## List of Important Mathematics Formulas for RRB Group D & ALP/ Technician Exam

### 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 and 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 and subtract the product from the cost.*

- Profit and Loss Based on Selling Price

(i) To find the profit and the cost when the selling price and the percent profit are given, multiply the selling price by the percent profit and subtract the result from the selling price.

(ii) To find the loss and the cost when the selling price and the percent loss are given, multiply the selling price by the percent loss and subtract the result from the selling price.

(iii) To find the selling price when the cost and the percent loss are given, add the percent loss to 100% and divide the cost by this sum.

(iv) To find the selling price when the profit and the percent profit are given, or to find the selling price when the loss and the percent loss are given, divide the profit or loss by the percent profit or loss.

- 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%
- 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
- 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, When a person leaves the group and another person joins the group in place of that person
- 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

### Partnership

P_{1}: P_{2} = C_{1}×T_{1}: C_{2}×T_{2}_{}Here, P_{1} = Profit for Partner 1.

C_{1} = Capital by Partner 1.

T_{1} = Time period for which Partner 1 invested his capital.

P_{2} = Profit for Partner 2.

C_{2} = Capital by Partner 2.

T_{2} = 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

- 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 90^{0}.

(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 =

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

Perimeter=2π(R+r)

#### Rhombus

Rhombus is a quadrilateral whose all sides are equal.

The diagonals of a rhombus bisect each other at 90^{0}

Area (a) of a rhombus

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

Product of its diagonals

since d^{2}_{2 }

since d^{2}_{2 }

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

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.,

Where, l = b – a if b > a = a – b if a > b

And

Height (h) of the trapezium

#### 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 r
^{2.} - 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 = 4a^{2}

Surface Area: S = 6s^2

Diagonal (d) = s√3

#### 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.

## Important PDFs of Mathematics for Railways Exams 2018

- Simple Interest & Compound Interest PDF: Download Now (English PDF)
- Simple Interest & Compound Interest PDF: Download Now (हिंदी PDF)
- Percentage PDF: Download Now(English PDF)
- Percentage PDF: Download Now(हिंदी PDF)
- List of Important Mathematics Formulas PDF: Download Now
- Time & Work (Shortcut Approach) PDF: Download Now
- Speed, Time & Distance PDF: Download Now

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