Aptitude cheat sheet

Divisibility rules let you test whether a number divides another without doing the long division — a huge time-saver in speed tests. Master the rules for 2, 3, 4, 5, 6, 8, 9 and 11 first; the rest are combinations.

HCF is the largest number that divides all given numbers; LCM is the smallest number they all divide. HCF handles “greatest common” situations, LCM handles “meeting again / next common occurrence” situations.

A percentage is just a fraction with denominator 100. Convert tricky percentages to fractions (25% = 1/4) to compute mentally. For “increase then decrease” questions, use successive-change instead of step-by-step.

Everything hangs off one idea: Profit = SP − CP (positive) or Loss = CP − SP (negative). Percentages are always on Cost Price unless stated. Discount is always on Marked Price. Learn to move between CP, SP, MP with the multiplier trick: SP = CP × (1 ± P%/100).

A ratio compares two quantities (a : b). When the same ratio is scaled by a constant, you get direct proportion (y = kx); when one doubles and the other halves, it is inverse (y = k/x). “Divide in ratio” means splitting a total proportionally to the parts.

Simple interest charges a fixed amount every year (only on the original principal). Compound interest charges interest on the interest too — the amount grows like a multiplier every period. One multiplier per year: ×(1 + R/100); split periods for half-yearly/quarterly.

The triangle “Distance = Speed × Time” unlocks everything. Convert units first (km/h → m/s = ×5/18). For trains, add lengths when crossing. For boats, the stream adds to downstream speed and subtracts upstream. For two objects, add speeds when moving opposite, subtract when moving the same way.

Turn “time to finish” into “work per unit time” (1/time). Add the rates when working together; subtract an emptying pipe’s rate. Total work is often best set to the LCM of the times — then everyone works in whole numbers.

Average = total ÷ count. When values have weights, multiply each by its weight first. For a journey with two legs at different speeds, the average speed is NOT the arithmetic mean — it is 2v₁v₂/(v₁+v₂) when distances are equal.

Alligation finds the ratio in which two things at different costs mix to hit a target mean cost: ratio = (dearer − mean) : (mean − cheaper). Draw the “cross” and the proportions fall out immediately — no algebra needed.

Learn the perimeter (a length) and area (a square-unit) for each shape; most other figures build from the rectangle and triangle. Two triangles sharing an altitude have areas proportional to their bases — the trick behind many questions.

Trig only needs the three ratios (sin, cos, tan) and their reciprocals. In right triangles, pick the ratio that involves the two sides you know. For height-and-distance questions, tanθ = height ÷ distance almost always cracks it.

The identities let you factorise and expand at a glance — the backbone of simplification questions. If you recognise a² − b² or (a+b)² hiding in an expression, the problem collapses in one line.