Volume

Volume is the quantity of three dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains.Volume is often quantified numerically using the cubic meter. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

 

                              (a is the length of the side of each edge of the cube)
     Volume of a Cube = a*a*a

rectangular prism:

                                               (a, b, and c are the lengths of the 3 sides)
         Volume of a Rectangular Prism = a*b*c

 A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.
 

                     ('h' is height of the cylinder and 'r' is the radius of base)  
Volume of a Cyinder = pi*r*r*h 

prism:

                                                           ( 'h' height and 'b' base area ) 

Volume of a prism = b*h

 In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base. A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

                                                      ( 'h' height and 'b' base area )
Volume of a pyramid = 1/3*b*h 
 
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

                            ( where 'r' radius of the base and 'h' is height)      
   Volume of a Cone = 1/3*pi*r*r*h 

A sphere is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball. Like a circle, which, in geometric contexts, is in two dimensions, a sphere is defined mathematically as the set of points that are all the same distance r from a given point in three-dimensional space. This distance r is the radius of the sphere, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.

                                                            (r is radius of circle)
Volume of a Sphere = 4/3*pi*r*r*r

Volume of a ellipsoid = (4/3) pi r1 r2 r3

   

Surface Areas

Surface Area Formulas In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.The surface area of a solid object is a measure of the total area that the surface of an object occupies
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

                         (a is the length of the side of each edge of the cube)
 Surface Area of a Cube = 6*a*a
in words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a. Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

Rectangular prism:

                               (a, b, and c are the lengths of the 3 sides)
Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac
In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.
 
Surface Area of Any Prism

                                            (b is the shape of the ends)
Surface Area of any prism = Lateral area + Area of two ends.where,Lateral area is equal to the perimeter of shape b * L.in generally,
Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

A sphere is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball. Like a circle, which, in geometric contexts, is in two dimensions, a sphere is defined mathematically as the set of points that are all the same distance r from a given point in three-dimensional space. This distance r is the radius of the sphere, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.

                                                               (r is radius of circle)
Surface Area of a Sphere = 4 pi* r *r

A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.

                            (h is the height of the cylinder, r is the radius of the top)
Surface Area of a Cylinder = 2 pi* r*r+ 2 pi* r* h
Surface Area = Areas of top and bottom +Area of the side = 2(Area of top) + (perimeter of top)* height = 2(pi r*r ) + (2 pi r)* h
In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.
You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r 2 ). Since there is both a top and a bottom, that gets multiplied by two.The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.Add those two parts together and you have the formula for the surface area of a cylinder.
Surface Area = 2(pi r 2 ) + (2 pi r)* h

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
Surface area of the cone = pi*r*r + pi*r*l
In words,the lateral surface area of a right circular cone is pi*r*l where 'r' is the radius of the circle at the bottom of the cone and 'l' is the lateral height of the cone (given by the pythagorean theorem l= square root of (r*r+h*h)where h is the height of the cone). The surface area of the bottom circle of a cone is the same as for any circle, pi*r*r.

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base. A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.
The surface area of a pyramid is A=B+(P*L)/2 where B is the base area, P is the base perimeter and L is the slant heightl= square root of (r*r+h*h) where h is the pyramid altitude and r is the radius of the base.

Areas and Perimeters

Area is simply defined as the size of a surface.Area is the quantity that expresses the extent of a two-dimensional  figure or shape, in the plane. The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle.
A perimeter is a path that surrounds a two-dimensional shape. The word comes from the Greek peri (around) and meter (measure). The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. Calculating the perimeter has considerable practical applications. The perimeter can be used to calculate the length of fence required to surround a yard or garden. The perimeter of a wheel (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter.
Note: "ab" means "a" multiplied by "b". "a 2 " means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements.

square:In geometry, a square is a regular quadrilateral , which means that it has four equal sides and four equal angles  (90- degree angles, or right angles). Below shown figure is square where,all sides are with length of 'a'.

