Unit V – Mathematical Reasoning and Aptitude
1. Types of Reasoning
Reasoning is the intellectual process of thinking logically to make inferences, decisions, or predictions from available information. In the context of mathematical and logical aptitude, reasoning is classified into several types that enable problem-solving and analytical thinking. The most common types include:
Deductive Reasoning: Deductive reasoning involves starting from general principles or premises and arriving at a specific conclusion. For instance, if we know that “all humans are mortal” and “Socrates is a human,” we can logically conclude that “Socrates is mortal.” This type of reasoning is deterministic and widely used in formal logic, proofs, and mathematical problem-solving.
Inductive Reasoning: Inductive reasoning begins with specific observations or examples and progresses towards a general conclusion. For example, observing that “the sun rises in the east every day” allows us to infer that “the sun always rises in the east.” Inductive reasoning is probabilistic and often used in scientific investigations and pattern recognition.
Analogical Reasoning: Analogical reasoning involves identifying relationships between two pairs of entities. For example, “Finger is to Hand as Toe is to Foot” demonstrates the relationship between components and their wholes. This type is especially useful in solving puzzles and analogy-based questions in aptitude tests.
Abductive Reasoning: Abductive reasoning seeks the most plausible explanation for observed facts. For instance, if we find wet streets, we may infer that “it probably rained.” This type of reasoning is common in problem-solving where information is incomplete, such as in detective work or diagnostic scenarios.
Causal Reasoning: Causal reasoning identifies cause-and-effect relationships. For example, “Smoking leads to lung disease” shows a causal link. In aptitude and reasoning assessments, causal reasoning tests a candidate’s ability to understand relationships and anticipate outcomes logically.
Inductive reasoning: "Every exam I have prepared for has been held on a Monday. The next exam is likely on a Monday."
2. Number Series, Letter Series, Codes and Relationships
These problems focus on recognizing patterns, sequences, and relationships between numbers or letters. They form a core part of aptitude tests and help develop analytical skills.
Number Series: Number series involve sequences of numbers where each number follows a defined rule, such as addition, subtraction, multiplication, division, or a combination of these. For example, in the series 2, 4, 8, 16, …, each number is obtained by multiplying the previous number by 2. Such series train candidates to identify patterns quickly and accurately.
Letter Series: Letter series follow a pattern of alphabets. For example, A, C, F, J, …, follows the pattern +2, +3, +4, +5, giving the next letter as O. Letter series improve the ability to recognize logical sequences and enhance memory skills.
Codes: Codes involve substituting letters or numbers according to a defined pattern or formula. For example, if A=1, B=2, …, then CODE = 3+15+4+5 = 27. Coding-decoding problems are used to assess logical thinking, attention to detail, and computational ability.
Relationships: These problems test a candidate's ability to understand relationships between entities, including analogies and cause-effect. For instance, "Finger is to Hand as Toe is to Foot" demonstrates a component-whole relationship. Identifying these connections develops higher-order reasoning skills.
Letter series: B, E, I, N, ? (Next letter = T)
Code: If CAT = 3+1+20 = 24, then DOG = 4+15+7 = 26
Relationships: Leaf : Tree :: Petal : Flower
3. Mathematical Aptitude
Mathematical aptitude is the skill to solve practical arithmetic and quantitative problems quickly and accurately. It is extensively tested in competitive exams and assessments. The key areas include:
Fractions
Fractions represent parts of a whole and require operations such as addition, subtraction, multiplication, and division. Understanding fractions is essential for advanced arithmetic problems.
Example 2: 7/8 × 2/3 = 14/24 = 7/12
Time & Distance
Time, speed, and distance are interconnected through the formula: Distance = Speed × Time. Solving such problems develops numerical reasoning and logical application skills.
Example 2: Time = Distance / Speed. Distance = 120 km, Speed = 40 km/h → Time = 3 h
Ratio, Proportion and Percentage
Ratio compares two quantities, proportion expresses equality of two ratios, and percentage represents a part per hundred. These concepts are used in a wide range of mathematical problems.
Example 2: If a product costs 200 and is sold for 250, profit % = ((250–200)/200)*100 = 25%
Profit and Loss
Profit occurs when Selling Price (SP) exceeds Cost Price (CP), and loss occurs when CP exceeds SP. Formulas include: Profit % = ((SP–CP)/CP)*100, Loss % = ((CP–SP)/CP)*100. Real-life examples include trade and commerce scenarios.
