Numbers
Numbers are mathematical symbols by which we express date, time, distance, position, quantity etc.
We use ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to write any number.
Like 346562232, 3465452155, 4003444656 etc.
Numbers System
Number system deals with the study of different types of numbers. In this chapter, we will study about the categorization of different types of numbers.
Numbers Types
Numbers are classified according to their type. The first type of numbers we ever learned about: the counting numbers or the natural numbers and the next type of numbers is whole numbers.

Natural Numbers: The counting numbers (1, 2, 3, 4, . . . . .) are called natural numbers.

Whole Numbers: Natural numbers including zero (0, 1, 2, 3, 4,.....) are called whole numbers.

Even and Odd Numbers: Numbers which are divisible by 2 are called even numbers. For example, 2, 4, 6, 8, 10,..... etc.

Numbers which are not divisible by 2 are called odd numbers. For example, 1, 3, 5, 7, 9, 11, .....etc.

Prime and Composite Numbers: Whole numbers which have only two factors 1 and the number itself are called prime numbers. For example, 2, 3, 5, 7, 11 etc.

Whole numbers which have at least one factor other than 1 and the number itself are called composite numbers. For example, 4, 6, 8, 9, 12 etc....

Twin Primes: Two prime numbers are called twin primes if there are only one composite number between them. For example, pair of twin primes between 1 and 20 are: (3, 5); (5, 7); (11, 13) and (17, 19).

Integers: A negative number is a real number that is less than zero, which represents opposite. Integers are set of whole numbers and negative numbers.

Successor and Predecessor of a Number: Successor of a number is obtained by adding 1 to the number. For example, successor of 243 is 243 + 1 = 244.

Predecessor of a number is obtained by subtracting 1 from the number. For example. The predecessor of 243 = 243 - 1 = 242.
Hindu-Arabic Numeral System
Hindu-Arabic Numeral System is also known as Indian System of Numeration. This system is based on the following place value chart.
Place Value Chart

Introduction
We live in the world of numbers. We see them every day, on clocks, in sports, and all over the news. Algebra is all about figuring out the numbers we don't see. In this chapter, we will study about basic algebra and simple problems based on it.
Variables
Algebra is the branch of mathematics that uses letters in place of some unknown numbers we use numerals to represent numbers in arithmetic, in a similar way in Algebra, letters of the alphabet are used to represent numbers and these letters are called variables. Let us look at the puzzle of an unknown number.
\[?7=6\]
Here, we see, \[137=6\]
In algebra, we do not used blank boxes, we use a letter (say a, b, c, x, y, z, etc.). So, we can write it as: \[x7=6\]
Where, x is a variable or unknown number. Now, \[x7=6,\] or, \[\left( x7 \right)=6,\text{ }x0=7+6\] or \[x0=13\] . Or, \[x=13\] Clearly, here, \[x=13\] stands for the unknown number given in the box above.
We use letter for unknown number because, it is easier to write x than drawing empty boxes and also it is easier to say x than say empty box. If there are several empty boxes, or several unknown, we use different letters for different unknown.
Equation with One Variable
A linear equation in one variable has a single unknown quantity represented by a letter. The process of finding out the variable value that makes the equation true is called 'solving’ the equation. Clearly, an equation is a statement that two quantities are equivalent. For example, \[x+3=5\]means that when we added 3 to an unknown value ‘x’ the answer is equal to 5.

Example:

Solve for
Solution: Given equation is: \[x19=34\]
Now, add 19 to both side of the equation: \[\left( x19 \right)+19=34+19\]
Or, \[x+0=53,\] or \[x=53\]
Verification: \[x19=34,\] or \[5319=34\] .
Equation and Formula
We know that, perimeter of rectangle \[\text{=2}\!\!\times\!\!\text{ }\left(\text{length + breadth} \right)\text{=2 }\!\!\times\!\!\text{ }\left( \text{l + b} \right)\]
Then, variable 'l' stands for the length of rectangle and 'b' stands for the breadth of rectangle. Clearly, the letters ‘l’ and ‘b’ will represent different numbers for different rectangles.