Area and perimeter of the square is can be calculated by using below formula
                                            Area = a*a
                                    Perimeter = 4a

Rectangle:It can also be defined as a rectangle in which two adjacent sides have equal length and with four equal angles  (90- degree angles, or right angles). Below shown figure is rectangle where,sides are with length of 'a' and breadth 'b'.

     

Area and perimeter of the rectangle is can be calculated by using below formula
                                            Area = a*b
                                    Perimeter = 2(a+b)

 

parallelogram: In Euclidean geometry, a parallelogram is a (non self-intersecting) quadrilateral  with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean Parallel Postulate and neither condition can be proven without appealing to the Euclidean Parallel Postulate or one of its equivalent formulations. The three-dimensional counterpart of a parallelogram is a parallelepiped.

 Area and perimeter of the parallelogram is can be calculated by using below formula
                                            Area =b*h
                                    Perimeter = 2(a+b)

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid in American and Canadian English but as a trapezium in English outside North America. The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides (if they are not parallel; otherwise there are two pairs of bases)
trapezoid = h/2 (b 1 + b 2 )

 Area and perimeter of the trapezoid is can be calculated by using below formula
                                            Area =(b1+b2)*h/2
                                    Perimeter = 2(b1+b2)

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.
A circle may also be defined as a special eclipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter, using calculus of variations.

Area and perimeter of the circle is can be calculated by using below formula ( pi = 3.14)
                                            Area = pi*r*r
                                    Circumference = 2*pi*r or pi*d

An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how 'elongated' it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.

Area and perimeter of the ellipse is can be calculated by using below formula
                                            Area = pi*r1*r2
                                    Perimeter =

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted .
perimeter of the any triangle is can be calculated by using below formula
                  Perimeter = a+b+c

In an equilateral triangle all sides have the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.
 Area of equilateral triangle:                                                                           
                                     
In an isosceles triangle, two sides are equal in length. An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length; this fact is the content of the isosceles triangle theoram, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tertrakis square tiling, is isosceles.

.'.Area of isosceles triangle = 1/2*(b*h)

In a scalene triangle, all sides are unequal, and equivalently all angles are unequal. An equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles since no angles are equal.
The shape of the triangle is determined by the lengths of the sides. Therefore the area can also be derived from the lengths of the sides.
 By Heron's formula:A=√s(s-a)(s-b)(s-c) ( square root for whole terms)
where 'S' equal to (a+b+c)/2 is the semiperimeter, or half of the triangle's perimeter.

