# Chapter 4 Mathematical Reasoning And Aptitude (UGC NTA NET JRF Teaching and Research Aptitude Book)

## Reasoning And Its Types

Reasoning is one of the highest order of thinking, it is stepwise thinking and mental recognition of cause and eff ect relationships. Reasoning is a productive thinking in which insight and past experiences are required.

Reasoning is a factor of intelligence. Reasoning is a process in which pre-knowledge, experiences, insight and understanding of relationship are used to solve the problem. The ability to reason is closely related to intelligence.

It goes in the direction given to the learner, i.e., it is always goal-oriented. It is creative and refl ective in nature. Reasoning ability develops gradually. It means that experiences are also helpful in developing reasoning power along with intelligence. There may be more than one logic to draw an inference, i.e., reasoning is multi-dimensional.

Reasoning entails the following steps. Numerous philosophical mathematicians and psychologists have given the following six steps for reasoning.

1. Identification of the problem 2. Defining the problem 3. Formation of hypothesis 4. Collection of data 5. Tabulating and systematizing data 6. Verification/evaluation of hypothesis

## Types Of Reasoning

Aristotle had given an extended, systematic treatment of the methods of human reasoning. The three methods were deductive, inductive and abductive reasonings.**1. Deductive reasoning: **It is also known as analytical reasoning as it deals with objects by looking at its component parts. This type of reasoning can also be called deductive reasoning. Formal logic has been described as the science of deduction. The concepts of syllogism has been explained in Chapter 6.**2. Inductive reasoning: **It is also known as ‘synthetically reasoning’ that deals with a class of objects by looking at the common properties of each object in the class. The method is called inductive reasoning. The study of inductive reasoning is generally carried out within the field known as ‘informal logic’ or ‘critical thinking’.**3. Abductive reasoning: **Abductive reasoning is considered as the third form of reasoning. It is somewhat similar to inductive reasoning. It takes its clues from the term ‘guessing’, since conclusions drawn here are based on probabilities. Here, it is presumed that the most plausible conclusion is also the correct one.** Example: Major premise: **The container is filled with yellow pebbles.

**Minor premise:**Bobby has a yellow pebble in his hand.

**Conclusion:**The yellow pebble in Bobby hand was taken out of the container.

By abductive reasoning, the possibility that Bobby took the yellow pebble from the container is reasonable, though it is purely based on the speculation. Anyone could have given the yellow pebble to Bobby, or probably Bobby could have bought a yellow pebble at a retail store. Therefore, abducing that Bobby took the yellow pebble, from the observation of ‘the yellow pebble filled container’ may lead to a false conclusion. Unlike deductive and inductive reasoning, abductive reasoning is not commonly used for psychometric testing.

Here, we can discuss other forms of reasoning as well.

**Hypothetical Syllogism (Modus Ponens)**

A syllogism is simply a three line argument that consists of exactly two premises and a conclusion. A hypothetical syllogism is a syllogism that conations at least one hypothetical or conditional (if-then) premise. So that such type of deductive reasoning is also known as conditional reasoning. This pattern of reasoning is also known as

*modus ponens*. The four varieties of

*modus*

*ponens*are as under.

Chain Argument

Chain argument consists of three conditional statements that link together. Here is an example of a chain argument.

If I do not appear in exam, then I will fail in graduation.

If I fail in graduation, then I will lose my time and money. Therefore, if I do not appear in exam, then I will lose my time and money.

Modus Tollens

These are sometimes called ‘denying the consequences’ because they consist of one conditional premise, a second premise that denies (asserts to be false) the consequences of the conditional and a conclusion that denies to be antecedent of the conditional.

Here is an example of modus tollens argument.

If we are in Manchester, then we are in Gujarat.

We are not in Gujarat. Therefore, we are not in Manchester.

Denying the Antecedent Argument

In such argument, first premise that denies (i.e., asserts to be false) the antecedent of the conditional and a conclusion that denies the consequent of the conditional.

Here, is an example.

If we are in Chandigarh, then we are in North.

We are not in Chandigarh. Therefore, we are not in North.

We can notice that the premises in above examples are true and the conclusion is false. The pattern of reasoning of this argument is not logically reliable.

Affirming the Consequent

This pattern of reasoning is also faulty and affirming the consequent.

For example, if we are on Venus, then we are in solar system.

We are in the solar system. Therefore, we are on Venus.

Such pattern of reasoning has true premises and a false conclusion; it is clear that affirming the consequent is not logically reliable.