Area of rectangle \[\text{=length }\!\!\times\!\!\text{ breadth=I }\!\!\times\!\!\text{ b}\]

The formula, distance = speed\[\times \]time is written as \[D=S\times T\], by making use of the letter 'D' for distance, 'S' for speed and 'V for time.

The relation between temperatures in Fahrenheit and Celsius is: \[C=\frac{5}{9}~\left( F32 \right)\]

Here, the letter 'C’ is used to denote the temperature in degrees in the Celsius scale and 'F' is used to denote the temperature in degrees in the Fahrenheit scale.

Increasing interest about this segment of reasoning.

Improving the general awareness.

Increasing the word power.

Introduction
In these types of questions different characters/numbers/letters) are arranged in a matrix with one term missing or characters are arranged in a wide range of geometrical figures. The characters in such arrangement follow a certain pattern and you are required to identify that pattern so that you can substitute the question mark (?) with a suitable character.
Such questions can be solved as series (numbers/letters) are done. No particular and specific rules are applied in such questions. Although you must keep the following tips in your mind:

Find the missing number in the given number matrix.

4

9

2

3

?

7

8

1

6

(a) 7 (b) 8
(c) 9 (d) 5
(e) None of these
Answer (d)
Explanation: It is a magical square starting from 1 to 9 and sum of each diagonal/ row or column is equal to 15.
2. What is the missing number in the given series below?
(a) 32 (b) 36
(c) 42 (d) 56
(e) None of these
Answer: (b)
Explanation: Pattern followed in is \[Q=P\times 2+2\]
Hence, number in lower part of the fourth circle is \[~17\times 2+2=34+2=36\]..
3. Which number will replace the question mark in the figure given below?
more...

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To understand the logic of figures.

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What is Non Verbal Reasoning?
Non verbal reasoning is a figure based reasoning. It has no language at all. To solve non-verbal problems one has to find out the pattern of pictorial presentation in the given figure. To get more clear concept about non verbal reasoning, let us see the types of problems coming before you.
Types of Problems
(a) Problems Based on Mirror Image
In a mirror image, left part of an object becomes right part and right part becomes left part. Remember the rule given below.
Left Hand Side (L.H.S.) \[\underset{{}}{\leftrightarrows}\] Right Hand Side (R.H.S.)

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Concept of Direction
In our day to day life we make our concept of direction after seeing the position of the sun. In fact, this is truth that sun rises in the East and goes down in the West. Thus, when we stand facing sunrise then our front is called East while our back is called West. At this position our left hand is in the northward and the right hand is in the southward. Let us see the following direction map that will make your concept more clear.
Direction Map:
Note:
On paper, North is always on the top while South is always at the bottom.
Concept of Turn
Left turn = Anti clockwise turn
Right turn = Clockwise turn
Let us understand it through pictorial presentation:-
Important Points Regarding Directions:
(1) If our face is towards North, then after left turn our face will be towards West while after right turn it will be towards East.
(2) If our face is towards South, then after left turn our face will be towards East and after right turn it will be towards West.
(3) If our face is towards East, then after left turn our face will be towards North and after right turn it will be towards South.
(4) If our face is towards West, then after left turn our face will be towards South and after right turn it will be towards North.
(5) If our face is towards North-West, then after left turn our face will be towards South-West and after right turn it will be towards North-East.
(6) If our face is towards South-West, then after left turn our face will be towards South-East and after right turn it will be towards North-West.
(7) If our face is towards South-East, then after left turn our face will be towards North-East and after right turn it will be towards South-West.
(8) If our face is towards North East, then after left turn our face will be towards North West and after right turn it will be towards South East.
Concept of Minimum Distance
Minimum distance between initial and last point
\[{{h}^{2}}=\text{ }{{b}^{2}}+{{p}^{2}}\], where
h = Hypotenuse
b = Base
p = Perpendicular
Remember this important rule is known as ‘Pythagoras Theorem’.
Example 1: Pinki starts moving from a point P towards East. After walking some distance she turns her left. Now, her direction more...

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Improving the logical ability.

To be perfect in solving problems.