Seating arrangement practice problems

Seating Arrangement- Questions
I. P, Q, R, S, T, U, V & W are eight friends sitting around a circle facing towards the centre.
i) W is to the immediate left of P but is not the neighbour of T and S ii) U is to the immediate right of Q and V is neighbour of T
iii) R is between T and U
1. What is the position of V?
a. To the immediate right of W
b. Between T and R
c. Third to the right of U
d. Second to the left of S
e. None of these
2. What is the position of S?
a. Between Q and U
b. To the immediate left of P
c. Second to the right of U
d. To the immediate left of Q
e. None of these
3. Which of the following statements is true?
a. U is the neighbour of V
b. V is between W and T
c. W is between P and S
d. T is between U and Q
e. None of these 
Study the following information carefully and answer the questions
II. A, B, C, D, E, F, G and H are sitting around a circle, facing the centre. A sits fourth to the right of H while second to the left of F. C is not the neighbor of F and B. D sits third to the right of C. H never sits next to G.
1. Who among the following sits between B and D?
a. G.   b. F   c. H   d. A   e. C
2. Which of the following pairs sit between H and G?
a. BH       b. EF      c. CE      d. DB
e. None of these
3. Four of the following five are alike in a certain way based on their positions in the seating arrangement and so form the group. Which is the one that does not belong to that group?
a. AE b. HF c. BD d. GE e. CH
4. Who is to immediate right of A?   a. C b. D c. G d. Data inadequate
e. None of these
5. Who sits second to the right of B?
a. A.      b. C.       c. D       d. E
e. None of these
6. Which is the position of B with respect to C?
I. Second to the right
II. Sixth to the left
III. Third to the left
IV. Fifth to the right
a. Only II
b. Only II and III
c. Only I and IV
d. Data inadequate
e. Both III and IV
III. Mohan and Suresh study in the same class. Mohan has secured more marks than Suresh in the terminal examination. Suresh's rank is seventh from top among all the students in the class. Which of the following is definitely true?
1. Mohan stood first in the terminal examination
2. There is at least one student between Mohan and Suresh in the rank list
3. There are at the most five students between Mohan and Suresh in the rank list
4. Suresh is five ranks lower than Mohan in the rank list
5. None of these
IV. In a row of girls facing North, Rinky is 10th to the left to Pinky who is 21st from the right end. If Minky who is 17th from the left end is 4th to the right of Rinky. How many girls  are there in the row?
1. Data inadequate
2. 44
3. 37
4. 43
5. None of these 
V. In a row of 40 girls, when Kamal was shifted to her left by 4 places her number from the left end of the row became 10. What was the number of Sujata from the right end of the row if Sujata was three places to the right of Kamal's original position? 
1. 22    2. 26.     3. 25.    4. 23
5. None of these
VI. Eight friends A, B, C, D, E, F, G and H are sitting around a circle facing the centre. E is third to the left of G who is to the immediate right of B who is third to the left of A. H is second to the right of F who is not an immediate neighbour of E. D is not an immediate neighbour of B.
1. Who is second to the right of B? a. F b. A c. H d. D e. None of these
2. Which of the following pairs has the first person to the immediate left of second person?
a. GB b. AF c. CE d. HD e. None of these
3. Which of the following is the correct position of B with respect to D?
a. Second to the right
b. Second to the left
c. Third to the right
d. Third to the left
e. None of these
4. Who sits between A and D?
a. F    b. E    c. G.    d. B     e. H
5. What is E's position with respect to C?
a. To the immediate right
b. To the immediate left
c. Second to the right
d. Cannot be determined

Questions to practise

1 Angle between hands at 8:30 ?
2. A watch gains 5 seconds in 5 minutes and was set right at 10 AM. What time will it show at 8 PM on the same day?
3. A watch loses 1 minutes every hour and was set right at 9 AM on a Monday. When will it show the correct time again?
4. What is the day on 23 October 2014 ?
5. A man can row a distance 50 km upstream in 5 hrs. If he rows the same distance down the stream in 3 hrs. then speed of man ?
6.  A man can row in still water at 4 km/h. In a stream flowing at 2 km/h, if it takes 3 hours more in upstream than to go downstream for the same distance, then the distance swims by person ?
7. A tank can be filled by a tap in 20 minutes and by another tap in 60 minutes. Both the taps are kept open for 10 minutes and then the first tap is shut off. After this, how much time required to fill tank will be completely ?
8. What is the time taken by the two trains to meet from first train side, that start at their points: A for the first train and B for the second train that travels at a speed of ‘4’ km/hr and ‘9’ km/hr respectively.The distance between A and B is 13 km?
9. A train crosses by a stationary man standing on the platform in 7 seconds and passes by the platform completely in 28 seconds. If the length of the platform is 330 meters, what is the length of the train?
10. A motorcar does a journey in 10 hrs, the first half at 21 kmph and the second half at 24 kmph. Find the distance?
11. A train travelling 25 kmph leaves Delhi at 9 a.m. and another train travelling 35 kmph starts at 2 p.m. in the same direction. How many km from away from starting point they will meet  together ?
12. A person travelled a distance with 5 km/hr then, he will take 2 hrs more.if he travels with 7.5 km/hr then,he will reach 2 hrs earlier.the distance traveled by person ?

