Formulae for quantitative and DI section

Simple Average: Sum of elements

Number of elements

Percentage Change: Final value – Initial value

Initial value

Simple Interest: Principle * Rate * Time

100

Compound Interest: P * (1+R/100)n – P

Amount = Principle + Interest

Profit: SP – CP Loss: CP – SP

Percentage Profit: Profit * 100 Percentage loss: Loss *100

CP CP

Discount: Marked Price – Selling price

Discount percentage: Discount * 100

Marked Price

If articles worth Rs. x are bought and articles worth Rs. y are obtained free along with x articles, then the discount is equal to y and discount percentage is given by: y * 100 x+y Successive discounts: When a discount of a % is followed by a discount of b% then Total discount= (a+b - ab/100) % Ratios If a : b = c : d, then a : b = c : d = (a + c) : (b + d) If a < b, then for a positive quantity x, a+x > a and a-x < a b+x b b-x b

If a > b, then for a positive quantity x, a+x < a and a-x > a b+x b b-x b

Alligations The ratio of the weights of the two items mixed will be inversely proportional to the deviation of attributes of these two items from the average attribute of the resultant mixture.

W1 = (x2-x1) W2 (x-x1)

Time speed and distance Speed = Distance

Time Important conversion factors 1 km/hr = 5 m/s and 1 m/s = 18 km/hr

18 5

Average speed: Total distance

Total Time

Relative speed: For trains

Time: Sum of the lengths = L1+L2

Relative speeds S1+ S2

For boats and streams

S (downstream) = S boat + S stream

S (upstream) = S boat - S stream

Time and work

Number of days to complete a work: 1

Work done in one day

HCF of Fractions: HCF of numerators of all fractions

LCM of denominators of all fractions

LCM of Fractions: LCM of numerators of all fractions

HCF of denominators of all fractions

Algebraic formulae

(a + b) (a − b) = a2 − b2 (a + b)2 = a2 + 2ab + b2 (a − b)2 = a2 − 2ab + b2 (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a − b)3 = a3 − 3a2b + 3ab2 − b3 a3 + b3 = (a + b)(a2 − ab + b2) a3 − b3 = (a − b)(a2 + ab + b2) Arithmetic progression Tn = a + (n – 1) d Sn = n [2a + (n – 1) d} 2

Arithmetic mean: x1+x2+x3…….xn

n

Geometric progression

Tn = ar n - 1

Sn = a (rn – 1)

( r – 1)

S∞ = a

1 - r

Sum of first n natural numbers

1+2+3+4+.......n = n (n+1)

2

Sum of the squares of the first n natural numbers

12+ 22+ 32 + 42………..n2 = n (n + 1) (2n + 1)

6

Sum of the cubes of the first n natural numbers

13+ 23+ 33 + 43………..n3 = n (n + 1) 2

2

Factorial n! = 1 * 2 * 3 * … * (n − 1) * n n! = n * (n − 1)! Permutations nPr = n!

(n - r)!

Combinations nCr = n!

(n - r)! r!

Important Properties nCr = n C n – r

nC0 + nC1 + nC2 +……..+ nCn = 2n

Probability

Probability of an event: Number of favorable outcomes

Number of total outcomes

Odds in favor: Number of favorable outcomes

Number of unfavorable outcomes

Odds against: Number of unfavorable outcomes

Number of favorable outcomes

Geometry concepts

Circle

The perpendicular from the center of a circle to a chord of the circle bisects

the chord. In the figure below, O is the centre of the circle and OM

perpendicular AB. Then, AM = MB.

Equal chords are equidistant from the centre. Conversely, if two

chords are equidistant from the centre of a circle, they are equal. In the following figure, two chords of a circle, AB and CD, intersect at

point P.

Then, AP * PB = CP * PD.

The angle subtended by an arc of a circle at the centre is double the

angle subtended by it at any point on the remaining part of the

circumference.

In the above figure, a=2b.