Interest and Discounting
Interest can be simple or compound. Simple Interest (SI) = (P × R × T)/100, and Compound Interest (CI) = P(1 + R/100)^T – P. Discounting involves determining present or future value of money.
CI = 1000*(1+5/100)^2 – 1000 = 102.50
Averages
Average is calculated by dividing the sum of all observations by the number of observations. It is widely used in statistics, data interpretation, and daily calculations.
Formulas and Key Notes
- SI = (P × R × T)/100
- CI = P(1 + R/100)^T – P
- Profit % = ((SP – CP)/CP) × 100
- Loss % = ((CP – SP)/CP) × 100
- Speed = Distance / Time
- Average = Sum of numbers / Count
- Percentage = (Part / Total) × 100
- Ratio = a:b = a/b
Unit V – Mathematical Reasoning and Aptitude
1. Types of Reasoning
Reasoning is the ability to think logically and solve problems systematically. There are various types:
- Deductive: General to specific. Example: "All humans are mortal. Socrates is a human. Therefore, Socrates is mortal."
- Inductive: Specific to general. Example: "Every cat I’ve seen has whiskers. Therefore, all cats have whiskers."
- Analogical: Relationship comparison. Example: "Hand is to glove as foot is to sock."
- Abductive: Most likely explanation. Example: "The street is wet. Likely reason: it rained."
- Causal: Identifying cause-effect. Example: "Smoking can cause lung disease."
Example 2: Inductive reasoning: "The sun rose every day. Therefore, the sun will rise tomorrow." (Shows probability)
Practice Questions
- Which reasoning starts with a general rule to conclude a specific fact?
A) Inductive
B) Deductive
C) Analogical
D) Abductive
Answer: B) Deductive - Inductive reasoning moves from:
A) Specific to General
B) General to Specific
C) Analogy to Conclusion
D) Cause to Effect
Answer: A) Specific to General - Analogical reasoning focuses on:
A) Cause
B) Relationships
C) Probability
D) Random facts
Answer: B) Relationships - Abductive reasoning is mainly about:
A) Certain facts
B) Likely explanation
C) Statistics
D) Patterns
Answer: B) Likely explanation - Causal reasoning identifies:
A) Analogy
B) Cause and Effect
C) Sequence
D) Probability
Answer: B) Cause and Effect - Example of Deductive: "All mammals have lungs. Whale is a mammal. Whale has lungs?"
A) Yes
B) No
C) Sometimes
D) Cannot say
Answer: A) Yes - Which reasoning is used to predict trends?
A) Deductive
B) Inductive
C) Analogical
D) Abductive
Answer: B) Inductive - Example of Analogical reasoning: "Tree is to Forest as Star is to ...?"
A) Sky
B) Galaxy
C) Moon
D) Planet
Answer: B) Galaxy - Abductive reasoning is commonly used in:
A) Detective work
B) Routine arithmetic
C) Memorization
D) Following instructions
Answer: A) Detective work - Which reasoning type deals with “if…then” logic?
A) Deductive
B) Inductive
C) Analogical
D) Abductive
Answer: A) Deductive
2. Number Series, Letter Series, Codes and Relationships
These problems test pattern recognition and logical thinking. Key types:
- Number Series: Identify the pattern to find the missing number. Example: 3, 6, 12, 24, ? (Pattern: ×2 → 48)
- Letter Series: Example: B, E, I, N, ? (Pattern: +3, +4, +5 → next = T)
- Codes: Represent letters or words using numbers or symbols. Example: If A=1, B=2, … CAT = 3+1+20=24
- Relationships: Understand associations. Example: Finger : Hand :: Toe : Foot
Letter series: A, C, F, J, ? (Pattern: +2, +3, +4 → Next = O)
Practice Questions
- Find the next number: 5, 10, 20, 40, ?
A) 60
B) 80
C) 100
D) 90
Answer: B) 80 - Letter series: B, D, G, K, ?
A) O
B) N
C) M
D) P
Answer: A) O - Code: If A=1, B=2, C=3, then BAD = ?
A) 5
B) 6
C) 7
D) 8
Answer: C) 7 - Relationship: Hand : Glove :: Foot : ?
A) Shoe
B) Sock
C) Sandal
D) Boot
Answer: B) Sock - Number series: 1, 4, 9, 16, ?
A) 20
B) 25
C) 30
D) 36
Answer: B) 25 - Letter series: A, E, I, M, ?