Number System

n this first article of our topic ‘Numbers’, we deal with ‘Numbers’ themselves. What are numbers ? Where do we see and meet them ? Are they of different types ? Can we learn these distinct types ?
Well, numbers are all around us, floating all over the place. They demand our attention and careful evaluation. A number here or there can make a huge difference. In face a misplaced zero or decimal is the difference between one being poor or rich. Such being their importance, it is critical we begin with the most basic concepts in Mathematics: types of numbers.
 In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. '0'  is called insignificant digit.
Types of Numbers

• Natural numbers
• Whole numbers
• Integers
• Positive Integers
• Negative Integers
• Non-negative Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
• Even Numbers
• Odd Numbers
• Prime Numbers
• Composite Numbers 

-Natural numbers : Counting numbers 1, 2, 3, 4, 5... are known as natural numbers. The set of all natural numbers can be represented by N= {1, 2, 3, 4, 5...}
-Whole numbers : If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 ... are called whole numbers.The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5...}. Every natural number is a whole number but 0 is a whole number which is not a natural number.
-Integer : All counting numbers and their negatives including zero are known as integers.
The set of integers can be represented by Z or I = {...-4,-3,-2,-1, 0, 1, 2, 3, 4 ...}
-Positive Integers :The set I = {1, 2, 3, 4...} is the set of all positive integers. Positive integers and natural numbers are synonyms.
-Negative Integers : The set I = {-1, -2, -3...} is the set of all negative integers. 0 is neither positive nor negative. Non-negative Integers : The set {0, 1, 2, 3...} is the set all non-negative integers.
-Rational Numbers : The numbers of the form p/q, where p and q are integers and q ≠ 0, are known as rational numbers, e.g. 4/7, 3/2, -5/8, 0/1, -2/3, etc. The set of all rational numbers is denoted by Q. i.e. Q ={x:x =p/q; p, q belong to I,q≠0}. Since every natural number ‘a’ can be written as a/1, every natural number is a rational number. Since 0 can be written as 0/1 and every non-zero integer ‘a’ can be written as a/1, every integer is a rational number. Every rational number has a peculiar characteristic that when expressed in decimal form is expressible rather in terminating decimals or in non-terminating repeating decimals. For example, 1/5 =0.2, 1/3 = 0.333...22/7 = 3.1428704287,
 8/44 = 0.181818...., etc. The recurring decimals have been given a short notation as 0.333.... = 0.3
4.1555... = 4.05, 0.323232...= 0.32.
-Irrational Numbers : Those numbers which when expressed in decimal from are neither terminating nor repeating decimals are known as irrational number, e.g. √2, √3, √5, π, etc.
Note that the exact value of is not 22/7. 22/7 is rational while π irrational number. 22/7 is approximate value of π. Similarly, 3.14 is not an exact value of it.
-Real Numbers : The rational and irrational numbers combined together are called real numbers, e.g.13/21, 2/5, -3/7;√3, 4 +√2, etc. are real numbers.The set of all real numbers is denoted by R.
Note that the sum, difference or product of a rational and irrational number is irrational, e.g. 3+√2, 4 -√3, 2/3-√5, 4√3,-7√5 are all irrational.
-Even Numbers : All those numbers which are exactly divisible by 2 are called even numbers.
e.g.2, 6, 8, 10, etc., are even numbers.
-Odd Numbers : All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1,3, 5, 7 etc., are odd numbers.
-Prime Numbers : A natural number other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers. 2 is the only even number which is prime number. Two numbers which have only 1 as the common factor are called co-relatively prime to each other, e.g. 3 and 5 are co-primes.
Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.
-Composite Numbers : Natural numbers greater than 1 which are not prime, are known as composite
numbers. For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers. The number 1 is neither a prime number nor composite number.

