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List of Formulas for Quant of SSC

CHSL Tier I

Fundamental Arithmetical Operations

1. Averages

An average is the sum of a list of entities divided by the number of entities in the list.

Where, SE = sum of entities, nE = number of entities, AE = Average of entities.

Handy Trick: When a set of numbers are in arithmetic progression, the average is

simply half the sum of the first and last numbers in the list.

Tip: The average will always lie in the middle of all values. It cannot be less than the

smallest number in the list or more than the largest number in the list.

2. Percentages

Percent means “for every 100” or “out of 100”. The (%) symbol is a quick way to write a

fraction with a denominator of 100.

Handy Trick 1: To express (p/q) as a percent

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For e.g.

Read Quick Conversion of Fractions to Percentage and Its Application

Handy Trick 2: When the price of any article increases by x% and consumption of

that article increases by y %. Then, increment in the expenditure is given by

Handy Trick 3: When the price of an article increases or decreases by x% while

expenditure remains the same, the consumption correspondingly decreases or increases

by

Tip: Given a constant expenditure, if price increases, consumption would decrease. To

make the entire fraction above (i.e. consumption) less, we need a bigger denominator, so

we use ‘+’ in the denominator. When price decreases, consumption increases, so we

need smaller denominator and we use ‘-‘ instead.

Handy Trick 4: When the price of an article is first increased by x% and then

decreased by y%, then the final price change is

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3. Ratio and Proportion

Ratio:- The ratio of two quantities p and q, in the same units, is the fraction p/q and we

write it as p : q.

In the ratio p : q, we call ‘p’ as the first term or antecedent and ‘q’, the second term or

consequent.

Proportion:- The equality of two ratios is called proportion.

If p : q = r : s, we write p : q :: r : s and we say that p, q, r, s are in proportion.

Types of Ratios:-

1. Duplicate Ratio:- The ratio of the squares of two numbers.

2. Triplicate Ratio:- The ratio of the cubes of two numbers.

3. Sub-duplicate Ratio:- The ratio of the square roots of two numbers.

4. Sub-triplicate Ratio:- The ratio of the cube roots of two numbers.

5. Inverse Ratio:- The ratio in which the antecedent and consequent change their

places.

Types of Proportions:-

Fourth Proportion:- If p : q = r : s, then ‘s’ is called the fourth proportional to p, q and r.

Third Proportion:- If third proportion of p and q is ‘r’, then p : q = q : r

Mean Proportional: If mean proportion of p and r is ‘q’, then we have the following

relation, p : q = q : r

⇒ q =√pr

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Ratios can be simplified using the following principles:

Then,

Invertendo

Alternendo

Componendo

Dividendo

Componendo – Dividendo

4. Interest

Simple Interest:- If the interest on a sum borrowed for a certain period is reckoned

uniformly, then it is called simple interest. The formula for simple interest is

Where P = principle value, R = rate of interest, T = time in years.

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Compound Interest:- Compound interest is interest added to the principal of a

deposit or loan so that every installment of interest also earns interest. Compound

interest may be contrasted with simple interest, where interest is not added to the

principal (there is no compounding). The formula for compound interest is

Where, A = the future value of the investment/loan, including interest;

P = the principal investment amount (the initial deposit or loan amount);

R = the annual interest rate (decimal);

n = the number of times that interest is compounded per year;

t = the number of years the money is invested or borrowed for.

Handy Trick 1: Compound Interest = A – P

Handy Trick 2: If interest rates are different for successive years then

Installments in Simple Interest:-

Formula for installment calculation:

Where P = Principal Amount, n = number of installments, R = rate of interest, x=

amount of each installment

Installments in Compound Interest:-

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Where, P = each equal installment, R = rate of interest per annum (or per specified

period),

T = time, say 4 years (or 4 specified terms).

Tip: If T = n years (or specified terms), then there will be n brackets.

Population Formula:-

Here, P = original Population, P’ = Population after ‘n’ years, R = Rate of annual change

5. Profit and Loss

Tips:

Cost price (CP) is the price at which an article is purchased.

Selling price (SP) is the price at which an article is sold.

