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List of Important Formulas of Mathematics

Speed, Distance & Time

• Speed = distance/time

• Time = distance/ Speed

• Distance = (Speed * Time)

• Distance = Rate x Time

• Rate = Distance/Time

• Convert from kph (km/h) to mps(m/sec):

x km/hr=x∗(5/18) m/sec

• Convert from mps(m/sec) to kph(km/h):

x m/sec= X*(18/5) km/h

• If the ratio of the speeds of A and B is a : b,

then the ratio of the times taken by then to

cover the same distance is :1/a : 1/b or b:a

• Suppose a man covers a certain distance at x

km/hr and an equal distance at y km/hr.

Then,

the average speed during the whole journey

is :- 2xy/(x + y)

• When speed is constant distance covered by

the object is directly proportional to the time

taken. ie; If Sa=Sb then Da/Db = Ta/Tb

• When time is constant speed is directly

proportional to the distance travelled. ie; If

Ta=Tb then Sa/Sb=Da/Db

• When distance is constant speed is inversely

proportional to the time taken ie if speed

increases then time taken to cover the

distance decreases. ie; If Da=Db then Sa/Sb

= Tb/Ta

• If the speeds given are in Harmonic

progression or HP then the corresponding

time taken will be in Arithmetic progression

or AP

• If the speeds given are in AP then the

corresponding time taken is in HP

• If two objects are moving in same direction

with speeds a and b then their relative speed

is |a-b|

• If two objects are moving is opposite direction

with speeds a and b then their relative speed

is (a+b)

Profit & Loss

• Cost Price is the price at which an article is

purchased, abbreviated as C.P.

• Selling Price is the price at which an article is

sold, abbreviated as S.P.

• If the Selling Price exceeds the Cost Price,

then there is Profit.

• Profit or gain = SP – CP

• Profit % = Profit/(C P)×100

• S P = (100+gain % )/100 ×C P

• C P = 100/(100+gain %)×S P

• If the overall Cost Price exceeds the selling

price of the buyer then he is said to have

incurred loss.

• Loss = C P – S P

• Loss % = LOSS/(C P)×100

• S P = (100-loss %)/100×C P

• C P = 100/(100-loss %)×S P

• Profit and Loss Based on Cost Price

(i) To find the percent gain or loss, divide the amount

gained or lost by the cost price & multiply it by 100.

(ii) To find the loss and the selling price when the

cost and the percent loss are given, multiply the cost

by the percent & subtract the product from the cost.

• Discount = M P – S P

• Discount %, D% = (Discount) / (M P) ×100

Percentage

• If we have to convert percentage into fraction

then it is divide by 100.

• If we have to convert fraction into percentage

we have to multiple with 100.

• If the price of a commodity increases by R%,

then the reduction in consumption so as not

to increase the expenditure is: [R/ (100 + R)]

x 100%

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• If the price of a commodity decreases by R%,

then the increase in consumption so as not to

decrease the expenditure is: [R/ (100 - R)] x

100%

• Let the population of a town be P now and

suppose it increases at the rate of R% per

annum, then:

1.Population after n years = P(1 + R/100)n

2.Population n years ago =P/(1 + R/100)n

• Let the present value of a machine be P.

Suppose it depreciates at the rate of R% per

annum. Then:

1. Value of the machine after n years = P (1

- R/100)n

2. Value of the machine n years ago = P/[(1

- R/100)]n

3. If A is R% more than B, then B is less

than A by= [R/ (100 + R)] x 100%

4. If A is R% less than B, then B is more

than A by= [R/ (100 - R)] x 100%

Note: For two successive changes of x% and

y%, net change = {x + y +xy/100}%

Average

Formula:

• Average: = (Sum of observations / Number

of observations).