Some more properties on natural numbers  :

• Sum of 1st n Natural number i.e 1 + 2 + 3 + …. + n = {n (n + 1)} /2
• Sum of squares of 1st n natural number i.e { n(n+l)(2n+l) }/6
• Sum of cubes of 1st n Natural number i.e  [{n(n+1)}/2]^2
• Sum of squares of 1st n Odd natural number i.e {n (2n -1 )(2n + 1)}/3
• Sum of squares of 1st n Even natural number i.e {2n(n+1)(2n+1)}/3
• Sum of cubes of 1st n Odd natural number i.e  n^2 (2n^2 -1)
• Sum of cubes of 1st n Even natural number i.e  2 [n (n + 1)]^2  

Let’s consider any digit (Say 2) and check how many times it appears in different ranges

• From 1 – 9, the digit 2 is used 1 time
• From 1 – 99, the digit 2 is used 20 times.
• From 1 – 999, the digit 2 is used 300 times
• From 1 – 9999, the digit 2 is used 4000 times and so on

This is a generalization we can use for any number between 1 to 9. So in case a question asks you how many 3s between 1 to 99, you can simply say 20. How did we arrive at this conclusion?
Well, there are two series of three’s here: 3, 13,…33,…93 = ten 3s at the unit’s place. 30,31…39= ten 3s at the ten’s place. So a total of 20 threes make an appearance here.

6 when multiplied with 7 give 4 and 2
6 x 7 = 42
66 x67 = 4422
666 x 667 = 444222
6666 x 6667 = 44442222
For multiplication such as the above : The number of digits in both the numbers is same , the 2^nd number has one 6 less than the number of  6’s the 1^st number has . The number of 4’s and 2’s is equal to the number of 6’s in the 1^st number

Cyclic numbers
142857 is called as the cyclic number, since its digits are rotated around when multiplied by any number from 1 to 6
142857 x 1 = 142857
142857 x 5= 714285
142857 x 4= 571428
142857 x 6 = 857142
142857 x 2= 285714
142867 x 3 = 428571

Magical 1
3 x 37 = 111
33 x 3367 = 111,111
333 x 333667 = 111,111,111
3333 x 33336667= 111,111,111,111
33333 x 3333366667= 111,111,111,111,111
333333 x 333333666667 = 111,111,111,111,111,111
Consider the L.H.S of the above numbers. The number of 3s in both the numbers are same and 6 is one less than the number of 3 and there is only one 7. Then in the final product, N would appear three times the number of threes in the first number on the LHS.

Some Other Properties of Numbers
Squares: There is a definite relationship between the unit digits when square of a number is considered, we  will see that one by one.
If unit digit of a perfect square is I then ten’s digit has to be Even (e.g. 81, 121, 441 etc)
If unit digit of a number is 2 then it can’t be a perfect square.
If unit digit of a number is 3 then it can’t be a perfect square.
If unit digit of a perfect square is 4 then ten’s digit has to be Even (e.g. 64,144 etc)
If unit digit of a perfect square is 5 then ten’s digit has to be 2 (e.g. 25, 225, 625 etc)
If unit digit of a perfect square is 6 then ten’s digit has to be Odd and multiple of 3.
If unit digit of a number is 7 & 8 then it can’t be a perfect square
If unit digit of a perfect square is 9 then ten’s digit has to be Even (e.g. 49, 169 etc)
If unit digit of a perfect square is 0 then ten’s digit has to be 0 (e.g. 100,400,900 etc)
Hence if a number end with 2, 3, 7 or 8 then it can’t be perfect square.
Properties of Integers :  We have studied about integers, the entities which can be shown on the number line is called integers there are some properties of Integers in Nature. Usually these properties are used in the GRE examination

• Among all the numbers positive and negative whole numbers are included in the integers
• Zero is neither rational nor irrational but it is an Integer
• There are some numbers which follow one another which are called as consecutive integers
• There is an Integer to which a number is multiplied the value will never change and that integer is 1