What is Coding-Decoding?
Let us start it with an interesting story.
Suppose you and your papa like ice-cream very much. But your mummy does not want you two to have it because you both catch cold very easily. Then you and your papa make a secret plan to use the word 'Chocolate’ for ice-cream. Now, whenever you feel like eating ice-cream you say to your papa that you want to eat chocolates. Mummy hears it and thinks that you are really demanding chocolates. Therefore, she gives you permission to go out with papa and enjoy chocolates. Then you and your papa go out, eat ice-cream and comeback.
Do you think what happens here? Here, you coded the word 'Ice-cream' with another word 'Chocolate'. Only you and papa know about this code, when you say that you want to eat 'Chocolate' then your papa hears you and easily decodes it that you want to eat ice-cream. This can be presented as below.
Ice-cream \[\xrightarrow{Coded\,\,as}\] Chocolate \[\xrightarrow{Decoded\,\,as}\] Ice-cream
How to Decode?
In reasoning, words, letters and numbers are coded according to a certain rule. While solving problems, students have to identify that particular rule 1st and then the same rule is applied to decode other coded words, letters, number etc. The types of coding decoding problems will give you more clear concept about it. But before coming to the actual problems, you must remember the positions of letters in English alphabet in forward order that will help you in solving problems of coding-decoding.
Let us see the positions:
Table 1:

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What is Analogy?
Simple meaning of analogy is similarity. But, in terms of reasoning, the meaning of analogy is logical similarity between two or more things. This similarity may be on the basis of properties, kinds, traits, shapes etc.

Example:

(i) Student: School:: Patient: Hospital
Explanation: A ‘Student’ goes to ‘School’ in the same way a ‘Patient’ goes to ‘Hospital’. In other words, school (place to take education) is a proper place for a student and in the same way hospital (place to get treatment) is a proper place for a patient.
1st pair- Student: School (person and proper place relationship).
2nd pair- Patient: Hospital (person and proper place relationship).
Clearly, both pairs show similar relationship in a logical way. Hence, both pairs are analogous or it is said that both pairs exhibit analogy.
(ii) Good: Bad: : Tall : Short
Explanation: 1st pair - Good: Bad (opposite relationship)
2nd pair - Tall: Short (opposite relationship)
Clearly, both pairs show similar relationship (opposite relationship).
Hence, both pairs exhibit analogy.
Types of Problems
(a) Problems Based on Synonymous Relationship
In such problems, the words given in one pair have same meaning and the same relationship is found in another pair of words.
Example 1: Right: Correct:: Fat: Bulky
Explanation: 1st pair – Right: Correct (Synonymous relationship).
2nd pair - Fat: Bulky (synonymous relationship).
Example 2: Brave: Bold:: Wrong: Incorrect
Explanation: 1st pair - Brave: Bold (Synonymous relationship).
2nd pair - Wrong: Incorrect (Synonymous relationship).
Commonly Asked Question

Select the pair which is related in the same way as the pair of words given in the question.

Tough: Hard: : ______: ______
(a) Rich: Wealthy (b) Rich: Poor
(c) Tall: Short (d) True: False
(e) None of these
Answer: (a)
Explanation: 'Tough' and 'Hard' are synonymous words. In the same manner 'Rich' and 'Wealthy' are synonymous words.
Rest of the options is incorrect because words in option (b), (c) and (d) have opposite meanings and option (e) is useless because of the correctness of option (a).

‘Start’ is related to ‘Begin’ in the same way as ‘joy’ is related to…………..

(a) Right (b) False
(c) True (d) Delight
(e) None of these
Answer: (d)
Explanation: ‘Start’ and ‘Begin’ have same meaning. Similarly, ‘Joy’ and ‘Delight’ have same meaning.
Rest of the options is incorrect because of the correctness of option (d).
(b) Problems Based on Opposite Relationship
In such problems, the pair of words given in the question are opposite to each other in meaning (antonyms). Similarly, the pair of words in the answer must be opposite in meaning.
Example 1: Poor: Rich:: Weak : Strong
Explanation: 'Poor' is opposite to ‘Rich’ and in the same way more...

