Quantitative Reasoning-1 (GE-104)
MODULE-1
What is Quantitative Reasoning?
Quantitative reasoning (QR) is the ability to use mathematical concepts and techniques to solve
real-world problems, analyse data, and make informed decisions. It involves the application of
mathematical tools like statistics, algebra, and logic to analyse situations and draw conclusions
based on data.
What is Mathematics?
The word mathematics is derived from the Greek word mathematikos, which means “inclined
to learn.” Literally speaking, to be mathematical is to be curious, open-minded, and interested
in always learning more! Today, we tend to look at mathematics in three different ways: as the
sum of its branches, as a way to model the world, and as a language.
➢ Branches of Mathematics
• logic—the study of principles of reasoning
• arithmetic—methods for operating on numbers (addition, subtraction, multiplication, and
division).
• algebra—methods for working with unknown quantities
• geometry—the study of size and shape
• trigonometry—the study of triangles and their uses
• probability—the study of chance
• statistics—methods for analysing data
• calculus—the study of quantities that change.
➢ Mathematics as a Way to Model the World
Mathematics also may be viewed as a tool for creating models, or representations of real
phenomena. For example, a road map is a model that represents the roads in some region.
Today, mathematical modeling is used in nearly every field of study for example business
management, engineering, computer science, artificial intelligence, physics and chemistry etc.
➢ Misconceptions about Mathematics
• Misconception One: Math Phobia (fear of mathematics)
• Misconception Two: Math requires a Special Brain (but the reality is that nearly
everyone can do mathematics. All it takes is self-confidence and hard work.)
• Misconception Three: the Math in Modern Issues Is too complex (the advanced
mathematical concepts underlying many modern issues are too complex for the
average person to understand.)
, • Misconception Four: Math Makes you Less Sensitive (Some people believe that
learning mathematics will somehow make them less sensitive to the romantic and
aesthetic aspects of life. But many people find beauty and elegance in mathematics
itself.)
• Misconception Five: Math Makes no allowance for creativity (but mathematics is not
just about rules and calculations; it is a deeply creative field. e.g. applying
mathematics is like the creative process of building a home.)
• Misconception Six: Math Is Irrelevant to My Life
• Misconception Seven: Math Provides exact answers
• Misconception Eight: It’s oK to Be “Bad at Math”
➢ Write down the Contributions of English and Muslim Mathematicians
in History.
➢ Different types of numbers and their use in daily life activities.
1. Natural Numbers (N)
• Definition: Counting numbers starting from 1 (1, 2, 3, ...).
• Use in Daily Life:
o Counting objects (e.g., apples, books).
o Keeping track of people (e.g., participants in an event).
2. Whole Numbers (W)
• Definition: Natural numbers plus zero (0, 1, 2, 3, ...).
• Use in Daily Life:
o Representing quantities where zero is included (e.g., measuring items,
inventory counts).
3. Integers (Z)
• Definition: Whole numbers that can be positive, negative, or zero (..., -3, -2, -1, 0,
1, 2, 3, ...).
• Use in Daily Life:
o Accounting for debts (negative integers) and credits (positive integers).
o Temperature changes (e.g., below or above zero).
4. Rational Numbers (Q)
• Definition: Numbers that can be expressed as the ratio of two integers (e.g., 1/2,
3/4, 0.75).
• Use in Daily Life:
o Financial transactions (e.g., prices, discounts).
, o Cooking (e.g., measuring ingredients).
• Terminating or recurring or repeating: Terminating decimals are decimal
numbers that have a finite number of digits after the decimal point. e.g. 1/2, 1/4 etc.
• Non-terminating or non-repeating: Non-terminating decimals are decimal
numbers that continue indefinitely without stopping e.g. 1/3, 2/3.
5. Irrational Numbers (𝑸′ )
• Definition: Numbers that cannot be expressed as a simple fraction.
• Use in Daily Life:
o Geometry and architecture (using π in calculations for circles).
o Natural phenomena (e.g., the golden ratio in art and design).
• Non-terminating or non-repeating: Non-terminating decimals are decimal
numbers that continue indefinitely without stopping e.g. √2, π.