Order of Operation
As like BODMAS in CAT exam, GRE believes in PEMDAS and this stands for
P = Parentheses
E = Exponents
M= Multiplication
D = Division
A = Addition
S= Subtraction
TOOLTIP 1:   What is the approximate value of N= [{2(4 +5)³ x 4}/10 -23]
Step by step
[{2(9)³ x 4}/10 -23] => [{2(729) x 4}/10 -23] => [{1458 x 4}/10 -23] => [5832/10 -23] => 560.2 = 560 approximate Consecutive Integers: The integers which follow one another are called consecutive integers
For example 3,4,5,6 are consecutive integers. An individual random number never be a consecutive integer.
Some more points about Consecutive Numbers
TOOLTIP 1
If there is odd number of digits in the  set of consecutive numbers like set of three consecutive numbers (4,5,6,) or 5 digits are there say (3,4,5,6,7). Then in this case the sum of all integers always divisible by the number of digits of set
For example 2+3+4 = 9 divisible by number of digits i.e. 3
For example 1+2+3+4+5 = 15 divisible by 5
TOOLTIP 2
On the other hand ,If there is even number of digits in  set of consecutive numbers like set of four consecutive numbers (4,5,6,7) or 6 digits are there say (3,4,5,6,7,8). Then in this the sums of all integers is never be divisible by the number of digits of the set
For example 2+3+4+5 = 14 never divisible by number of digits i.e. 4
For example 1+2+3+4+5+6 = 21 never divisible by 6

Number systems-II

The concept of fractions, though a simple one, can be often confused. Having not solved questions based on this simple concept, students often tend to confuse the problems. These questions can throw up the occasional challenge and it makes sense to practice these questions from this area.
Definition : Technically, fraction is defined as part of the whole. The most common example of a fraction that comes to mind is half. When we say give me half of something, we are essentially demanding ½ part of it, in others word, ½ is the fractional representation for half.
Fractions are nothing else than the numerator divided by denominator, that is they occur in the form X/Y where X is the numerator and y is the denominator.
Remember : The numerator represents how many parts of that whole are being considered. To remember simply, numerator is the top number of the fraction that represents the numbers of that that are to be chosen. The denominator represents the total number of parts created from the whole, in other words it is the bottom number representing the total number of parts created.
Example of Fractions : ½,2/3,3/4, and the numbers which are in the form of x/y
Types of Fractions:
Proper Fraction: When Numerator < Denominator, then the fraction is called as proper fraction. For example: 2/3, 4/5, 6/7 etc.
Improper fractions: When Numerator > Denominator, then the fraction is called as improper fraction. For example: 5/3, 7/5, 19/7 etc.
Mixed Numbers: when a natural number combines with a fraction that is called a mixed number. For Example: 2¹/₂ ,3⁴/₅  etc
Tool tip 1: Basic Applications of Fractions 1. Fraction help us determine the part of any number ¾ part of 56 = ¾ x 56 = 42, 4/5 part of 90 = 4/5 x 90 =24
2. You can be asked to represent a number in the form of fraction. For example, you can be asked to represent 15 as a fraction of 450. This can be done as follows: 15/450 = 1/30
We have solved the above example and it can be easily seen that 15 is our numerator and 450 is our denominator
3. Always remember that the major quantity from which we have to extract something is the denominator. For example, when we say 4/5, we are essentially extracting four parts out of five.
