CAT Quant formula sheet

r% = r / 100           x% of y = y% of x
% change = (new − old) / old × 100
+r% → × (1 + r/100)    −r% → × (1 − r/100)
two changes: a + b + ab/100
r% more → r/(100 + r) less     r% less → r/(100 − r) more
1/6 = 16.67%   1/7 = 14.29%   1/8 = 12.5%   1/9 = 11.11%   1/11 = 9.09%

Profit % = (SP − CP) / CP × 100       SP = CP × (1 + p/100)
MP = CP × (1 + m/100)    SP = MP × (1 − d/100)
markup m, discount d → profit = m − d − md/100     MP/CP = (100 + p)/(100 − d)
discounts a, b → single = a + b − ab/100
false weight w for 1000 → gain = (1000 − w) / w
buy x get y free → discount = y / (x + y)
CP of n = SP of x → profit = (n − x) / x
same SP, ±r% → loss = r²/100 %

SI = PRT/100            A = P(1 + R/100)^T
k times a year: R/k per period, kT periods
CI − SI, 2 yr = PR²/100²    3 yr = PR²(300 + R)/100³
SI doubles in 100/R yr     CI doubles in ≈ 72/R yr
Loan = x/(1 + r) + x/(1 + r)²
Depreciation: P(1 − R/100)^n
CAGR = (F/I)^(1/n) − 1  <  simple annual rate = total % ÷ n

a : b → a + b parts;  one part = total ÷ (a + b)
a:b and b:c → match b with the LCM;  A/C = (A/B)(B/C)
kA = mB = nC  →  A : B : C = 1/k : 1/m : 1/n
a : b = c : d ⇔ ad = bc;  mean prop. √(ab)
a/b = p/q ⇔ (a + b)/(a − b) = (p + q)/(p − q)
y ∝ x: y/x fixed;  y ∝ 1/x: xy fixed;  part fixed: y = a + bx
a/b = c/d = (a + c)/(b + d)     duplicate a² : b²   sub-duplicate √a : √b
partnership: share ∝ capital × months

Average = sum ÷ n = assumed + (sum of deviations) ÷ n
Joins: A + (x − A)/(n + 1)    Leaves: A − (x − A)/(n − 1)
Replace: change = (in − out) ÷ n
Weighted: (n₁a₁ + n₂a₂)/(n₁ + n₂);  distances in ratio n₂ : n₁
Equal distances: 2ab/(a + b)    Equal times: (a + b)/2
Equally spaced: average = (first + last)/2 = middle term
Ages: same group, t years on → average + t

mean = (c₁q₁ + c₂q₂) / (q₁ + q₂)
q₁ : q₂ = (c₂ − mean) : (mean − c₁)
water = 0% solution, pure substance = 100%
remove x of V, n times: V × (1 − x/V)ⁿ left;  different draws: multiply each (1 − xᵢ/V)
sold at CP: profit % = water / milk × 100

job = LCM of days     rate = job ÷ days     rates add, days do not
two people: ab / (a + b)
pairs: (A+B) + (B+C) + (C+A) = 2(A+B+C)
efficiency ratio = inverse of time ratio
M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂
together: abc/(ab + bc + ca)     A alone from A+B (x) and B (y): xy/(y − x)
emptying pipe or leak = negative rate;  fill a, empty b → ab/(b − a)

D = S × T     D fixed: speed ratio = inverse time ratio
km/h × 5/18 = m/s
equal distances: average = 2ab / (a + b);  three: 3abc / (ab + bc + ca)
relative speed: a + b opposite, a − b same way
train crossing: (total length to clear) / relative speed
boats: D = b + s, U = b − s;  b = (D + U)/2, s = (D − U)/2
track: first meeting = L / relative speed