If 2 tangents are drawn to a circle from an exterior point,

the length of two tangent segments are equal. Also, the

line joining the exterior point to the centre of the circle

bisects the angle between the tangents.

The angle that a tangent to a circle makes with a chord drawn

from the point of contact is equal to the angle subtended by

that chord in the alternate segment of the circle. In the figure

above, PA is the tangent at point A of the circle and AB is the

chord at point A. Hence, angle BAP = angle ACB.

hypotenuse

A

B C base

perpendicular

Triangle

Pythagoras Theorem

For a right angled triangle

AC2 = AB2 + BC2

Properties of a Triangle

The sum of the two sides is always greater than the third side:

a + b > c, a +c >b, b + c > a

The sum of three angles of a triangle always equals 180o

Exterior angle is equal to sum of interior opposite angles

Area of a Triangle

1 * base * height

2

abc where R = circumradius

4r

r * s where r = inradius and s= a+b+c

3

Medians of a triangle

Medians are lines that join the vertex to

the midpoint of the opposite side. In the

figure AF, BD and CE are the medians. The

point where the three median intersect is

called the centroid. O is the centroid in

the figure. Area ABF = Area AFC = Area ABC

2

The centroid divides the median internally in the ratio of 2:1

Apollonius theorem: AB2+AC2 = (2AF2+BF2) or

BC2+BA2 = 2(BD2+DC2) or BC2+AC2 =

2(EC2+AE2)

Altitudes are the perpendiculars from the

vertex to the opposite side. In the figure

given alongside. AN, CE and BF are the

altitudes. H is the orthocentre

Internal angle bisector of a triangle

In the figure given along side Ad, Be and Cf are

the internal angle bisector and I is theincentre of

the triangle.

Internal bisector theorem: AB = BD

AC CD

Cone Cylinder

Sphere Cube

Quadrilaterals

A parallelogram is a quadrilateral whose opposite sides are parallel.

A rhombus is a parallelogram which has four equal sides.

A trapezoid (or trapezium) is a quadrilateral whose one pair of

opposite sides are parallel and one pair of opposite sides are

nonparallel.

Area of a quadrilateral where diagonals intersect at right angles is

½ *product of diagonals

Area of a quadrilateral when one diagonal and lengths

of the perpendicular from its opposite vertices are

given

= ½ * diagonal * sum of lengths of perpendiculars

In the figure: Area = ½ * PR * (SX+QY)

Area of a trapezoid: ½ X sum of parallel sides

X distance between parallel sides.

In the figure alongside:

Area = ½ * (AB+CD) * CM

Area of a parallelogram: Base * height

Area of a Rhombus: ½ * product of Diagonals

Area of a Rectangle: Length * Breadth

Diagonal of a Rectangle: Length2 + Breadth2

Perimeter of a Square: 4 * length

Diagonal of a Square: 2 * side

Area of a Square: √Side2

Area of a Square: ½ * Diagonal2

VERBAL COMPENDIUM – Do not confuse:

adoptive with adopted: children are adopted, but parents are adoptive.

adverse, 'unfavourable, bad', with averse, which means 'strongly disliking or opposed to', as in I am not averse to helping out.

affect and effect: affect means 'make a difference to', whereas effect means 'a result' or 'bring about (a result)'.

amoral with immoral: amoral means 'not concerned with morality', while immoral means 'not conforming to accepted standards of morality'.

appraise with apprise: appraise means 'assess', while apprise means 'inform'.

augur, 'be a sign of (a likely outcome)', with auger (a tool used for boring).

censure with censor: censure means 'express strong disapproval of', whereas censor means 'suppress unacceptable parts of (a book, film, etc.)'.

climactic, 'forming a climax', with climatic, which means 'relating to climate'.

complacent, 'smug and self-satisfied', with complaisant, which means 'willing to please'.

complement, 'a thing that enhances something by contributing extra features', with compliment, which means 'an expression of praise' or 'politely congratulate'.