If SP > CP, there is a profit or gain

If CP > SP, there is a loss.

Gain or Profit = SP – CP

Loss = CP – SP

Loss or gain is always reckoned on CP

Discount = Marked Price or List Price – Selling Price

Discount is always provided on the List Price

Formulas:

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When SP is greater than CP

When CP is greater than SP

Handy Trick: When a person sells two items at the same Selling Price, one at a gain of

x%, and the other at a loss of x%, then the seller always incurs a loss expressed as:

6. Time and Work

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Tips

If A can do a piece of work in n days, work done by A in 1 day = 1/n

If A does 1/n work in a day, A can finish the work in n days

Handy Trick 1: If M1 persons can do W1 work in D1 days working T1 hours per day and

M2 persons can do W2 work in D2days working T2 hours per day, then the relationship

between them is:

Handy Trick 2: If A can do a piece of work in ‘p’ days and B can do the same in ‘q’ days

then A and B together can finish it in (in days)

More Tips:

If A is ‘x’ times as good (efficient) as B in work, then ratio of work done by A and B = x : 1

Ratio of time taken to finish a work by A and B = 1 : x

That is, A will take (1/x)th of the time taken by B to do the same work.

Some More Tips: All of these things are valid for Pipes and Cistern also.

7. Speed, Time and Distance

Distance = Speed × Time

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Average Speed:- The average speed of an object tells you the (average) rate at which it

covers any distance. Average speed is a measure of the distance traveled in a given

period of time. It is sometimes referred to as the distance per time ratio.

Average speed formula helps you calculate the average speed for a set of different

distances d1, d2….. dn if their corresponding different time intervals t1, t2,….tn are given.

Handy Trick 1: Say, a car travels at S1 kmph on a trip in t1 hours and at S2 kmph on

return trip in t2 hours. What is the average speed for the entire trip?

For this type of problem, don’t fall in the trap of just averaging the 2 speeds. Overall

average speed is not simply (S1+S2)/2.

Total average speed is simply = Total distance/Total time

Tip: This is for 2 speeds. You can extend this to 3 speeds. Simply use harmonic mean.

Average speed is not the arithmetic mean, just harmonic mean.

Problems on trains:-

Problems on trains and ‘Time and Distance’ are almost the same. The only difference is

we have to consider the length of the train while solving problems on trains.

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Handy Trick 1: Time taken by a train of length of ‘L’ meters to pass a stationary pole

or standing man or a signal is equal to the time taken by train to cover L meters.

Handy Trick 2: Time taken by a train of length of L meters to pass a stationary object

of length P meters is equal to the time taken by the train to cover (L + P) meters.

Handy Trick 3:

If two trains are moving in the same direction and their speeds are x kmph and y kmph

(x > y) then their relative speed is (x – y) kmph.

If two trains are moving in opposite direction and their speeds are x kmph and y kmph

then their relative speed is (x + y) kmph.

Handy Trick 4:

If two trains of p meters and q meters are moving in same direction at the speed of x

m/s and y m/s (x > y) respectively then time taken by the faster train to overtake slower

train is given by

If two trains of p meters and q meters are moving in opposite direction at the speed of x

m/s and y m/s respectively then time taken by trains to cross each other is given by

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Boat and Stream:-

Stream: Moving water of the river is called stream.

Still Water: If the water is not moving then it is called still water.

Upstream: If a boat or a swimmer moves in the opposite direction of the stream

then it is called upstream.

Downstream: If a boat or a swimmer moves in the same direction of the stream

then it is called downstream.

Let the speed of a boat in still water be ‘x’ kmph and the speed of the stream be ‘y’ kmph,

then

Speed downstream = (x + y) kmph

Speed upstream = (x – y) kmph

Handy Trick 5: Let the speed downstream be a kmph and the speed upstream be b

kmph, then

8. Progressions:

Progressions are a form of series. There are three major types. These are Arithmetic,

Geometric and Harmonic progressions. We even have combinations of progressions

such as Arithmetic-Geometric progression). An arithmetic progression (AP) is a series

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where each term differs from the previous term by a common difference.