beats by x m in L:  A : B = L : (L − x)     by x m or t s: B's speed = x/t
dead heat: start = margin     chains: multiply ratios
track: first meeting = L/(a − b) same way, L/(a + b) opposite
spots (p : q lowest terms): p − q same way, p + q opposite
all at start: LCM of lap times     all together: LCM of pair meeting times
clock angle = |30H − 5.5M|     M = (2/11)(30H ± θ)
12 h: 11 overlaps, 11 opposites, 22 right angles
gains g min/h: real = shown × 60/(60 + g)

a₁/a₂ ≠ b₁/b₂ → one solution;  = b-ratio ≠ c-ratio → none;  all equal → infinite
ax + by = c in integers: needs gcd(a, b) | c; step x by b, y by a
× or ÷ by a negative → flip the inequality
|x − a| = distance from x to a
|x − a| < r ↔ a − r < x < a + r      |x − a| > r ↔ outside
Σ|x − pᵢ| is least at the median of the pᵢ
AM ≥ GM ≥ HM;  GM² = AM × HM;  a + b fixed → aᵐbⁿ greatest at a/m = b/n
|x + y| ≤ |x| + |y|    |x − y| ≥ ||x| − |y||    (a + x)/(b + x) moves a/b towards 1

x = (−b ± √D) / 2a,   D = b² − 4ac
D < 0 none · D = 0 equal · D > 0: rational if D is a perfect square, else p ± √q
α + β = −b/a   αβ = c/a   (α − β)² = (α + β)² − 4αβ
roots α, β  →  x² − (α + β)x + αβ = 0
cubic ax³ + bx² + cx + d:  Σα = −b/a   Σαβ = c/a   αβγ = −d/a
a > 0: negative between the roots, positive outside
vertex x = −b/2a, extreme value −D/4a

domain: no √(negative), no ÷ 0, log only of positives
f(g(x)): g first.   f⁻¹: swap x and y, mirror in y = x
even: f(−x) = f(x)   odd: f(−x) = −f(x)
a·f(x − h) + k: right h, up k, stretch a;   f(−x): flip in y-axis   −f(x): flip in x-axis
|f(x)| folds up;  f(|x|) mirrors the right half
min of max(rising, falling) at the crossing
a + b ≥ 2√(ab), equality at a = b

aᵐ × aⁿ = aᵐ⁺ⁿ    aᵐ ÷ aⁿ = aᵐ⁻ⁿ    (aᵐ)ⁿ = aᵐⁿ    a⁻ⁿ = 1/aⁿ
compare powers: match the index (HCF) or the base
1/(√a − √b) = (√a + √b)/(a − b)
log_b x = y ⟺ bʸ = x;   x > 0, b > 0, b ≠ 1
log xy = log x + log y    log xⁿ = n log x
log_a b = log b / log a    log_a b × log_b a = 1
digits of N = ⌊log N⌋ + 1     log 2 ≈ 0.301   log 3 ≈ 0.477   log 7 ≈ 0.845

AP: tₙ = a + (n − 1)d     Sₙ = n/2 × (a + l) = average × count
GP: tₙ = a·rⁿ⁻¹     Sₙ = a(rⁿ − 1)/(r − 1)     S∞ = a/(1 − r), |r| < 1
AM ≥ GM ≥ HM     GM² = AM × HM     middle term: AP 2b = a + c,  GP b² = ac,  HP b = 2ac/(a + c)
Σk = n(n + 1)/2   Σk² = n(n + 1)(2n + 1)/6   Σk³ = (Σk)²
1/(k(k + d)) = (1/d)(1/k − 1/(k + d))

(a + b)ⁿ = Σ nCr aⁿ⁻ʳ bʳ (r = 0 … n)     n + 1 terms     nCr = nC(n − r)
Tᵣ₊₁ = nCr aⁿ⁻ʳ bʳ     kth term: r = k − 1
xᵏ or constant term: power of x in Tᵣ₊₁ = k or 0; r must be whole, 0 ≤ r ≤ n
Middle: n even → T(n/2 + 1); n odd → two middle terms     largest nCr in the middle
Σ coefficients = f(1)     Σ nCr = 2ⁿ     odd places = even places = 2ⁿ⁻¹
Even powers: [f(1) + f(−1)]/2     odd powers: [f(1) − f(−1)]/2
(dm + 1)ⁿ leaves 1 ÷ d     (dm − 1)ⁿ leaves (−1)ⁿ     11ⁿ ends like 1 + 10n
(1 + x)ⁿ ≈ 1 + nx + n(n − 1)x²/2 for small x

