GATE Aptitude Master Formula Cheatsheet
Complete high-yield mathematical formulas, shortcuts, and theorems for last-minute GATE revision
50+ High-Yield Formulas
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Percentages, Profit & Loss
9 Formulas Percentage Increase / Decrease
[(New Value - Old Value) / Old Value] × 100% Baseline Shift (A is x% more than B)
B is [x / (100 + x)] × 100% less than A Consumption Reduction for Same Budget
Reduction % = [r / (100 + r)] × 100% Successive Percentage Changes (x% and y%)
Net % Change = x + y + (x × y)/100 Profit Percentage
Profit % = [(Selling Price - Cost Price) / Cost Price] × 100% Loss Percentage
Loss % = [(Cost Price - Selling Price) / Cost Price] × 100% Discount Percentage
Discount % = [(Marked Price - Selling Price) / Marked Price] × 100% Dishonest Dealer Profit (Using Faulty Weight)
Gain % = [Error / (True Weight - Error)] × 100% Effective Single Discount from D1 and D2
Net Discount = D1 + D2 - (D1 × D2)/100 ⚖️
Ratio, Proportion & Mixtures
6 Formulas Duplicate & Triplicate Ratios
Duplicate of a:b = a²:b² | Triplicate of a:b = a³:b³ Compounded Ratio of (a:b) and (c:d)
Compounded Ratio = (a × c) : (b × d) Mean Proportional between A and B
Mean Proportional = √(A × B) Third Proportional to A and B
C = B² / A Allegation Formula (Price of Mixture)
(Cheaper Qty / Dearer Qty) = (Dearer Price - Mean Price) / (Mean Price - Cheaper Price) Repeated Liquid Replacement
Final Pure Qty = Initial Qty × (1 - x / V)^n 🚀
Speed, Time, Distance & Trains
8 Formulas Unit Conversion (km/h to m/s)
1 km/h = 5/18 m/s | 1 m/s = 18/5 km/h Average Speed (Equal Distances)
Average Speed = 2xy / (x + y) Average Speed (3 Equal Distances x, y, z)
Avg Speed = 3xyz / (xy + yz + zx) Relative Speed (Same Direction)
V_rel = V1 - V2 Relative Speed (Opposite Direction)
V_rel = V1 + V2 Time to Cross Platform (Length L_train + L_plat)
Time = (L_train + L_platform) / Speed Boats in Upstream & Downstream
Upstream = B - S | Downstream = B + S | Boat Speed B = (D + U)/2 Circular Track First Meeting (Opposite Direction)
Time = Track Length / (V1 + V2) ⏱️
Time and Work / Pipes & Cisterns
6 Formulas Combined Work Time (A in x days, B in y days)
Combined Time = (x × y) / (x + y) days Three Workers Combined (x, y, z days)
Combined Time = (xyz) / (xy + yz + zx) days Work Efficiency Relation
Work Done = Efficiency × Time | Efficiency ∝ 1 / Time Man-Day-Hour Formula
(M1 × D1 × H1) / W1 = (M2 × D2 × H2) / W2 Inlet & Outlet Pipes Combined Fill Time
Net Fill Rate = 1/Inlet - 1/Outlet Alternate Days Work Cycle
1 Cycle (2 days) = (1/A + 1/B) part completed 🔢
Number System & Remainder Theorems
9 Formulas HCF × LCM Property
HCF(A, B) × LCM(A, B) = A × B HCF & LCM of Fractions
HCF = HCF(Numerators) / LCM(Denominators) Cyclicity of Digits mod 10
Cyclicity of 2,3,7,8 = 4 | 4,9 = 2 | 0,1,5,6 = 1 Fermat’s Little Theorem
a^(p-1) ≡ 1 (mod p) where p is prime and gcd(a,p) = 1 Euler’s Totient Function φ(n)
φ(n) = n × (1 - 1/p1) × (1 - 1/p2)... Number of Factors of N = p^a × q^b
Total Factors = (a + 1)(b + 1) Sum of Factors of N = p^a × q^b
Sum = [(p^(a+1) - 1)/(p - 1)] × [(q^(b+1) - 1)/(q - 1)] Remainder of (x + 1)^n divided by x
Remainder is always 1 for all n Remainder of (x - 1)^n divided by x
Remainder is 1 (if n is even) or (x - 1) (if n is odd) 📐
Algebra, Logarithms & Series
9 Formulas Symmetric Quadratic Identity 1
x² + 1/x² = (x + 1/x)² - 2 Symmetric Quadratic Identity 2
x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) Sum of First n Natural Numbers
S_n = n(n + 1) / 2 Sum of Squares of n Numbers
S_n² = n(n + 1)(2n + 1) / 6 Sum of Cubes of n Numbers
S_n³ = [n(n + 1) / 2]² Arithmetic Progression (AP)
T_n = a + (n - 1)d | S_n = (n/2)[2a + (n - 1)d] Geometric Progression (GP)
T_n = a·r^(n-1) | S_n = a(r^n - 1)/(r - 1) | S_inf = a/(1 - r) Logarithm Base Change Formula
log_b (a) = log_c (a) / log_c (b) | log_a (b) = 1 / log_b (a) Logarithm Product & Quotient Rules
log(x·y) = log(x) + log(y) | log(x/y) = log(x) - log(y) 🔺
Geometry & Mensuration 2D & 3D
10 Formulas Equilateral Triangle Area & Height
Area = (√3 / 4) × a² | Height = (√3 / 2) × a Inradius of Equilateral Triangle
r = a / (2√3) Circumradius of Equilateral Triangle
R = a / √3 Heron’s Formula (Scalene Triangle)
Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2 Rhombus Area & Side Length
Area = (d1 × d2) / 2 | Side = √[(d1/2)² + (d2/2)²] Circle Area, Perimeter & Sector
Area = π r² | Circumference = 2πr | Sector Area = (θ/360) × π r² Cube Surface Area & Volume
Total Surface Area = 6a² | Volume = a³ | Diagonal = a√3 Cylinder Surface Area & Volume
Curved SA = 2πrh | Total SA = 2πr(h + r) | Volume = π r² h Cone Surface Area & Volume
Slant Height l = √(r² + h²) | Total SA = π r(l + r) | Vol = (1/3)π r² h Sphere Surface Area & Volume
Surface Area = 4π r² | Volume = (4/3)π r³ 🎲
Probability, Combinatorics & Statistics
10 Formulas Permutation vs Combination
nPr = n! / (n - r)! | nCr = n! / [r! (n - r)!] Circular Permutations (n distinct objects)
Ways = (n - 1)! Probability of Event E
P(E) = Favorable Outcomes / Total Sample Space N(S) Complementary Probability
P(E') = 1 - P(E) Conditional Probability P(A|B)
P(A|B) = P(A ∩ B) / P(B) Bayes’ Theorem
P(A|B) = [P(B|A) × P(A)] / P(B) Standard Dice Sum Shortcut (2 Dice)
Ways to get Sum S = 6 - |7 - S| for S ∈ [2, 12] Mean, Median, Mode Relation
Mode = 3 × Median - 2 × Mean Variance & Standard Deviation
Var(X) = E[X²] - (E[X])² | SD(X) = √Var(X) Scaling Property of Variance
Var(kX) = k² Var(X) | SD(kX) = |k| SD(X)