9. Number System:

It is a chapter where problems can be made as tough as possible to challenge the

candidates to the fullest. The trick is to find a method to make calculations easier. There

are numerous tricks to solve these problems. Following are just a few of them-

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Mensuration Formulas

1. Mensuration Formulas for RECTANGLE

Area of Rectangle = Length × Breadth.

Perimeter of a Rectangle = 2 × (Length + Breadth)

Length of the Diagonal = √(Length2 + Breadth2)

2. Mensuration Formulas for SQUARE

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Area of a Square = Length × Length = (Length)2

Perimeter of a square = 4 × Length

Length of the Diagonal = √2 × Length

3. Mensuration Formulas for PARALLELOGRAM

Area of a Parallelogram = Length × Height

Perimeter of a Parallelogram = 2 × (Length + Breadth)

4. Mensuration Formulas for TRIANGLE

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Area of a triangle=(1/2)(Base × Height)=(1/2)(BC×AD)

For a triangle with sides measuring a, b and c, respectively:

Perimeter = a + b + c

s = semi perimeter = perimeter/2 = (a+b+c)/2

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Area of Triangle, A=

(This is also known as “Heron’s formula”)

Area of isosceles triangle =

(Where a = length of two equal side, b = length of base of isosceles triangle.)

Area of an equilateral triangle =

(Where, a is the side of an equilateral triangle)

5. Mensuration Formulas for TRAPEZIUM

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Area of a trapezium = (1/2) × (sum of parallel sides) × (distance between parallel

sides)

= (1/2) × (AB+DC) × AE

Perimeter of a Trapezium = Sum of All Sides

6. Mensuration Formulas for RHOMBUS

Area of a rhombus=(1/2)×Product of diagonals

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Perimeter of a rhombus = 4 × l

(where l = length of a side)

7. Mensuration Formulas for CIRCLE and SEMICIRCLE

In the following formulae, r = radius and d = diameter of the circle

Area of a circle = πr2= (πd2)/4

Circumference of a circle = 2πr = πd

Circumference of a semicircle = πr

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Area of semicircle =(πr2)/2

Length of an arc = (2πrθ)/360, where θ is the central angle in degrees.

Area of a sector = (1/2) × (length of arc) × r = (πr2θ)/360

8. Mensuration Formulas for CUBOID

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In the following formulae, l = length, b = breadth and h = height

Total surface area of cuboid = 2 (lb + bh + lh)

Length of diagonal of cuboid= √(l2+b2+h2)

Volume of cuboid = l × b × h

9. Mensuration Formulas for CUBE

In the following formulae, a = side of a cube

Volume of cube = a3

Total surface area of cube = 6a2

Length of Leading Diagonal of Cube = a√3

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10. Mensuration Formulas for CONE

In the following formulae, r = radius of base, l = slant height of cone and h = height of

the cone (perpendicular to base)

Slant height of a cone = l =√(h2+r2 )

Curved surface area of a cone = C = π × r × l

Total surface area of a cone = π × r × (r + l)

Volume of right circular cone =1/3 πr2h

11. Mensuration Formulas for CYLINDER

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In the following formulae, r = radius of base, h = height of cylinder

Curved surface area of a cylinder = 2πrh

Total surface area of a cylinder = 2πr(r + h)

Volume of a cylinder = πr2h

12. Mensuration Formulas for SPHERE

In the following formulae, r = radius of sphere, d = diameter of sphere

Surface area of a sphere = 4πr2 = πd2

Volume of a sphere = (4/3) πr3 = (1/6)πd3

13. Mensuration Formulas for HEMISPHERE

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In the following formulae, r = radius of sphere

Volume of a hemisphere =(2/3)πr3

Curved surface area of a hemisphere = 2πr2

Total surface area of a hemisphere = 3πr2

14. Mensuration Formulas for HOLLOW CYLINDER

Hollow cylinder made by cutting a smaller cylinder of same height and orientation out

of a bigger cylinder.

Volume of hollow cylinder = πh(R2– r2)

(Where, R = radius of cylinder, r = radius of cavity, h = height of cylinder)

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15. Mensuration Formulas for FRUSTUM OF A RIGHT CIRCULAR CONE

Frustum is created when a plane cuts a cone parallel to its base.