• If a person travels a distance at a speed of x

km/hr and the same distance at a speed of y

km/hr then the average speed during the

whole journey is given by-

• If a person covers A km at x km/hr and B km

at y km/hr and C km at z km/hr, then the

average speed in covering the whole distance

is- When a person leaves the group

and another person joins the group in place

of that person then-

• If the average age is increased, Age of new

person = Age of separated person +

(Increase in average × total number of

persons)

• If the average age is decreased, Age of new

person = Age of separated person -

(Decrease in average × total number of

persons)

• When a person joins the group- In case of

increase in average, Age of new member =

Previous average + (Increase in average ×

Number of members including new member)

• When a person joins the group- In case of

decrease in average, Age of new member =

Previous average - (Decrease in average ×

Number of members including new member)

• In the Arithmetic Progression there are two

cases when the number of terms is odd and

second one is when number of terms is even.

(i) So when the number of terms is odd the average

will be the middle term.

(i) when the number of terms is even then the

average will be the average of two middle terms.

Algebra

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Partnership

P1: P2 = C1×T1: C2×T2

Here, P1 = Profit for Partner 1.

C1 = Capital by Partner 1.

T1 = Time period for which Partner 1 invested his

capital.

P2 = Profit for Partner 2.

C2 = Capital by Partner 2.

T2 = Time period for which Partner 2 invested his

capital.

Time, work & wages

1. Work from Days:

• If A can do a piece of work in n days, then A’s

n days work is=1/n

• No. of days = total work / work done in 1 day

• Days from Work: If A’s 1 day’s work =1/n

then A can finish the work in n days.

2. Relationship between Men and Work.

• More men ------- can do -------> More work

• Less men ------- can do -------> Less work

3. Relationship between Work and Time

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• More work -------- takes------> More Time

• Less work -------- takes------> Less Time

4. Relationship between Men and Time

• More men ------- can do in -------> Less

Time

• Less men ------- can do in -------> More

Time

5. If M1 persons can do W1 work in D1 days and

M2 persons can do W2 work in D2 days, then

6. If M1 persons can do W1 work in D1 days for

h1 hours and M2 persons can do W2 work in D2 days

for h2 hours, then

7. If A can do a work in ‘x’ days and B can do the

same work in ‘y’ days, then the number of days

required to complete the work if A and B work

together is

8. If A can do a work in ‘x’ days and A + B can do

the same work in ‘y’ days, then the number of days

required to complete the work if B works alone is

Perimeter, Area & Volume Rectangle

A four-sided shape that is made up of two pairs of

parallel lines and that has four right angles;

especially: a shape in which one pair of lines is longer

than the other pair.

The diagonals of a rectangle bisect each other and

are equal.

Area of rectangle = length x breadth = l x b

OR Area of rectangle = if one sides (l)

and diagonal (d) are given.

OR Area of rectangle = if perimeter (P)

and diagonal (d) are given.

Perimeter (P) of rectangle = 2 (length + breadth) =

2 (l + b).

OR Perimeter of rectangle = if one

side (l) and diagonal (d) are given.

Square

A four-sided shape that is made up of four straight

sides that are the same length and that has four right

angles.

The diagonals of a square are equal and bisect each

other at 900.

(a) Area (a) of a square

Perimeter (P) of a square

= 4a, i.e. 4 x side

Length (d) of the diagonal of a square

Circle

A circle is a path travelled by a point which moves

in such a way that its distance from a fixed point

remains constant.

The fixed point is known as center and the fixed

distance is called the radius.

(a) Circumference or perimeter of circle =

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where r is radius and d is diameter of circle

(b) Area of circle

is radius

is circumference

circumference x radius

(c) Radius of circle =

Sector:

A sector is a figure enclosed by two radii and an

arc lying between them.

here AOB is a sector

length of arc AB= 2πrΘ/360°

Area of Sector ACBO=1/2[arc AB × radius] = πr ×

r×Θ/360°

Ring or Circular Path:

R=outer radius

r=inner radius

area=π(R2-r2)

Perimeter=2π(R+r)

Rhombus

Rhombus is a quadrilateral whose all sides are

equal.

The diagonals of a rhombus bisect each other at

900

Area (a) of a rhombus

= a * h, i.e. base * height

Product of its diagonals

since d2

2

since d22

Perimeter (P) of a rhombus

= 4a, i.e. 4 x side

Where d1 and d2 are two-diagonals.

