Quantitative Aptitude

A percentage is a fraction with denominator 100. Learn the fraction–percentage pairs (1/8 = 12.5%, 1/6 ≈ 16.67%, 1/12 ≈ 8.33%) so you can do percentage maths mentally.

Successive change: if a value changes by a% and then by b%, the net change is a + b + ab/100%. For example, +20% then −20% gives 20 − 20 − 4 = −4%.

💡 Shortcut: after a fall of x%, the rise needed to get back to the original value is x/(100 − x) × 100%. After a 20% fall you need a 25% rise.

Profit & Loss

Profit % and loss % are always calculated on cost price. Discount is always calculated on marked price. Selling at a loss of x% means SP = CP × (100 − x)/100.

Time, Speed & Distance

To convert km/h to m/s, multiply by 5/18. A train crossing a pole covers its own length. A train crossing a platform covers its own length plus the platform's. For equal distances at speeds x and y, the average speed is the harmonic mean 2xy/(x + y), not the simple average.

Formula & rules sheet

12 topics · 48 formulas and rules. Tap a topic to open or close it.

Percentages4

Net change = a + b + ab/100 %Two successive changes of a% and b%. Use a negative value for a decrease.
% change = (New − Old) / Old × 100
Rise needed after x% fall = x / (100 − x) × 100 %To get back to the original value.
Price ↑ r% ⇒ cut use by r / (100 + r) × 100 %Keeps spending unchanged.

Profit & Loss4

Profit % = (SP − CP) / CP × 100Profit and loss % are always on cost price.
SP = CP × (100 + P%) / 100For a loss, use (100 − L%).
SP = MP × (1 − d/100)Discount is always on the marked price.
False-weight gain % = (True − False) / False × 100E.g. 900 g for 1 kg: 100/900 = 11 1/9%.

Simple & Compound Interest3

SI = P × R × T / 100
A = P (1 + R/100)ⁿ, CI = A − PCompounded annually for n years.
CI − SI (2 years) = P (R/100)²

Ratio & Proportion3

a : b = c : d ⇔ a × d = b × cCross-multiplication.
Mean proportional of a and b = √(ab)
Partnership: profit share ∝ capital × time

Averages & Alligation3

Average = Sum / Count
Average of 1, 2, …, n = (n + 1) / 2
Cheaper : Dearer = (d − m) : (m − c)c = cheaper price, d = dearer price, m = mixture price.

Time & Work3

Together = ab / (a + b) daysWhen A takes a days and B takes b days.
M₁ × D₁ = M₂ × D₂For the same job: men × days stays constant.
Net rate = 1/a − 1/bA fill pipe (a hours) and an empty pipe (b hours) open together.

Time, Speed & Distance5

Speed = Distance / Time
km/h → m/s: × 5/18 m/s → km/h: × 18/5
Average speed = 2xy / (x + y)Two equal distances at speeds x and y.
Crossing time = (L₁ + L₂) / relative speedAdd the speeds for opposite directions and subtract for the same direction. For a pole, L₂ = 0.
Downstream = b + s, Upstream = b − sb = boat speed in still water, s = stream speed.

Number System4

Zeros at end of n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …
N = pᵃ·qᵇ·rᶜ ⇒ factors = (a+1)(b+1)(c+1)From the prime factorisation of N.
HCF(a, b) × LCM(a, b) = a × bFor two numbers only.
Unit digits of 2, 3, 7, 8 repeat every 4 powersUse the power mod 4 (0 means the 4th). 4 and 9 repeat every 2 powers. 0, 1, 5, 6 never change.

Algebra5

(a ± b)² = a² ± 2ab + b²
a² − b² = (a + b)(a − b)
a³ + b³ = (a + b)³ − 3ab(a + b)
x² + 1/x² = (x + 1/x)² − 2
ax² + bx + c = 0: sum of roots = −b/a, product = c/a

Geometry & Mensuration6

Circle: Area = πr², Circumference = 2πr
Right triangle: circumradius = hypotenuse / 2
Equilateral triangle: Area = (√3 / 4) a²
Regular n-gon: interior angle = (n − 2) × 180° / n
Diagonals of an n-gon = n(n − 3) / 2
Cube: Surface area = 6a², Volume = a³

Permutations & Combinations4

nPr = n! / (n − r)!Order matters (arrangements).
nCr = n! / (r! (n − r)!)Order doesn't matter (selections).
Arrangements with repeats = n! / (p! q! …)p, q, … = how many times each letter repeats.
At least one = Total − None

Probability4

P(E) = favourable / total outcomesWhen all outcomes are equally likely.
P(not E) = 1 − P(E)
P(A or B) = P(A) + P(B) − P(A and B)
Independent events: P(A and B) = P(A) × P(B)

Tip: bookmark this page and revise it for 5 minutes before each practice session.