### Categorical Syllogism

They have been discussed in Unit 6 also. Here, the statements of the premises begin typically with ‘all’, ‘none’ or ‘some’ and conclusion start with ‘therefore’ or ‘hence’.**Argument from Definition**

An argument from definition, the conclusion is presented as being true by definition. Nand is a cardiologist. Therefore, Nand is a doctor. A straightforward relationship among cardiologist and doctor (two elements) is observed here. The conclusion of valid deductive reasoning contains no more information than the premises.**Argument by Elimination**

An argument by elimination seeks to logically rule out various possibilities until only a single possibility remains. It is like attempting questions in an examination.**Argument Based on Mathematics**

The main aim of Mathematics is to develop reasoning power of humanity. Here, the conclusion is claimed to depend largely or entirely on some mathematical calculation or measurement (perhaps in conjunction with one or more non-mathematical premises). Here, is an example. Twelve is greater than eight. Eight is greater than four. Therefore, twelve is greater than four.

## Inductive Reasoning

Let’s discuss inductive reasoning in more detail. The statement or proposition is based on general observations and experiences; such reasoning is called inductive reasoning. There is a strong contrast with deductive reasoning. Even in the strongest cases of inductive reasoning, the truth of the premises does not guarantee the truth of the conclusion. Rather, the conclusion of an inductive argument follows with some degree of probability. There may be more information in the conclusion of an inductive reasoning than that is already containing in the premises. Thus, this method of reasoning is applicative.

In inductive reasoning, a statement in one particular case will be true in all other cases in same serial order.

It may be applied generally to all such cases. Here, one can formulate generalized statement or principle and conclusions on the basis of certain facts and specific examples.

Once we have discussed inductive reasoning, now we can discuss six common patterns of inductive reasoning.**1. Inductive generalization: **It is an argument in which a generalization is claimed to be probably true based on the information about some members of a class. All inductive generalization claims that their conclusions are probable rather than certain.**2. Predictive argument: **It is a statement about what may or will happen in the future, here a prediction is defended with reasons. They are among the most common patterns of inductive reasoning. Here is an example, it has rained in Mumbai every June since weather records have been kept. Therefore, it will probably rain in Mumbai next June.

Nothing in the future is absolutely certain; argument containing predictions are usually inductive.**3. Argument from authority: **The conclusion is supported by presumed authority or witness who has said that the conclusion is true. We can never be absolutely certain that a presumed authority or witness is accurate or reliable.**4. Causal argument: **One of the most basic, most common and most important kinds of knowledge, we seek is knowledge of cause and effect. A causal argument asserts or denies that something is the cause of something else.**5. Statistical argument: **A statistical argument rests on statistical evidence, i.e, evidence that some percentage of some group has some particular characteristics.**6. Argument from analogy: **An analogy is a comparison of two or more things that are claimed to be alike in some relevant respect. Towards the end, we can say that inductive reasoning is informative because the conclusion of an inductive reasoning contains more information than is already contained in the premises.

Other Reasoning Types

We are using verbal and non-verbal symbols to communicate with others. There are four types of reasoning, which are explained as under.**1. Verbal reasoning: **Normally, we communicate with others by language and the language is a vehicle for reasoning. So reasoning without language or words-symbols is not possible. In verbal reasoning, we use linguistic symbols like words. Some verbal reasoning ability tests are available.**2. Non-verbal reasoning: **In competitive examinations, we observe that there is a part of reasoning in written paper containing some figures, graphs, drawings which can measure the non-verbal reasoning ability of the contestant.**3. Reasoning as propositions: **It is often difficult to determine whether a long and complex argument is valid or invalid just by reading. To analyse the parts of whole arguments and symbolize it to determine the validity of arguments, this logical process is known as propositional reasoning.

Here, is an example.

If *a *= 4 and *b *=7 *c *=? and *a *+ *c *= *b*

Then 4 + *c *=7 Therefore, *c *=3.**4. Automated reasoning: **It is basically an area of computer science. It understands different aspects of reasoning to allow the creation of computer software to reason completely or almost completely automatically. Sometimes, it is usually considered a subfield of artificial intelligence, but it also has strong connections to theoretical computer science and even philosophy.**5. Brain’s centre of reasoning: **The left hemisphere is dominant for most of the people; it controls written and spoken languages and mathematical calculations.

Prefrontal cortex is a farthest forward area in head and it is the associated area of frontal lobe. The left brain is said to be analytical, logical, mathematical hemisphere, concerned with cause and effect scientific thinking.

## Series Completion

A series may consist of a number series or a letter series. There are several such series, such as finding the missing numbers, replacing the wrong numbers, finding the missing letters, finding the wrong group of numbers or letters, to name a few.

### Number Series

Prime Number Series**Example 1**

2, 3, 5, 7, 11, 13, 17, … (a) 15 (b) 17 (c) 18 (d) 19*Solution*

The given series is a prime number series. The next prime number is 19.** Answer: **(d)

**Example 2**

2, 5, 11, 17, 23, 31, … (a) 33 (b) 37 (c) 41 (d) 43

*Solution*The prime numbers in this range are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, … Prime numbers have been written alternatively. Thus, after 31, the prime numbers are 37, 41, … Ignoring 37, the answer is 41.