6. Real Numbers (R)
• Definition: All rational and irrational numbers (including integers, fractions, and
decimals).
• Use in Daily Life:
o Measuring distances, weights, and temperatures.
o Any scenario involving continuous quantities (like time).
𝑹 = 𝑸 ∪ 𝑸′
7. Complex Numbers
• Definition: Numbers that have a real part and an imaginary part (e.g., 3 + 2i, where
iii is the imaginary unit).
• Use in Daily Life:
o Engineering and physics, particularly in signal processing and control
systems.
o Complex financial calculations in advanced economics.
, MODULE-2
Problem Solving Techniques
Unit M2.1 (Practical life scenarios involving fractions, Principles
to deal with fractions)
➢ Fraction
𝑎
A fraction is a mathematical expression in the form 𝑏, where a and b are any integers and
b≠0. It consists of two main parts: numerator and denominator. In this case, a is numerator
5 −2
and b is denominator. Example: 7 , etc.
9
• Addition of two factors with same and different denominator
• Subtraction of two factors with same and different denominator
• Multiplying the factors with same and different denominator
• Dividing the factors with same and different denominator
➢ Mixed Numbers
A mixed number is a way of representing a quantity that consists of both a whole number
3
and a fraction. Example: 2 7 .
➢ Decimal Fractions
A decimal fraction is a fraction where the denominator is a power of ten (such as 10, 100,
1000, etc.), and it is expressed in decimal notation.
• Types of Decimal Fractions:
Terminating Decimal Fractions: These have a finite number of decimal places. For
example:
o 0.5 (which is 5/10)
o 0.75 (which is 75/100 )
Repeating Decimal Fractions: These have a decimal part that repeats indefinitely.
For example:
o 1/3=0.333... (repeating the digit 3)
o 1/6=0.1666... (repeating the digit 6)
➢ Converting Between Fractions and Decimal Fractions:
• Fraction to Decimal: To convert a simple fraction to a decimal, divide the numerator
by the denominator. For example: 1/4=0.25
• Decimal to Fraction: To convert a decimal to a fraction, write it over the appropriate
power of ten, then simplify if necessary. For example: 0.75=75/100.
MODULE-1
What is Quantitative Reasoning?
Quantitative reasoning (QR) is the ability to use mathematical concepts and techniques to solve
real-world problems, analyse data, and make informed decisions. It involves the application of
mathematical tools like statistics, algebra, and logic to analyse situations and draw conclusions
based on data.
What is Mathematics?
The word mathematics is derived from the Greek word mathematikos, which means “inclined
to learn.” Literally speaking, to be mathematical is to be curious, open-minded, and interested
in always learning more! Today, we tend to look at mathematics in three different ways: as the
sum of its branches, as a way to model the world, and as a language.
➢ Branches of Mathematics
• logic—the study of principles of reasoning
• arithmetic—methods for operating on numbers (addition, subtraction, multiplication, and
division).
• algebra—methods for working with unknown quantities
• geometry—the study of size and shape
• trigonometry—the study of triangles and their uses
• probability—the study of chance
• statistics—methods for analysing data
• calculus—the study of quantities that change.
➢ Mathematics as a Way to Model the World
Mathematics also may be viewed as a tool for creating models, or representations of real
phenomena. For example, a road map is a model that represents the roads in some region.
Today, mathematical modeling is used in nearly every field of study for example business
management, engineering, computer science, artificial intelligence, physics and chemistry etc.
➢ Misconceptions about Mathematics
• Misconception One: Math Phobia (fear of mathematics)
• Misconception Two: Math requires a Special Brain (but the reality is that nearly
everyone can do mathematics. All it takes is self-confidence and hard work.)
• Misconception Three: the Math in Modern Issues Is too complex (the advanced
mathematical concepts underlying many modern issues are too complex for the
average person to understand.)
, • Misconception Four: Math Makes you Less Sensitive (Some people believe that
learning mathematics will somehow make them less sensitive to the romantic and
aesthetic aspects of life. But many people find beauty and elegance in mathematics
itself.)
• Misconception Five: Math Makes no allowance for creativity (but mathematics is not
just about rules and calculations; it is a deeply creative field. e.g. applying
mathematics is like the creative process of building a home.)