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What is Classification?
You must have in your mind what classification means. In fact, in classification, we take an element out of some given elements and the element to be taken out is different from the rest of the elements in terms of common properties, shapes, sizes, types, nature, colours, traits etc. In this way the remaining elements form a group and the element that has been taken out is not the member of that group, as this single element does not possess the common quality to be possessed by rest of the elements. For example, if we compare the animals like lion, cow, tiger, panther, bear and wolf, then we find that this is a group of animals. How do we classify then?
To understand this let us see the presentation given below:
Here/if we want to separate out one animal then definitely that animal will be cow because cow is the only animal in the group which is a domestic animal. Rest of the animals (lion, tiger, panther, bear and wolf) are wild animals. Hence, rest of the animals (lion, tiger, panther, bear & wolf) form a group of wild animals separating out the domestic animal (cow).
Similarly, out of 6 letters A, M, N X P & Q, we will take out A and form a group of 5 letters M, N, W, P & Q because out of given six letters only A is a vowel while remaining letters form a group of consonants.
Types of Classification
(a) Letter Based Classification.
Such classification is based on letters of English alphabet. So many groups of latter are given in the question in which one group is different from remaining group and hence, the different group will be our answer.
Example 1: Find the odd one out of the following options.
(a) PQT (b) UVY
(c) DEH (d) UN
(e) FGJ
Explanation:
(a) (b)
(c) (d)
As it is clear that except option (d), all the other options have a gap of 2 letters between 2nd and third letter and the 1st two letters are in consecutive order, while in case of option (d), 1st two letters are in consecutive order but there is a gap of 3 letters between 2nd and third letter. This is the reason why option (d) separates itself out of the remaining options. Hence, option (d) is the correct answer.
Example 2: Following are given options and out of them 4 form a more...

What is a Number Series?
A number series is a sequence of many elements made of numbers only. Such sequence is formed by putting the numbers one after another from left to right.

Example

(i) 1 2 3 4 5 \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\mathbf{1}\,\,\mathbf{2}\,\,\mathbf{3}\,\,\mathbf{4}\,\,\mathbf{5}} \right]\]
(ii) 6 5 4 3 2 \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\mathbf{6}\,\,\mathbf{5}\,\,\mathbf{4}\,\,\mathbf{3}\,\,\mathbf{2}} \right]\]
(iii) 1 3 5 7 \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\mathbf{1}\,\,\mathbf{3}\,\,\mathbf{5}\,\,\mathbf{7}} \right]\]
(iv)\[\left( 1+1 \right)\text{ }\left( 1+2 \right)\text{ }\left( 1+3 \right)\] \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\left( \mathbf{1}\,\,\mathbf{+}\,\,\mathbf{1} \right)\mathbf{ }\left( \mathbf{1 + 2} \right)\mathbf{ }\left( \mathbf{1 + 3} \right)} \right]\]
(v) \[\left( 1\times 1 \right)\,\,\left( 1\times 2 \right)\,\,\left( 1\times 3 \right)\] \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\left( \mathbf{1}\,\,\mathbf{\times }\,\,\mathbf{1} \right)\,\,\left( \mathbf{1}\,\,\mathbf{\times }\,\,\mathbf{2} \right)\,\,\left( \mathbf{1}\,\,\mathbf{\times }\,\,\mathbf{3} \right)} \right]\]
(vi) \[\left( 1\,\,\div \,\,1 \right)\,\,\left( 1\,\,\div \,\,2 \right)\,\,\left( 1\,\,\,\div \,\,3 \right)\] \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\left( \mathbf{1}\,\,\mathbf{\div }\,\,\mathbf{1} \right)\,\,\left( \mathbf{1}\,\,\mathbf{\div }\,\,\mathbf{2} \right)\,\,\left( \mathbf{1}\,\,\,\mathbf{\div }\,\,\mathbf{3} \right)} \right]\]
(vii) (4 - 1) (4 - 2) (4 - 3) \[\left[ \xrightarrow[\mathbf{Left}\,\,\mathbf{to}\,\,\mathbf{Right}]{\left( \mathbf{4 - 1} \right)\mathbf{ }\left( \mathbf{4 2} \right)\mathbf{ }\left( \mathbf{4 3} \right)} \right]\]
Note:
An element is a single member (identity) of a series. For example, in a number series 1, 2 15 8 12', each 1, 2, 15, 8 and 12 is an element.
Properties of Number Series
(1) A number series can be in forward or reverse order.
(2) A number series can be in random order.
(3) A number series must have more than one element.
(4) Numbers can be repeated in a number series.
(5) A single number series can have more than one series.
(6) A number series may have some arithmetical signs also.