**(c) Difference Series**

*Answer:***Example 3**

2, 5, 8, 11, 14, 17, …, 23, 26, (a) 19 (b) 21 (c) 20 (d) 18

*Solution*The difference between the numbers is 3 (17 + 3 = 20).

**(c)**

*Answer:***Example 4**

45, 38, …, 24, 17, 10, 3 (a) 31 (b) 34 (c) 38 (d) 29

*Solution*The difference between the consecutive numbers is 7 (38 – 7 = 31).

**(a) Multiplication Series**

*Answer:***Example 5**

2, 6, 18, 54, …, 486, 1,458 (a) 152 (b) 182 (c) 162 (d) 108

*Solution*The numbers are multiplied by 3 to get the next number (54 × 3 = 162).

**(c)**

*Answer:***Example 6**

3, 12, 48, …, 768, 3,072 (a) 192 (b) 216 (c) 512 (d) 72

*Solution*The numbers are multiplied by 4 to get the next number (48 × 4 = 192).

**(a) Division Series**

*Answer:***Example 7**

720, 120, 24, 6, …, 1 (a) 1 (b) 2 (c) 3 (d) 4

*Solution*720 divided by 6 = 120 120 divided by 5 = 24 24 divided by 4 = 6 6 divided by 3 = 2 2 divided by 2 = 1

**(b)**

*Answer:***Example 8**

32, 48, 72, …, 162, 243 (a) 84 (b) 96 (c) 108 (d) 132

*Solution*Each number is being multiplied by 3/2 to get the next number.

**(c) N 2 Series**

*Answer:***Example 9**

1, 4, 9, 16, 25, 36, …, 64 (a) 42 (b) 44 (c) 45 (d) 49

*Solution*The series is squares of 1, 2, 3, 4 and so on.

**(d)**

*Answer:***Example 10**

0, 4, 16, 36, 64, …, 144 (a) 100 (b) 84 (c) 96 (d) 120

*Solution*The series is squares of even numbers, such as 2, 4, 6, 8, 10 and 12. Hence, the answer is 102 = 100.

**(a) N 2 − 1 Series**

*Answer:***Example 11**

0, 3, 8, 15, 24, 35, 48, 63, … (a) 80 (b) 82 (c) 83 (d) None of the above

*Solution*The series is 12 – 1, 22 – 1, 32 – 1 and so on. The next number is 92 – 1 = 80.

**(a)**

*Answer:*

*Alternative solution*The differences between the numbers across the series are 3, 5, 7, 9, 11, 13, 15 and 17. The next number is 63 + 17 = 80.

N 2 + 1 Series

**Example 12**

2, 5, 10, 17, 26, 37, …, 65 (a) 50 (b) 48 (c) 49 (d) 51

*Solution*The series is 12 + 1, 22 + 1, 32 + 1 and so on. The next number is 72 + 1 = 50.

**(a) N 2 + N Series and N 2 − N Series**

*Answer:***Example 13**

0, 2, 6, 12, 20, 30, …, 56 (a) 36 (b) 40 (c) 42 (d) None of the above

*Solution*The series is 02 + 0, 12 + 1, 22 + 2, 32 + 3 and so on. The missing number is 62 + 6 = 42. The next number = 62 +

6 = 42.

**(c)**

*Answer:***First alternative solution**

The series is 0 × 1,1 × 2, ……. 1 × 2, 2 × 3, 3 × 4, 4 × 5 and 5 × 6 = 30. The next number is 6 × 7 = 42.

**Second alternative solution**

The series is 12 – 1, 22 – 2, 32 – 3, 42 – 4, 52 – 5, 62 – 6, 72 – 7, 82 – 8 and so on.

N 3 Series

**Example 14**

1, 8, 27, 64, 125, 216, … (a) 256 (b) 343 (c) 365 (d) 400

*Solution*The series is 13, 23, 33, etc. The missing number is 73 =

343.

**(b) N 3 + 1 Series**

*Answer:***Example 15**

2, 9, 28, 65, 126, 217, 344, … (a) 513 (b) 362 (c) 369 (d) 361

*Solution*The series is 13 + 1, 23 + 1, 33 + 1 and so on. Thus, the missing number is 83 + 1 = 513.

Answer: (a)

## Classificati On

The questions based on classification are based on similarity or dissimilarity between a number of items or objects. Some objects are grouped together on the basis of some common characteristics. The candidate has to identify that characteristic and separate out the object that does not belong to the group. This test is also known as ‘Odd Man Out’.

### Choosing the Odd Word

In these types of problems, some words belong to the real word. They have certain common features except the odd one.**Example 44**

Choose the word that is least like the other words in a group? (a) Calendar (b) Date (c) Day (d) Month** Answer: **(a)

*Explanation*All other words are parts of a calendar.

