √7 ≈ 2.646
• π ≈ 3.1416 (22/7) · e ≈ 2.718
• 1/7 = 14.28%, 1/9 = 11.11%, 1/11 = 9.09%, 1/12 = 8.33%, 1/13 = 7.69%
• Product constancy: if one factor ↑ x%, other ↓ [x/(100+x)]×100% to keep product same
• Compare fractions: cross-multiply, or compare each to a benchmark like ½
• Squares to know: 11²–20² and 25², 35² (n5² = n(n+1) then 25)

Remainders & Cyclicity (advanced)8

• Euler's totient φ(N): for N = pᵃqᵇ, φ = N(1−1/p)(1−1/q)
• Euler's theorem: a^φ(N) ≡ 1 (mod N) when gcd(a,N) = 1
• Fermat's little: a^(p−1) ≡ 1 (mod p) for prime p, gcd(a,p)=1
• Wilson's theorem: (p−1)! ≡ −1 (mod p) for prime p
• Last digit: powers cycle with period 1, 2 or 4
• Last two digits of large powers: reduce exponent mod 20 (for many bases)
• Digital root (repeated digit sum) ≡ number (mod 9)
• Chinese Remainder: unique solution mod (m·n) when m, n coprime

Geometry — special triangles & theorems9

• 45-45-90 sides 1 : 1 : √2 · 30-60-90 sides 1 : √3 : 2
• Angle bisector divides opposite side in ratio of the other two sides
• Median to hypotenuse = ½ × hypotenuse (right triangle)
• Centroid divides each median in 2 : 1 from the vertex
• Inradius r = Area / s · Circumradius R = abc / (4·Area)
• Equilateral side a: R = a/√3, r = a/(2√3), so R = 2r
• Apollonius: AB² + AC² = 2(AD² + BD²), D = midpoint of BC
• Ptolemy (cyclic quad): AC·BD = AB·CD + AD·BC
• Stewart, basic proportionality (Thales): DE ∥ BC ⇒ AD/DB = AE/EC

Counting — advanced6

• Derangements Dₙ = n!(1 − 1/1! + 1/2! − … ± 1/n!)
• D₂=1, D₃=2, D₄=9, D₅=44
• Partitions of N into consecutive integers = (number of odd divisors of N) − 1
• Perfect-square factors of N = Π(⌊eᵢ/2⌋ + 1) over prime powers eᵢ
• Paths in grid (only right/up), m×n = ⁽ᵐ⁺ⁿ⁾Cₘ
• Handshakes among n people = ⁿC₂ = n(n−1)/2