Example

Look at the following:
(i) 1 2 3 4 5 6 7 (Forward order series)
(ii) 7 6 5 4 3 2 1 (Reverse order series)
Commonly Asked Question

Find the next number.

4 5 6.................
(a) 7 (b) 8
(c) 3 (d) 2
(e) None of these
Answer: (a)
Explanation: Option (a) is correct because the series goes as following:
\[4+1=5\]
\[5+1=~6\]
\[6+1=7\]
Rest of the options is incorrect because of the correctness of option (a).
Note: This problem is based on forward order series.

After how many numbers does 2 come in the series given below? (Count from left)

1 5 8 2 0 3
(a) 5 (b) 2
(c) 1 (d) 3
(e) None of these
Answer: (d)
Explanation: Option (d) is correct,
Let us see:
As 2 is the 4th number from left, it comes after the 3 numbers 1, 5 and 8.
Rest of the options is incorrect because of the correctness of option (d).
3. Which of the following is not a number series?
(a) 4 5 (b) 9
(c) 1 2 3 (d) 3 4 5 6
(e) None more...

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Introduction
Ranking is based on the arrangement of things in a particular order. The arrangement may be on the basis of their position, size, age etc.
Position Series Test
In this series, questions are asked about the positions of the persons from up or down/or from left or right etc. Some important types are as given below:
Order and Ranking Concepts:
Formulas to determine the positioning of a person
(1) \[\text{Left+Right=Total+1}\]
(2) \[\text{Left=Total+1--Right}\]
(3) \[\text{Right=Total+1--left}\]
(4) \[\text{Total=left+Right--1}\]
Note: The above formulas are only for a single person's position.

Example:

3rd from left
3rd from right
Total \[=3+31\]
Same for vertical and Horizontal

\[\text{Total+1=top+Bottom}\]

\[Top=Total+1Bottom\]

\[Bottom=Total+1Top\]

\[\text{Total=Top+Bottom--1}\]

In a row of 40 students, A is 13th from the left end, find the rank from right end.

Explanation: Total = 40
A's rank from right side \[\text{=Total+1--left}\]
\[=40+113\]
\[=27+1\]
\[=28\]
2. In a line of girls, if Kamla's position from the left is 15th and from the right her position is 17th, how many girls are there in the line?
Explanation: Total no. of girls = Kamla's position from the left + her position from the right - 1
\[=15+17-1=31\]
3. In a line of girls Nivedita's position from the left is 18th and Priti’s position from the right is 22nd. If there are 5 girls between them, what is the total number of girls in the line?
Explanation: In this question there are two possible positions.
Position I.
\[\therefore \]Total no. of girls \[=18+5+22=45\]
Position II.
\[\therefore \] Total no. of girls \[=22+18-\] (5 + Priti + Nivedita)
\[=22+18-7=33\]
4. In a line of girls, Juli's position from the left is 10th while Lalli's position from the right is 16th. When they interchange their positions, Juli’s position becomes 20th from the left, then what will be the position of Lalli from the right?
Explanation: Original position:
Since Juli's new position after interchange is 20th from the left.
\[\therefore \] Total no. of girls in the line \[=20+15=35\]
Hence, Lalli's position from the right \[=\left( 35-10 \right)+1={{26}^{th}}\]
Types of Order and Ranking
Type-1

Total number of persons = {(sum of positions of same person from both sides i.e. left and right side) - 1}

OR

Position of a person from opposite side = {(Total no. of persons - Position of same person from given side) +1}

Example:

In a row of persons, position of A from left side of the row is 27th and position more...