• Misconception Six: Math Is Irrelevant to My Life
• Misconception Seven: Math Provides exact answers
• Misconception Eight: It’s oK to Be “Bad at Math”
➢ Write down the Contributions of English and Muslim Mathematicians
in History.
➢ Different types of numbers and their use in daily life activities.
1. Natural Numbers (N)
• Definition: Counting numbers starting from 1 (1, 2, 3, ...).
• Use in Daily Life:
o Counting objects (e.g., apples, books).
o Keeping track of people (e.g., participants in an event).
2. Whole Numbers (W)
• Definition: Natural numbers plus zero (0, 1, 2, 3, ...).
• Use in Daily Life:
o Representing quantities where zero is included (e.g., measuring items,
inventory counts).
3. Integers (Z)
• Definition: Whole numbers that can be positive, negative, or zero (..., -3, -2, -1, 0,
1, 2, 3, ...).
• Use in Daily Life:
o Accounting for debts (negative integers) and credits (positive integers).
o Temperature changes (e.g., below or above zero).
4. Rational Numbers (Q)
• Definition: Numbers that can be expressed as the ratio of two integers (e.g., 1/2,
3/4, 0.75).
• Use in Daily Life:
o Financial transactions (e.g., prices, discounts).
, o Cooking (e.g., measuring ingredients).
• Terminating or recurring or repeating: Terminating decimals are decimal
numbers that have a finite number of digits after the decimal point. e.g. 1/2, 1/4 etc.
• Non-terminating or non-repeating: Non-terminating decimals are decimal
numbers that continue indefinitely without stopping e.g. 1/3, 2/3.
5. Irrational Numbers (𝑸′ )
• Definition: Numbers that cannot be expressed as a simple fraction.
• Use in Daily Life:
o Geometry and architecture (using π in calculations for circles).
o Natural phenomena (e.g., the golden ratio in art and design).
• Non-terminating or non-repeating: Non-terminating decimals are decimal
numbers that continue indefinitely without stopping e.g. √2, π.
6. Real Numbers (R)
• Definition: All rational and irrational numbers (including integers, fractions, and
decimals).
• Use in Daily Life:
o Measuring distances, weights, and temperatures.
o Any scenario involving continuous quantities (like time).
𝑹 = 𝑸 ∪ 𝑸′
7. Complex Numbers
• Definition: Numbers that have a real part and an imaginary part (e.g., 3 + 2i, where
iii is the imaginary unit).
• Use in Daily Life:
o Engineering and physics, particularly in signal processing and control
systems.
o Complex financial calculations in advanced economics.
, MODULE-2
Problem Solving Techniques
Unit M2.1 (Practical life scenarios involving fractions, Principles
to deal with fractions)
➢ Fraction
𝑎
A fraction is a mathematical expression in the form 𝑏, where a and b are any integers and
b≠0. It consists of two main parts: numerator and denominator. In this case, a is numerator
5 −2
and b is denominator. Example: 7 , etc.
9
• Addition of two factors with same and different denominator
• Subtraction of two factors with same and different denominator
• Multiplying the factors with same and different denominator
• Dividing the factors with same and different denominator
➢ Mixed Numbers
A mixed number is a way of representing a quantity that consists of both a whole number
3
and a fraction. Example: 2 7 .
➢ Decimal Fractions
A decimal fraction is a fraction where the denominator is a power of ten (such as 10, 100,
1000, etc.), and it is expressed in decimal notation.
• Types of Decimal Fractions:
Terminating Decimal Fractions: These have a finite number of decimal places. For
example:
o 0.5 (which is 5/10)
o 0.75 (which is 75/100 )
Repeating Decimal Fractions: These have a decimal part that repeats indefinitely.
For example:
o 1/3=0.333... (repeating the digit 3)
o 1/6=0.1666... (repeating the digit 6)
➢ Converting Between Fractions and Decimal Fractions:
• Fraction to Decimal: To convert a simple fraction to a decimal, divide the numerator
by the denominator. For example: 1/4=0.25
• Decimal to Fraction: To convert a decimal to a fraction, write it over the appropriate
power of ten, then simplify if necessary. For example: 0.75=75/100.
