Quantitative aptitude formula sheet: every formula worth memorising, on one page

This page collects the formulas the quantitative section actually tests, in one place, so the final weeks of preparation can be spent checking recall rather than hunting through a textbook. It is deliberately terse. Every formula here is explained properly — with worked examples, the reasoning behind it, and the traps set around it — in the topic article linked from each section. Learn from those; revise from this.

One rule before the formulas: there is no calculator in prelims and none at all in RRB NTPC or SSC, and no formula sheet in any of them. Everything below has to be on recall, which is a different and higher standard than “I understand it”. Test that with the practice sets and a timed mock, not by rereading.

Number series and basic sums

Quantity	Formula
Sum of first n natural numbers	n(n + 1) / 2
Sum of squares 1² … n²	n(n + 1)(2n + 1) / 6
Sum of cubes 1³ … n³	[n(n + 1) / 2]²
Sum of first n odd numbers	n²
Sum of first n even numbers	n(n + 1)
Number of factors of N = aˣ·bʸ·cᶻ	(x + 1)(y + 1)(z + 1)

Squares to 30, cubes to 15, and the fraction-to-percentage table are the single highest-return things to memorise in this whole section; they are set out in full, with the memorisation method, in calculation speed.

Percentage

Quantity	Formula
x% of y	xy / 100
Percentage change	(change / original) × 100
Two successive changes of a% then b%	a + b + ab/100
If A is x% more than B, then B is less than A by	[x / (100 + x)] × 100 %
If A is x% less than B, then B is more than A by	[x / (100 − x)] × 100 %

The sign matters in the successive-change formula: a 20% rise then a 20% fall is 20 − 20 − 400/100 = −4%, a net loss, not zero. Full treatment in percentage, profit and averages.

Profit, loss and discount

Quantity	Formula
Profit %	(SP − CP) / CP × 100
Loss %	(CP − SP) / CP × 100
Selling price	CP × (100 + profit%) / 100
Discount %	(MP − SP) / MP × 100
Two successive discounts a% and b%	a + b − ab/100 (net discount)
Same item sold at +x% and −x% (equal SP)	always a loss of (x/10)² %

The last row is the trap that is set every cycle: selling two articles at the same price, one at a gain of x% and one at a loss of x%, is never break-even — it is always a loss of (x/10)² per cent on the whole transaction.

Simple and compound interest

Quantity	Formula
Simple interest	P × R × T / 100
Amount under compound interest	P × (1 + R/100)ᵀ
CI − SI for 2 years	P × (R/100)²
CI − SI for 3 years	P × (R/100)² × (3 + R/100)
Half-yearly compounding	rate R/2, time 2T

These two-year and three-year difference formulas are the ones worth having instantly, because the question is very often “the difference between CI and SI is ₹X, find the principal”. Worked through in simple and compound interest.

Ratio, proportion and partnership

Quantity	Formula
Proportion	a : b = c : d means a × d = b × c
Dividing amount A in ratio m : n	m/(m+n) × A and n/(m+n) × A
Partnership profit share	in the ratio of (capital × time) for each partner
Mean proportional between a and b	√(ab)

Average, mixture and alligation

Quantity	Formula
Average	sum of observations / number of observations
Average speed over equal distances at x and y	2xy / (x + y)
Alligation (ratio of quantities)	(dearer − mean) : (mean − cheaper)
New average when one value replaces another	old average ± (difference / number)

Alligation is one rule that answers a surprising range of questions — mixtures, average age, average price, even some data interpretation — and it is covered with diagrams in mensuration, mixtures and counting.

Time, speed and distance

Quantity	Formula
Speed	distance / time
Convert km/h to m/s	× 5/18
Convert m/s to km/h	× 18/5
Relative speed, opposite directions	sum of the two speeds
Relative speed, same direction	difference of the two speeds
Train crossing a pole	time = length of train / speed
Train crossing a platform	time = (train + platform length) / speed
Boat downstream speed	boat + stream
Boat upstream speed	boat − stream
Boat speed and stream speed	½(down + up) and ½(down − up)

All of these, with the ratio method that avoids most of the arithmetic, are in time, speed, distance and work.

Time and work

Quantity	Formula
One day’s work if the job takes a days	1 / a
A and B together (a and b days alone)	ab / (a + b) days
Work equivalence	M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂
Pipes and cisterns	inlet fills (+), outlet empties (−); add the rates

The reliable method is to set the total work to the LCM of the given days so every rate becomes a whole number — it removes almost all the fractions.

Mensuration — two dimensions

Shape	Area	Perimeter / other
Rectangle	l × b	2(l + b); diagonal √(l² + b²)
Square	a²	4a; diagonal a√2
Triangle	½ × base × height	—
Triangle (three sides)	√[s(s−a)(s−b)(s−c)], s = (a+b+c)/2	Heron’s formula
Equilateral triangle	(√3 / 4) × a²	height = (√3 / 2) × a
Circle	π r²	circumference 2π r
Parallelogram	base × height	—
Rhombus	½ × d₁ × d₂	d₁, d₂ are the diagonals
Trapezium	½ × (sum of parallel sides) × height	—

Mensuration — three dimensions

Solid	Volume	Surface area
Cube	a³	6a²; diagonal a√3
Cuboid	l × b × h	2(lb + bh + hl); diagonal √(l²+b²+h²)
Cylinder	π r² h	curved 2π r h; total 2π r(r + h)
Cone	⅓ π r² h	curved π r l; total π r(r + l); slant l = √(r² + h²)
Sphere	4/3 π r³	4π r²
Hemisphere	⅔ π r³	curved 2π r²; total 3π r²

Use π = 22/7 when the radius is a multiple of 7 and 3.14 otherwise; the paper is written to make one of them cancel cleanly.

Algebra identities

Identity	Expansion
(a + b)²	a² + 2ab + b²
(a − b)²	a² − 2ab + b²
a² − b²	(a + b)(a − b)
(a + b)³	a³ + 3a²b + 3ab² + b³
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)
a³ + b³ + c³ − 3abc	(a + b + c)(a² + b² + c² − ab − bc − ca)

A useful pair for the “find x + 1/x” family: if x + 1/x = k, then x² + 1/x² = k² − 2, and x³ + 1/x³ = k³ − 3k. These appear almost every year in SSC.

Progressions

Sequence	nth term	Sum of n terms
Arithmetic (first a, common difference d)	a + (n − 1)d	n/2 × [2a + (n − 1)d] = n/2 × (first + last)
Geometric (first a, common ratio r)	a × r⁽ⁿ⁻¹⁾	a(rⁿ − 1) / (r − 1), for r ≠ 1

Permutations, combinations and probability

Quantity	Formula
Permutations of r from n	nPr = n! / (n − r)!
Combinations of r from n	nCr = n! / [r! (n − r)!]
Relationship	nPr = nCr × r!
Probability of an event	favourable outcomes / total outcomes

Whether a question is a permutation or a combination comes down to one test: does the order matter? “Arrangements” and “ranks” are permutations; “selections”, “committees” and “handshakes” are combinations.

How to use this sheet

1. Do not read it top to bottom hoping to absorb it. Recognition is not recall, and the exam tests recall.
2. Cover the right-hand column and write each formula from the topic name alone. The ones you cannot produce in a few seconds are your revision list; ignore the rest.
3. Return to the topic article for anything you get wrong, because a formula you memorise without understanding is one you will misapply the moment the question is phrased unusually.
4. Prove it under time. A timed mock tells you which formulas are fast and which still need the rough sheet — and that gap, not the list of formulas, is what separates a good quantitative score from an average one.