Quantitative Aptitude — Cheat Sheet
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Cheat Sheet · AiCanCode.org
Quantitative Aptitude
Formula Sheets14 topicsQuick revision reference
1
Percentages
- ✓x% of y = (x/100) × y
- ✓Increase/decrease %: ((New − Old) / Old) × 100
- ✓If A is x% more than B → B is (100x / (100 + x))% less than A
- ✓Successive changes x% then y%: net = x + y + (xy/100) %
- ✓Common fractions: 1/8 = 12.5%, 1/6 = 16.67%, 1/3 = 33.33%, 1/7 ≈ 14.28%
2
Ratio & Proportion
- ✓a : b = a/b; a : b :: c : d ⇒ a×d = b×c (product of extremes = means)
- ✓If a/b = c/d then (a+b)/(a−b) = (c+d)/(c−d) (componendo–dividendo)
- ✓Mean proportional between a and b = √(ab)
- ✓To divide N in ratio a : b : c → parts are Na/(a+b+c), Nb/(…), Nc/(…)
3
Averages
- ✓Average = (sum of observations) / (number of observations)
- ✓Average speed (equal distances) = 2xy / (x + y)
- ✓New average when one value replaced: shift = (new − old) / count
- ✓Average of first n natural numbers = (n + 1) / 2
4
Profit, Loss & Discount
- ✓Profit% = (Profit / CP) × 100; Loss% = (Loss / CP) × 100
- ✓SP = CP × (100 + Profit%) / 100
- ✓Discount% = (MP − SP) / MP × 100 (on Marked Price)
- ✓Two successive discounts a%, b%: net = a + b − ab/100 %
- ✓False weight gain% = (Error / (True − Error)) × 100
5
Simple & Compound Interest
- ✓SI = P × R × T / 100
- ✓Amount (CI) = P × (1 + R/100)^T; CI = Amount − P
- ✓CI − SI for 2 years = P × (R/100)²
- ✓Half-yearly CI: rate = R/2, time = 2T
- ✓Amount doubles in T years at R% (SI) when R × T = 100
6
Time, Speed & Distance
- ✓Speed = Distance / Time; km/h → m/s: × 5/18
- ✓Relative speed: same direction = |a − b|, opposite = a + b
- ✓Train crossing a pole: t = length / speed
- ✓Train crossing a platform: t = (train + platform length) / speed
- ✓Boats: downstream = b + s, upstream = b − s (b = boat, s = stream)
7
Time & Work
- ✓If A does a job in n days → 1 day work = 1/n
- ✓A and B together: time = (a × b) / (a + b) days
- ✓Work = Rate × Time; more workers ⇒ proportionally less time
- ✓M1·D1·H1 / W1 = M2·D2·H2 / W2 (men–days–hours–work)
8
Pipes & Cisterns
- ✓Inlet fills 1/x per hour; outlet empties 1/y per hour
- ✓Both open: net rate = 1/x − 1/y (per hour)
- ✓Same as Time & Work, with a leak treated as negative work
9
Numbers, LCM & HCF
- ✓Product of two numbers = HCF × LCM
- ✓Divisibility: 3 → digit sum ÷ 3; 9 → digit sum ÷ 9; 11 → alt. sum ÷ 11
- ✓Number of factors of N = (a+1)(b+1)… where N = pᵃ·qᵇ…
- ✓Sum of first n numbers = n(n+1)/2; of squares = n(n+1)(2n+1)/6
10
Permutations & Combinations
- ✓nPr = n! / (n − r)! (order matters)
- ✓nCr = n! / (r!(n − r)!) (order does not matter)
- ✓nCr = nC(n−r); nC0 = nCn = 1
- ✓Circular permutations of n objects = (n − 1)!
11
Probability
- ✓P(E) = favourable outcomes / total outcomes, 0 ≤ P ≤ 1
- ✓P(A or B) = P(A) + P(B) − P(A and B)
- ✓Independent events: P(A and B) = P(A) × P(B)
- ✓P(not E) = 1 − P(E); Conditional: P(A|B) = P(A∩B)/P(B)
12
Mensuration (2D & 3D)
- ✓Circle: area = πr², circumference = 2πr
- ✓Triangle: area = ½ × base × height; Heron: √(s(s−a)(s−b)(s−c))
- ✓Cuboid: volume = l×b×h, surface = 2(lb+bh+hl)
- ✓Cylinder: volume = πr²h, CSA = 2πrh; Sphere: volume = 4/3 πr³
- ✓Cone: volume = 1/3 πr²h, slant l = √(r²+h²)
13
Progressions (AP & GP)
- ✓AP nth term: aₙ = a + (n−1)d; sum = n/2 × (2a + (n−1)d)
- ✓AP sum = n/2 × (first + last)
- ✓GP nth term: aₙ = a·r^(n−1); sum = a(rⁿ−1)/(r−1), r ≠ 1
- ✓GP infinite sum (|r| < 1) = a / (1 − r)
14
Mixtures & Alligation
- ✓Alligation: (dearer − mean) : (mean − cheaper) = qty(cheaper) : qty(dearer)
- ✓Repeated replacement: final = initial × (1 − x/V)ⁿ after n draws of x from V
- ✓Mean price = total value / total quantity
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