prime test: divide by primes up to √n;  primes > 3 are 6k ± 1
squares end in 0, 1, 4, 5, 6, 9;  n consecutive integers → divisible by n!
8: last 3 digits;  11: alternate-digit difference;  7: tens + 5 × units
N = pᵃ qᵇ rᶜ  →  (a + 1)(b + 1)(c + 1) factors
sum of factors = (1 + … + pᵃ)(1 + … + qᵇ)(1 + … + rᶜ)
φ(N) = N(1 − 1/p)(1 − 1/q)…;  co-prime factor pairs = 2^(m − 1)
HCF: lower powers of common primes   LCM: higher powers of all
HCF × LCM = A × B (two numbers)
repeat and meet → LCM    cut into equal pieces → HCF
zeros of n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …

N = n × q + r,  0 ≤ r < n;  r ⇔ r − n
(a ± b), (a × b): work with the remainders, reduce again
unit digits: cycles of 1, 2 or 4 → power mod 4 (0 → 4th)
…a1^n ends in [a × n]1;  76, 25 stay; 24^even → 76
2^10 → 24, 2^20 → 76 (last two digits)
Fermat a^(p−1) → 1;  Euler a^φ(n) → 1 (no common factor)
Wilson (p − 1)! → p − 1  (p prime)
hunt for a^k → ±1;  p(x) ÷ (x − a) leaves p(a)
base b digits = remainders of ÷ b, read bottom up

Parallels: corresponding =, alternate =, co-interior sum 180°
A + B + C = 180°   exterior = sum of opposite interiors
|a − b| < c < a + b
c² = a² + b² right, > obtuse, < acute
45-45-90 → 1 : 1 : √2   30-60-90 → 1 : √3 : 2   30-30-120 → 1 : 1 : √3
Area = ½bh = ½ab sin C = √(s(s−a)(s−b)(s−c)) = rs = abc/(4R)
Similar: sides k, areas k²   same height: areas ∝ bases
Mid-point: DE = ½BC   bisector: BD/DC = AB/AC
Apollonius: AB² + AC² = 2(AD² + BD²)
G cuts medians 2 : 1   O, H outside if obtuse   O, G, H collinear, OG : GH = 1 : 2

Chord: r² = d² + (c/2)²
∠ at centre = 2 × ∠ at circumference;  semicircle → 90°
Same segment: equal angles;  cyclic quad: opposite ∠s = 180°
Tangent ⟂ radius;  PT = PT′ = √(d² − r²)
Common tangents: direct √(d² − (r₁ − r₂)²), transverse √(d² − (r₁ + r₂)²)
Apart 4, touching outside 3, cutting 2, touching inside 1, inside 0
PA × PB = PC × PD;  PT² = PA × PB;  tangent-chord ∠ = alternate segment ∠
Arc = θ/360 × 2πr;  sector = θ/360 × πr²
Interior sum = (n − 2) × 180°;  exterior = 360°/n
Diagonals = n(n − 3)/2
Rhombus, kite ½d₁d₂;  trapezium ½(a + b)h;  parallelogram bh
Cyclic quad: √((s − a)(s − b)(s − c)(s − d))
Regular hexagon: (3√3/2)a², diagonals 2a and a√3

Cuboid: lbh, 2(lb + bh + hl), √(l² + b² + h²)     Cube: a³, 6a², a√3
Cylinder: πr²h, curved 2πrh, total 2πr(r + h)
Cone: ⅓πr²h, curved πrl, l = √(r² + h²)
Frustum: ⅓πh(R² + Rr + r²), curved π(R + r)L
Sphere: ⁴⁄₃πr³, 4πr²     Hemisphere: ⅔πr³, curved 2πr², total 3πr²
Prism: base × h     Pyramid: ⅓ × base × h, lateral ½ × perimeter × slant
Sphere in cube r = a/2     cube in sphere a√3 = 2R
Melting: volume before = volume after     Scaling: k, k², k³