continuous and continual: continuous primarily means 'without interruption', and can refer to space as well as time, as in the cliffs form a continuous line along the coast; continual, on the other hand, typically means 'happening frequently, with intervals

between', as in the bus service has been disrupted by continual breakdowns.

council, an administrative or advisory body, with counsel, advice or guidance.

definite ('certain, sure') with definitive, which means 'decisive and with authority'.

discreet, 'careful not to attract attention or give offence', with discrete, which means 'separate, distinct'.

egoism and egotism: it is egotism, not egoism, that means 'excessive conceit or self-absorption'; egoism is a less common and more technical word, for an ethical theory that treats self-interest as the foundation of morality.

exceptionable ('open to objection; causing disapproval or offence') with exceptional ('not typical' or 'unusually good').

fawn with faun: a fawn is a young deer, and a light brown colour; a faun is a Roman deity that is part man, part goat.

flaunt with flout; flaunt means 'display ostentatiously', while flout means 'openly disregard (a rule)'.

forego and forgo: forego means 'precede', but is also a less common spelling for forgo, 'go without'.

hoard with horde: a hoard is a store of something valuable; horde is a disparaging term for a large group of people.

the possessive its (as in turn the camera on its side) with the contraction it's (short for either it is or it has, as in it's my fault; it's been a hot day).

loath ('reluctant; unwilling') with loathe, 'dislike greatly'.

loose with lose: as a verb loose means 'unfasten or set free', while lose means 'cease to have' or 'become unable to find'.

militate, which is used in the form militate against to mean 'be an important factor in preventing', with mitigate, which means 'make (something bad) less severe'.

naturism (nudism) and naturist (a nudist) with naturalism and naturalist: naturalism is an artistic or literary approach or style; a naturalist is an expert in natural history, or an exponent of naturalism.

officious, 'asserting authority or interfering in an annoyingly domineering way', with official, which means 'relating to an authority or public body'.

ordinance, 'an authoritative order', with ordnance, which means 'guns' or 'munitions'.

perquisite and prerequisite: a perquisite is a special right or privilege enjoyed as a result of one's position; prerequisite is something that is required as a prior condition for something else; prerequisite can also be an adjective, meaning 'required as a prior condition'.

perspicuous, 'expressing things clearly', with perspicacious, which means 'having a ready understanding of things'.

principal, 'first in order of importance; main', with principle, which is a noun meaning chiefly 'a basis of a system of thought or belief'.

proscribe with prescribe: proscribe is a rather formal word meaning 'condemn or forbid', whereas prescribe means either 'issue a medical prescription' or 'recommend with authority'.

regretful, 'feeling or showing regret', with regrettable, which means 'giving rise to regret; undesirable'.

stationary and stationery: stationary is an adjective with the sense 'not moving or changing', whereas stationery is a noun meaning 'paper and other writing materials'.

titillate and titivate: titillate means 'excite', whereas titivate means 'adorn or smarten up'.

tortuous, 'full of twists and turns' or 'excessively lengthy and complex', with torturous, which means 'characterized by pain or suffering'.

turbid and turgid: turbid is generally used in reference to a liquid and means 'cloudy or opaque'; turgid tends to mean 'tediously pompous' or, in reference to a river, 'swollen, overflowing'.

unsociable with unsocial and antisocial: unsociable means 'not enjoying the company of or engaging in activities with others'; unsocial usually means 'socially inconvenient' and typically refers to the hours of work of a job; antisocial means 'contrary to accepted social customs and therefore annoying'.

venal ('susceptible to bribery; corruptible') with venial, which is used in Christian theology in reference to sin (a venial sin, unlike a mortal sin, is not regarded as depriving the soul of divine grace).

who's with whose; who's is a contraction of who is or who has, while whose is used in questions such as whose is this? and whose turn is it?

wreath and wreathe: wreath with no e at the end means 'arrangement of flowers', while wreathe with an e is a verb meaning 'envelop, surround, or encircle'.