In the following formulae, R = radius of the base of the frustum, r = radius of the top of

the frustum,

h = height of the frustum, l = slant height of the frustum

If a cone is cut by a plane parallel to the base of the cone, the lower part is called

the frustum of the cone.

Slant height of the frustum =l=√(h2+(R-r)2)

Curved surface area of frustum = π(R + r)l

Total surface area of frustum = π(R + r)l + π(R2 + r2)

Volume of the frustum=(1/3)πh(R2+r2+Rr)

16. Mensuration Formulas for PRISM

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Prism consists of two polygonal bases which are parallel to each other.

These bases are joined by lateral faces, which are perpendicular to the polygonal

bases.

The number of lateral faces is equal to the number of sides in the polygonal base.

Thus, the base of a prism could be of various shapes, namely, triangular,

quadrangular, pentagonal etc.

Volume of prism = Base area × height

Lateral surface area of prism = perimeter of base × height

Total surface area of prism = Lateral surface area + (2 × base area)

17. Mensuration Formula for PYRAMID

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Pyramid consists of a polygonal base and triangles at its sides. These triangles are

called faces.

The base could be of any shape, whereas the faces are generally isosceles

triangles.

All these triangular faces meet in a single point called the apex.

Total surface area of pyramid = base area + (number of sides × ½ × slant height

× base length)

Volume of pyramid = (1/3) × base area × height

Permutation and Combination

A permutation is an arrangement in a definite order of a number of objects taken some

or all at a time.

A permutation is an arrangement of n objects taken r at a time. It is given as

A combination is a selection of n objects taken r at a time. It is given as

Tip: 0! = 1

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Handy Trick 1: The number of permutations of ‘n’ different objects taken ‘r’ at a time,

where 0 < r ≤ n and the objects do not repeat, is given by:

Handy Trick 2: The number of permutations of ‘n’ different objects taken ‘r’ at a

time, where repetition is allowed, is nr.

Handy Trick 3: The number of permutations of ‘n’ objects, where ‘p’ objects are of the

same kind and rest are all different is

The number of permutations of n objects, where p1 objects are of one kind, p2 are of

second kind, …… pk are of kth kind and the rest, if any, are of different kind is

Handy Trick 4: The number of combinations of ‘n’ different things taken ‘r’ at a time,

denoted by , is given by

where 0 ≤ r ≤ n.

Handy Trick 5:

The number of ways to arrange ‘n’ distinct objects along a fixed circle is (n – 1)!

Tip: This works for circles where there is a definite ‘up’ and definite ‘down’. Like a

circular table. Else, for objects like garlands and strings, where up and down doesn’t

matter, it is 2(n – 1)!

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Probability

Probability is the chance of occurrence of an event.

Let S be the sample space and let E be an event.

Then, E ⊆ S

Mixture Problems

Handy Trick 1: Rule of Alligation

If two ingredients are mixed, then we can present it as below

∴ (Cheaper quantity) : (Dearer quantity) = (d – m) : (m – c).

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Handy Trick 2: Replacement of Part of Solution Formula

Suppose a container contains x units of liquid from which y units are taken out and

replaced by water. After n operations, the quantity of pure liquid is

Algebra Formula

These are the basic formulas that you should keep in mind while solving algebra

questions.

1. (a + b)2 = a2 + 2ab + b2; a2 + b2 = (a+b)2 −2ab

2. (a − b)2 = a2 − 2ab + b2; a2 + b2 = (a−b)2 + 2ab

3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

4. (a + b)3 = a3 + b3 + 3ab(a + b);

5. a3 + b3 = (a+b)3 −3ab(a + b)

6. (a − b)3 = a3 − b3 − 3ab(a − b);

7. a3 − b3 = (a−b)3 + 3ab(a − b)

8. a2 − b2 = (a+b)(a − b)

9. a3 − b3 = (a−b)(a2 + ab + b2)

10. a3 + b3 = (a+b)(a2 − ab + b2)

11. am . an = am+n

12. am/ an = am-n

13. (am)n = amn = (an)m

14. a0 = 1