Side (a) of a rhombus

Parallelogram

A quadrilateral in which opposite sides are equal and

parallel is called a parallelogram. The diagonals of a

parallelogram bisect each other.

Area (a) of a parallelogram = base × altitude

corresponding to the base = b × h

Area (a) of a parallelogram

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where a and b are adjacent sides, d is the length of

the diagonal connecting the ends of the two sides

and

In a parallelogram, the sum of the squares of the

diagonals = 2

(the sum of the squares of the two adjacent sides).

i.e.,

Perimeter (P) of a parallelogram

= 2 (a+b),

Where a and b are adjacent sides of the

parallelogram.

Trapezium (Trapezoid)

A trapezoid is a 2-dimensional geometric figure with

four sides, at least one set of which are parallel. The

parallel sides are called the bases, while the other

sides are called the legs. The term ‘trapezium,’ from

which we got our word trapezoid has been in use in

the English language since the 1500s and is from the

Latin meaning ‘little table.’

Area (a) of a trapezium

1/2 x (sum of parallel sides) x perpendicular

Distance between the parallel sides

i.e.,

Pathways Running across the middle of a

rectangle:

X is the width of the path

Area of path= (l+b-x)x

perimeter= 2(l+b-2x)

Outer Pathways

Area=(l+b+2x)2x

Perimeter=4(l+b+2x)

Inner Pathways

Area=(l+b-2x)2x

Perimeter=4(l+b-2x)

• If there is a change of X% in defining

dimensions of the 2-d figure then its

perimeter will also change by X%

• If all the sides of a quadrilateral are changed

by X% then its diagonal will also change by

X%.

• The area of the largest triangle that can be

inscribed in a semicircle of radius r is r2.

• The number of revolution made by a circular

wheel of radius r in travelling distance d is

given by number of revolution =d/2πr

• If the length and breadth of the rectangle are

increased by x% and y% then the area of the

rectangle will be increased by.

(x+y+xy/100)%

• If the length and breadth of a rectangle are

decreased by x% and y% respectively then

the area of the rectangle will decrease by:

(x+y-xy/100)%

• If the length of a rectangle is increased by

x%, then its breadth will have to be

decreased by (100x/100+x)% in order to

maintain the same area of the rectangle.

• If each of the defining dimensions or sides of

any 2-D figure is changed by x% its area

changes by :

x(2+x/100)%

where x=positive if increase and negative if

decreases.

Cube

s = side

Volume: V = s^3

Lateral surface area = 4a2

Surface Area: S = 6s^2

Diagonal (d) = s√3

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Cuboid

Volume of cuboid: length x breadth x width

Total surface area = 2 ( lb + bh + hl)

Right Circular Cylinder

Volume of Cylinder = π r^2 h

Lateral Surface Area (LSA or CSA) = 2π r h

Total Surface Area = TSA = 2 π r (r + h)

Right Circular Cone

l^2 = r^2 + h^2

Volume of cone = 1/3 π r^2 h

Curved surface area: CSA= π r l

Total surface area = TSA = πr(r + l )

Frustum of a Cone

r = top radius, R = base radius,

h = height, s = slant height

Volume: V = π/ 3 (r^2 + rR + R^2)h

Surface Area: S = πs(R + r) + πr^2 + πR^2

Sphere

r = radius

Volume: V = 4/3 πr^3

Surface Area: S = 4π^2

Hemisphere

Volume-Hemisphere = 2/3 π r^3

Curved surface area(CSA) = 2 π r^2

Total surface area = TSA = 3 π r^2

Prism

Volume = Base area x height

Lateral Surface area = perimeter of the base x

height

Pyramid

Volume of a right pyramid = (1/3) × area of the base

× height.

Area of the lateral faces of a right pyramid = (1/2)

× perimeter of the base x slant height.

Area of the whole surface of a right pyramid = area

of the lateral faces + area of the base.

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