Cheat sheet

Aptitude, puzzles & GD

Companion to the 250-question aptitude bank. Learn the rule, then drill the pattern.

Percentages · ratio · averages

Percentage change

Change % = (new − old)/old × 100.
Successive changes multiply: 1.20 × 0.80 = 0.96.
If A is x% more than B, B is x/(100+x) less than A.

Ratio

Make the common term equal before combining.
a : b = 3 : 4 and b : c = 2 : 5 → a : c = (3/4)(2/5).
Divide a total by the sum of ratio parts.

Averages

Total = average × count. New average after adding = (old total + new value)/(n+1).
AP average = (first + last)/2.
For equal distances, average speed is the harmonic mean.

Weighted average

(n₁x₁ + n₂x₂)/(n₁ + n₂). Never average two group averages without weighting by group size.

Profit, loss & interest

Profit %	profit / CP × 100 · SP = CP(1 + p) · CP = SP/(1 + p)
Discount	SP = MP(1 − d). Successive discounts a and b → 1 − (1−a)(1−b)
Mark-up + discount	Profit factor = (1 + mark-up)(1 − discount)
Simple interest	SI = PRT/100 · doubles in T years at R = 100/T
Compound interest	A = P(1 + R/100)^T · CI = A − P
CI − SI, 2 years	P(R/100)²

Time & work · pipes · boats

Work

Rate = 1/time. Add rates for people working together.
If A is twice as efficient as B, A's rate is 2× B's.
Work = men × days × hours (hold it constant when a crew changes).

Pipes

Net rate = fill rates − empty rates.
If a pipe fills in 6 h, its rate is 1/6 per hour.
Time = 1 / net rate.

Boats

Downstream = b + s · upstream = b − s.
b = (down + up)/2 · s = (down − up)/2.
Upstream time = twice downstream ⇒ b : s = 3 : 1.

Mixtures

Alligation ratio = (cost of high − mean) : (mean − cost of low).
Replacement n times: pure left = initial × (1 − removed/total)ⁿ.

Speed · time · distance · trains

D = S × T	km/h × 5/18 = m/s · m/s × 18/5 = km/h
Pole / person	distance = train length
Platform / bridge	distance = train + platform
Two trains	distance = L₁ + L₂ · opposite: S₁ + S₂ · same direction: S₁ − S₂
Average speed (equal distances)	2xy / (x + y), not (x + y)/2
Speed +20%	time falls by 1 − 1/1.2 = 16.67%

Series, codes, counting & clocks

Number series ladder

① differences (and 2nd differences) → ② ratios → ③ squares, cubes, primes, n² ± 1 → ④ alternating or two interleaved series.

Letters

EJOTY = 5, 10, 15, 20, 25 · opposite = 27 − position · codes: shift, opposite, reverse + shift, letter sums, sentence (compare shared words).

Counting

nPr = n!/(n−r)! · nCr = n!/[r!(n−r)!]
Arrangements with repeats: n!/(p! q! …)
Handshakes among n people = nC2.

Probability

P(A∪B) = P(A)+P(B)−P(A∩B) · P(A|B) = P(A∩B)/P(B)
Binomial: nCr p^r(1−p)^n−r · at least one = 1 − P(none).

Clocks

Angle = |30H − 5.5M| (take the smaller angle).
Hands coincide 22 times/day; right angles 22 times/12 h.

Calendars

Ordinary year = 1 odd day; leap year = 2. 100 years = 5 odd days; 400 years = 0.
Add the remainder of days ÷ 7 to the weekday.

Water jugs

T is reachable ⇔ gcd(A, B) divides T and T ≤ the bigger jug. 8 & 4 → 5 is impossible (gcd 4). Method: always pour in one direction (fill the big jug → pour into the small one → empty the small one), and keep a state table.

5 & 3 → 4: (5,0) (2,3) (2,0) (0,2) (5,2) (4,3). 8-5-3 split: 8→5, 5→3, 3→8, 5→3, 8→5, 5→3, 3→8 → (4,4,0).

AP

aₙ = a + (n − 1)d
Sₙ = n/2 · (first + last)

1 + … + n = n(n+1)/2
# terms = (last − first)/d + 1

GD: AI & IT companies

Facts (Nasscom, Feb 2026): FY26 revenue ~$315 bn (+6.1%) · headcount ~5.95 mn (+1.35 lakh, +2.3%), non-linear · AI revenue $10–12 bn (best case 5–6%) · 20 lakh+ upskilled in AI · campus hiring down.
3 lenses: business model (hours → outcomes, pyramid → diamond) · opportunity (AI integration, GCCs) · people (reskilling, moving up the stack).
Moves: open with a definition, a fact and a structure · "building on…" · bring in quiet members · summarise both sides plus a way forward.

Sources: raw.org · IndiaBIX · ETV Bharat · Business Today · Lessons 9–12