4. Extending the above concept, the quantity which is extracted is our numerator
Example: 15/450 = 1/30
15 is the numerator because we have extracted 15 from 450 and the denominator is 450 because 15 is extracted from 450 so we can say that 1/30^th part of 450 is 15
Tool tip 2: Properties of fractions
Property 1: If we multiply the numerator and denominator by same quantity, the basic value of fraction will never change.  For example: 4/5 x 5/5 = 20/25 = 4/5
Property 2: If there are two fractions a/b and c/d then a/b=c/d when ad=bc. For example 3/4 = 12/16 because 3 x 16 = 4 x 12
Property 3: A fraction with zero as the denominator is not defined.
Property 4: If the numerator of the fraction is zero, then the fraction equals zero.
Property 5: If the numerator and denominator of the fraction are equal, then the fraction is equal to one.
DECIMALS
Are all numbers integers? Well, the obvious answer to that question is a no. All numbers are not integers. Consider the case of 0.333333. What is this number? An integer? Well, it is a decimal. But wait are decimals? Decimals are nothing else but the values lying between two integers on the number line.
Remember the following decimal forms (showcasing how decimals look like on either side of the number line):
Decimals less than -1: -12.12, -9.13, -1.2 (the numbers are arranged in increasing order with -1.2 being the largest among all).
Decimals between -1 and 0: -0.52, -0.40, -0.04
Decimals between 0 to 1: 0.12, 0.14, 0.80
Decimals greater than 1: 2.13, 5.64, 7.83
Relating decimals to fraction: When we solve a fraction of the p/q form, it is not necessarily it would return an integral value.  When we left with a remainder, we ultimately convert it into decimal form. Some examples of Decimals are 4.5, 9.6, 6.78, and 99.98 , these all are decimal numbers 
Tool tip 1: Forming Decimals from Fractions
How we can write 4/5 in decimal form?
Since the number 4 is smaller than the 5, so the decimal value will be less than one. Multiple and divide both numbers by 10. We have: 40/ (10×5) Which effectively is (dividing the 40 by 5 first)
8/10. Thus, the final result is 0.8 So 4/5 = 0.8. 0.8 is the decimal form.
Tool tip 2: Adding Decimals While adding decimals, you should always write the decimals in a vertical column with the decimal points aligned vertically.
Add all these 0.567 +78+8.9+5.06+56
= 78.000 +  56.00 +   5.06   +  0.567 =  139.627
Tool tip 3: Subtracting Decimal
In addition we can write the numbers in any order. But while subtracting, we should preferably write the numbers in descending order and the vertical column with decimal points should be aligned to the same decimal points.
Let’s take an example: we have to subtract 5.06 from 0.567. We write it as: 5.06 - 0.567 = 5.507
This result is wrong because in case of subtraction we need equal digits in both the quantities, so these blank spaces filled with 0. So this can be done like as 5.060 - 0.567 =  4.493
This is the right approach for the question
Tool tip 4: Multiplying Decimals
As a first step, multiply the given integers in the normal form keeping the decimals aside.  The number of decimal places in the product is then equal to the total of the decimal places in the two decimals. It is as simple as that.
Consider the following example : 5.060 X 0.567  = 2.869020
First multiply 5060 with 567 and get the result as 2869020 and then move the decimal point to 6 places from the left, that is between 2 and 8. Effectively, we move it to six decimals places, our sum total of the decimal places in the two numbers.