AB = √(Δx² + Δy²)       midpoint = average of coordinates
m : n internal: (m·x₂ + n·x₁)/(m + n)   external: (m·x₂ − n·x₁)/(m − n)
centroid = (Σx/3, Σy/3)   incentre = (Σa·x/(a + b + c), Σa·y/(a + b + c))
m = Δy/Δx = tan θ;  ax + by + c = 0 → m = −a/b
y − y₁ = m(x − x₁)   x/a + y/b = 1   two-point: slope from both points
parallel: m₁ = m₂     perpendicular: m₁m₂ = −1 (a₁a₂ + b₁b₂ = 0)
angle between lines: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
point to line: |ax₁ + by₁ + c| / √(a² + b²)   parallel lines: |c₁ − c₂| / √(a² + b²)
area = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|;  area 0 ⇔ collinear
circle: (x − h)² + (y − k)² = r²;  general: centre (−g, −f), r = √(g² + f² − c)
reflect: x-axis (x, −y), y = x (y, x), point (a, b) → (2a − x, 2b − y)
|x| + |y| ≤ a → 2a²     |x|/a + |y|/b ≤ 1 → 2ab

sin = opp/hyp,  cos = adj/hyp,  tan = opp/adj = sin/cos
cosec = 1/sin,  sec = 1/cos,  cot = 1/tan
sin: 0, 1/2, 1/√2, √3/2, 1 at 0°, 30°, 45°, 60°, 90° (cos reversed)
sin² + cos² = 1,  1 + tan² = sec²,  1 + cot² = cosec²
sin(90° − θ) = cos θ,  tan(90° − θ) = cot θ,  sec(90° − θ) = cosec θ
Quadrants: All, Sin, Tan, Cos positive in I, II, III, IV
height = distance × tan(elevation);  depression = elevation from below
Same side: AB = h(cot α − cot β);  30° → 60° walk = 2 × remaining
Area = ½ab sin C = abc/(4R);  a/sin A = 2R
a² = b² + c² − 2bc cos A
sin(A ± B) = sin A cos B ± cos A sin B;  sin 2A = 2 sin A cos A

AND → ×       OR → +
nPr = n!/(n−r)!        n^r with repeats
Repeats: n!/(p! q! …)   Round table: (n−1)!
Necklace: (n−1)!/2      nCr = n!/(r!(n−r)!)
Together: glue          Apart: gaps b! × (b+1)Pg
Identical to r people: (n+r−1)C(r−1)
At least 1 each:       (n−1)C(r−1)
Different items to r people: rⁿ
Unnamed equal groups: divide by k!
D(n): 0, 1, 2, 9, 44, 265

P(E) = favourable / total
P(at least one) = 1 − P(none)
Independent: P(A and B) = P(A) P(B)
P(A or B) = P(A) + P(B) − P(A and B)
P(A | B) = P(A and B) / P(B)
E = Σ x·p        Odds a : b ⇔ P = a/(a+b)
Two dice: sum s in (s−1) or (13−s) ways
Exactly r of n: nCr pʳ(1 − p)ⁿ⁻ʳ     Bayes: P(A|B) = P(A) P(B|A) / P(B)

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Total = union + none
n(A∪B∪C) = ΣA − Σ(pairs) + n(A∩B∩C)
ΣA = E1 + 2E2 + 3E3     union = E1 + E2 + E3
Subsets of an n-set: 2ⁿ (proper 2ⁿ − 1)   (A ∪ B)′ = A′ ∩ B′
Two sets: both ∈ [max(0, a+b−T), min(a, b)]
Two sets: neither ∈ [max(0, T−a−b), T − max(a, b)]
Three sets (%): all three ≥ a + b + c − 200