A) Q
B) P
C) N
D) R
Answer: A) Q - Find missing in number series: 2, 3, 5, 8, 12, ?
A) 15
B) 18
C) 17
D) 20
Answer: B) 18 - Code: DOG = ? if A=1, B=2, …
A) 26
B) 25
C) 27
D) 24
Answer: C) 27 - Relationship: Leaf : Tree :: Petal : ?
A) Plant
B) Flower
C) Stem
D) Branch
Answer: B) Flower - Number series: 3, 7, 15, 31, ?
A) 63
B) 62
C) 61
D) 60
Answer: A) 63
3. Mathematical Aptitude
This section deals with practical arithmetic and algebra problems. Key areas include:
Fractions
Operations with fractions include addition, subtraction, multiplication, and division. Examples:
Example 2: 7/8 × 2/3 = 14/24 = 7/12
Practice Questions
- 1/2 + 2/3 = ?
A) 7/6
B) 4/5
C) 5/6
D) 3/4
Answer: A) 7/6 - 3/4 × 8/9 = ?
A) 1/3
B) 2/3
C) 2/3
D) 1/2
Answer: B) 2/3 - 5/6 – 1/3 = ?
A) 2/6
B) 1/2
C) 1/3
D) 5/6
Answer: B) 1/2 - 7/10 ÷ 7/5 = ?
A) 1/2
B) 1
C) 5/10
D) 2/1
Answer: A) 1/2 - 2/5 + 3/10 = ?
A) 7/10
B) 1/2
C) 1/3
D) 3/5
Answer: A) 7/10 - 4/7 × 7/8 = ?
A) 1/2
B) 1
C) 1/4
D) 2/7
Answer: A) 1/2 - 5/8 – 1/4 = ?
A) 1/2
B) 3/8
C) 1/4
D) 5/8
Answer: B) 3/8 - 9/10 ÷ 3/5 = ?
A) 1/2
B) 3/2
C) 2/3
D) 3/5
Answer: B) 3/2 - 1/3 + 2/9 = ?
A) 1/2
B) 5/9
C) 2/3
D) 1/3
Answer: B) 5/9 - 7/12 × 3/4 = ?
A) 7/16
B) 7/12
C) 21/48
D) 7/8
Answer: C) 21/48
Time & Distance
Speed, Distance, and Time are related by: Distance = Speed × Time
Example 2: Time = Distance/Speed. If Distance = 120 km, Speed = 40 km/h → Time = 3 h
Practice Questions
- A train travels 90 km in 2 hours. Speed = ?
A) 45 km/h
B) 50 km/h
C) 60 km/h
D) 40 km/h
Answer: A) 45 km/h - A car covers 150 km in 3 hours. Time to cover 300 km?
A) 4 h
B) 6 h
C) 5 h
D) 7 h
Answer: B) 6 h - If speed = 50 km/h and distance = 200 km, time = ?
A) 3 h
B) 4 h
C) 5 h
D) 6 h
Answer: C) 4 h - Train travels 120 km at 60 km/h, Time = ?
A) 1 h
B) 2 h
C) 3 h
D) 4 h
Answer: B) 2 h - A bus speed = 80 km/h, distance = 160 km, Time = ?
A) 1 h
B) 2 h
C) 3 h
D) 4 h
Answer: B) 2 h - Distance = 180 km, speed = 60 km/h, Time = ?
A) 2 h
B) 3 h
C) 4 h
D) 5 h
Answer: B) 3 h - Time = Distance / Speed. Distance = 240 km, speed = 60 km/h, time = ?
A) 3 h
B) 4 h
C) 5 h
D) 6 h
Answer: B) 4 h - Car travels 150 km in 2.5 h. Speed = ?
A) 50 km/h
B) 55 km/h
C) 60 km/h
D) 65 km/h
Answer: C) 60 km/h - A man walks 5 km in 1 hour. How far in 4 h?
A) 15 km
B) 20 km
C) 25 km
D) 30 km
Answer: B) 20 km - Speed = 90 km/h, Time = 3 h. Distance = ?
A) 270 km
B) 250 km
C) 200 km
D) 180 km
Answer: A) 270 km
Formulas & Abbreviations
- SI = (P × R × T)/100
- CI = P(1 + R/100)^T – P
- Profit % = ((SP – CP)/CP) ×100
- Loss % = ((CP – SP)/CP) ×100
- Speed = Distance / Time
- Average = Sum of numbers / Count
- Percentage = (Part/Total) × 100
- Ratio = a:b = a/b
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