cyclicity : 
 type 1 : where power of single digit is considered
The concept of cyclicity is used to identify the last digit of the number which is in the form of power like p^k. Let’s take an example to understand this:
Example 1 :  find the unit digit of 3⁵⁶.
Solution :  Now it’s a big term so we cannot find the last digit by doing 3 x 3 x 3 x 3 x 3……. 56 times so we use the concept of cyclicity
Step 1 : 3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
So now pay attention to the last digits we saw that the last digit repeats itself after a cycle of 4 and the cycle is 3 ,9,7,1,this is called the cyclicity of any number ,therefore when we need to find the unit digit of any number like 3ⁿ we just need to find the number on which the cycle of last digit ends .  And in the next step we will divide the power with the cyclicity

• if the remainder will be 1 then the unit digit will be 3
• if the remainder will be 2 then the unit digit will be 9
• if the remainder will be 3 then the unit digit will be 7
• if the remainder will be 0 then the unit digit will be 1

This is all about the cyclicity
Why the power is divided by number 4.
We will divide the power with 4 because cycle repeat itself after 4 values, and also we need to find the remainder which tells us the required values to complete the next cycle.
Now the main question was that how much is the last digit of 3⁵⁴
So we know the cycle repeats itself after 4 so we will divide the 54 with 4 ,so on dividing 54 by 4 the remainder becomes 2 .Now as we discussed above if the remainder is 2 the last digit would be 9, so in the end the unit digit of 3⁵⁴ is 9.
Type 2 :  where power of 2 and 3 digits number is to be considered
Example 2 : What will be the unit digit of 24⁴⁵ or 347⁴⁵
Solution : Lets take some example to understand it very clearly
We know that unit digit of 3 x 3 = 9
And the unit digit of 453 x 543 = 9
The main purpose of the above expression is that the unit digit of any multiplication depends upon the unit digit of numbers , whatever is the number big or small the unit digit always depends upon the multiplication of the last digit .
So the last digit of 24⁴⁵ can be found by 4⁴⁵
So the cyclicity of 4 is 2 because the cycle of last digit repeats after two values
4¹ = 4
4² =16
4³ = 64
So when we divide 45 with 2 then we will get the remainder as 1 and the last digit will be 4
Now come to the case number second unit digit of 347⁴⁵
The unit digit of this number can be find by the same method
The cyclicity of 7 is  4
7¹  = 7
7² = 49
7³ = 343
7⁴ = 2401
So on dividing 45 with 4 , 1 will be the remainder and the last digit would be 7
Type 3 : where _p^qr is to be considered
What will be the last digit of ₁₂²³⁴⁵
To find the last digit of this type of number we  will start the question from the base the base is 12. It means we will see the cyclicity of 2 because the last digit is depends upon the unit digit of 12. Lets do it step vise step
Before the steps we will write the last digits of
2¹ = 2
2² = 4
2³ = 8
2⁴ = 6
2⁵ = 2
Step 1: Now we know that cyclicity of last digit of 12 i.e 2 is 4 , hence the divide the power of 12 i.e 23⁴⁵  with 4
Step 2:  Now the remainder 23⁴⁵ /4 will determine the last digit.
Step 3:  The remainder will be 3 because we can write remainder of 23 /4 = 3 or -1 and -1 ⁴⁵ / 4 will give us remainder as -1 or 3
Hence in the end the last digit of ₁₂²³⁴⁵ is nothing but 12³  =  8.
How to find unit digit of a number :
For the concept of identifying the unit digit, we have to first familiarize with the concept of cyclicity. Cyclicity of any number is about the last digit and how they appear in a certain defined manner. Let’s take an example to clear this thing:
The cyclicity chart of 2 is:
2¹ =2
2² =4
2³ =8
2⁴=16
2⁵=32
Have a close look at the above. You would see that as 2 is multiplied every-time with its own self, the last digit changes. On the 4^th multiplication, 2⁵ has the same unit digit as 2¹. This shows us the cyclicity of 2 is 4, that is after every fourth multiplication, the unit digit will be two.
Cyclicity table:
The cyclicity table for numbers is given as below:
Number     Cyclicity
1                   1
2                  4
3                  4
4                  2
5                  1
6                  1
7                  4
8                  4
9                  2
10                1

How did we figure out the above?
Multiply and see for yourself. It’s good practice. Now let us use the concept of cyclicity to calculate the Unit digit of a number.
What is the unit digit of the expression  4⁴⁵?
Now we have two methods to solve this but we choose the best way to solve it i.e. through cyclicity
We know the cyclicity of 4 is 2
Have a look:
4¹ =4
4² =16
4³ =64
Here the 4 comes again to the end when the 4 raised to the power of 3 so it is clear that the cyclicity of 4 is 2.  Now with the cyclicity number i.e. with 2 divide the given power i.e. 45/2 what will be the remainder the remainder will be 1 so the when remainder was 1 what was the answer when 4 raised to the power one see first step , yes , 4
So the unit digit in this case is 4.
For checking whether you have learned the topic, think of any number like this, calculate its unit digit and then check it with the help of a calculator.
Lets solve another example:
The digit in the unit place of the number 7⁹⁵ X 3⁵⁸ is
A.  7
B.  2
C.  6
D.  4
Solution
The Cyclicity table for 7 is as follows:
7¹ =7
7² =49
7³ = 343
7⁴ = 2401
Let’s divide 95 by 4: the remainder is 3.
Thus, the last digit of 7⁹⁵ is equals to the last digit of 7³ i.e. 3.
The Cyclicity table for 3 is as follows:
3¹ =3
3² =9
3³ = 27
3⁴ = 81
3⁵ = 243
Let’s divide 58 by 4, the remainder is 2. Hence the last digit will be 9.