**Example 45**

Choose the word that is least like the other words in a group? (a) Peacock (b) Vulture (c) Sparrow (d) Swan

**(c)**

*Answer:*

*Explanation*Swan is the only water bird in the group.

### Choosing the Odd Pair of Words

In each of the following questions, five pairs of words are given, out of which the words in five pairs bear a certain common relationship. Choose the pair in which the words are differently related.**Example 46**

Choose the pair in which the words are differently related? (a) Man : Crowd (b) Cow : Herd (c) Sheep : Flock (d) Fish : Shoal** Answer: **(a)

*Explanation*In all other pairs, the second word is a collective group of the first.

**Example 47**

Choose the pair in which the words are differently related? (a) Joule : Energy (b) Ampere : Current (c) Angle : Degree (d) Pascal : Pressure

**(c)**

*Answer:*

*Explanation*In all other pairs, first is a unit to measure the second.

### Choosing the Odd Numeral

In each of the following questions, four numbers are given. Out of these, three are alike in a certain way except one.**Example 48**

Choose the number that is different from others in the group? (a) 139 (b) 177 (c) 144 (d) 183** Answer: **(c)

*Explanation*Number 144 is the only perfect square number in the group.

**Example 49**

Choose the number that is different from others in the group? (a) 127 (b) 345 (c) 361 (d) 514

**(b)**

*Answer:*

*Explanation*All other numbers except 361 are two more than the cube of a certain number.

### Choosing the Odd Numeral Pair or Group

Choose the odd numeral pair or group in each of the following questions.**Example 50**

Choose the number pair or group that is different from others? (a) 15 : 46 (b) 12 : 37 (c) 9 : 28 (d) 8 : 33** Answer: **(d)

*Explanation*In all other pairs, second number = (first number × 3) + 1

*.*

**Example 51**

Choose the number pair or group that is different from others? (a) 3, 5 (b) 7, 2 (c) 6, 2 (d) 1, 7

**(b)**

*Answer:*

*Explanation*In all other pairs, the sum of two numbers is 8.

### Choosing the Odd Letter Group

In each of the following questions, some groups of letters are given, where all of which, except one, share a common similarity.**Example 52**

Choose or find the odd letter group.

(a) BCD (b) NPR (c) KLM (d) RQP** Answer: **(b)

*Explanation*All other groups contain three consecutive letters of the alphabet series.

**Example 53**

Choose the group of letters that is different from others.

(a) KLM (b) IJK (c) PQR (d) RST

**(b)**

*Answer:*

*Explanation*No other group contains a vowel.

## Blood Relati Ons

The questions that are asked in this section depend upon relation. The candidate should have a sound knowledge of the blood relations in order to solve the questions. To remember easily, the relations may be divided onto two sides as given below in.

Blood Relations of Paternal and Maternal Sides

Relations of paternal side | |

Father’s father | Grandfather |

Father’s mother | Grandmother |

Father’s brother | Uncle |

Grandfather’s Son | Father or Uncle |

Grandfather’s only son | Father |

Father’s sister | Aunt |

Children of uncle | Cousin |

Wife of uncle | Aunt |

Children of aunt | Cousin |

Husband of aunt | Uncle |

Sister’s husband | Brother-in-law |

Wife’s brother | Brother-in-law |

Brother’s son | Nephew |

Brother’s wife | Sister-in-law |

Brother’s daughter | Niece |

Grandson’s or granddaughter’s daughter | Great granddaughter |

Mother’s or father’s son/daughter | Brother/sister |

Son’s wife | Daughter-in-law |

Relations of maternal side | |

Mother’s father | Maternal grandfather |

Mother’s mother | Maternal grandmother |

Mother’s brother | Maternal uncle |

Mother’s sister | Aunt |

Children of maternal uncle | Cousin |

Wife of maternal uncle | Maternal aunt |

Others | |

Children of same parents | Siblings |

Common term for husband and wife | Spouse |

### Developing a Family Relationship Tree

To develop a blood relation tree, some standard symbols may be used to tell about the relationships among the family members.

Suppose M is male and N is female. Some authors use the sign of + and – for indicating male and female.

Cousin is a common gender; it means that this relationship can be used for both male and female.

Condition | Sign |

When M is male. | +M |

When N is female. | -N |

M and N are married to each other. | M = N |

P and Q are siblings. | P « > |

A is the child of B. | A |

B | |

When A has two children B | A |

### Approach to Draw the Family Relations Diagram

To draw a family tree,**1. **First of all identify the males and the females, then according to generation, try to put each member at the appropriate position in the tree.**2. **Draw the diagram with relationships among family members using notations.**3. **Once the diagram is filled, the candidate can answer the given questions.