BASIC-MATHEMATICS-SOLVINGS-FORM1

Objectives: BASIC-MATHEMATICS-SOLVINGS

Mathematics Form 1 - Table of Contents

Table of Contents

Acknowledgements

Preface

Chapter One: Concept of Mathematics

  • Meaning of mathematics
  • Branches of mathematics
  • Relationship between mathematics and other subjects
  • Importance of mathematics

Chapter Two: Numbers I

  • Base ten numeration
  • Place value of a digit in a number
  • Total value
  • Writing numbers in words and in numerals
  • Natural and whole numbers
  • Even, odd, and prime numbers
  • Operations on whole numbers
  • BODMAS
  • Factors and multiples of numbers
  • Operations on integers

Chapter Three: Fractions

  • Fractions
  • Types of fractions
  • Equivalent fractions
  • Comparison of fractions
  • Operations on fractions

Chapter Four: Decimals and Percentages

  • Decimals
  • Types of decimals
  • Operations on decimals
  • Percentages

Chapter Five: Metric Units

  • Metric units of length
  • Metric units of mass
  • Metric units of time
  • Metric units of capacity

Chapter Six: Approximations

  • Rounding off numbers
  • Approximations in calculations
  • Significant figures
  • Decimal places

Chapter Seven: Introduction to Geometry

  • Points, lines, rays, line segments, and planes
  • Angles
  • Perpendicular lines
  • Transversals
  • Polygons and polygonal regions
  • Triangles
  • Quadrilaterals
  • Circles

Chapter Eight: Algebra

  • Algebraic expressions
  • Algebraic equations
  • Simultaneous equations
  • Inequalities with one unknown

Chapter Nine: Numbers II

  • Rational numbers
  • Irrational numbers
  • Real numbers
  • Absolute value of a real number

Chapter Ten: Ratios, Profit, and Loss

  • Ratios
  • Proportions
  • Profit and loss
  • Simple interest

Chapter Eleven: Coordinate Geometry

  • Coordinates of a point
  • Gradient of a straight line
  • Equation of a straight line
  • Graphing straight lines
  • Solving linear simultaneous equations graphically

Chapter Twelve: Perimeters and Areas

  • Perimeters of polygons
  • Circumference of a circle
  • Area
  • Areas of rectangles and squares
  • Area of a triangle
  • Areas of parallelograms and trapezia
  • Area of a kite
  • Area of a circle

Additional Sections:

  • Answers to Odd-Numbered Questions
  • Glossary
  • Bibliography
Chapter One: Concept of Mathematics – Questions & Answers

Chapter One: Concept of Mathematics
Questions & Answers

1. What is mathematics?

Answer: Mathematics is the study of numbers, shapes, quantities, patterns, and their relationships, often using logical reasoning and symbolic representations.

2. Give one definition of mathematics.

Answer: One definition: “Mathematics is the science of quantity, structure, space, and change.”

3. Why is mathematics described as a language of science?

Answer: Because it provides precise notation and tools for describing scientific phenomena in physics, chemistry, biology, economics, and more.

4. List two branches of mathematics.

Answer: Examples: Arithmetic, Algebra, Geometry, Trigonometry, Calculus, Statistics.

5. Define Arithmetic and Algebra.

Answer: Arithmetic deals with basic operations on numbers; Algebra generalizes arithmetic using symbols to represent numbers.

6. How is Geometry different from Trigonometry?

Answer: Geometry studies shapes and space, while Trigonometry focuses on relationships among angles and sides in triangles.

7. What is Calculus?

Answer: Calculus is the branch of mathematics dealing with rates of change (differentiation) and accumulation of quantities (integration).

8. Describe the relationship between mathematics and physics.

Answer: Physics uses mathematical models to describe natural laws and predict behavior of physical systems.

9. How does mathematics relate to economics?

Answer: Economics uses mathematics to model supply and demand, optimize resources, and analyze trends.

10. In what way is mathematics used in geography?

Answer: Mathematics helps measure distances, calculate areas, represent maps, and analyze spatial data.

11. Why is mathematics important in everyday life?

Answer: It helps in budgeting, cooking measurements, time planning, problem solving, and logical thinking.

12. Give a real‑world example of mathematics in trade.

Answer: Calculating change, VAT, discounts, profit margins, and interest on loans.

13. What role does mathematics play in medicine?

Answer: It’s used in medical imaging (like CT scans), dosage calculation, statistics in clinical trials.

14. Define Branches of mathematics and list three.

Answer: Branches are specialized areas. Examples: Number theory, Topology, Probability.

15. What is number theory?

Answer: The study of properties of integers, prime numbers, divisibility, and related structures.

16. Explain Probability and Statistics.

Answer: Probability studies chance, while Statistics deals with collecting, analyzing, and interpreting data.

17. How does mathematics link with computer science?

Answer: Algorithms, cryptography, data structures, and computational complexity use mathematical logic.

18. State two benefits of learning mathematics.

Answer: Enhances logical reasoning; improves analytical thinking and problem‑solving skills.

19. How can mathematics support engineering?

Answer: Engineers use calculus, differential equations, and linear algebra to design structures and systems.

20. What is the importance of mathematical thinking in daily decision‑making?

Answer: It helps evaluate options logically, compare quantities, and choose optimal solutions.

21. Define mathematical model.

Answer: A mathematical model is a representation using equations or algorithms to describe a real‑world system.

22. Give an example of a mathematical model.

Answer: The population growth model using P = P₀e^{rt} in biology or economics.

23. Why do students learn branches of mathematics separately?

Answer: To specialize understanding, master tools unique to each branch, and apply them effectively.

24. How has mathematics evolved historically?

Answer: From ancient counting systems (Egypt, Mesopotamia) to abstract theories in modern research.

25. Describe the role of mathematics in architecture.

Answer: It’s essential in structural design, proportions, symmetry, and calculations of loads.

26. How is mathematics applied in agriculture?

Answer: Used in yield prediction, land measurement, irrigation scheduling, and crop modeling.

27. Discuss mathematics in finance.

Answer: Used in interest calculation, annuities, amortization, risk evaluation, and portfolio theory.

28. What is logic in mathematics?

Answer: Mathematical logic studies principles of valid reasoning and proof structure.

29. How does mathematical logic link to computer science?

Answer: It underpins algorithm correctness, formal languages, and program verification.

30. Why is precision key in mathematics?

Answer: It avoids ambiguity and ensures reproducibility of results and proofs.

31. What do we mean by abstraction in mathematics?

Answer: Abstraction involves focusing on essential properties and ignoring irrelevant details.

32. How does abstraction help in problem‑solving?

Answer: It simplifies complex systems to core elements, making solutions universal and re‑usable.

33. How do branches like topology or number theory demonstrate abstraction?

Answer: They study overarching properties (like continuity or divisibility) beyond concrete numbers or shapes.

34. What does “pure mathematics” mean?

Answer: Pure mathematics is abstract study with no immediate application.

35. What does “applied mathematics” mean?

Answer: Applied mathematics uses mathematical methods to solve real‑world problems in science, engineering, business.

36. Give one example of pure mathematics research.

Answer: Research in prime gaps or Riemann Hypothesis falls under pure mathematics.

37. Give one example of applied mathematics.

Answer: Using differential equations to model fluid flow in engineering.

38. How does mathematics relate to art?

Answer: Concepts like symmetry, perspective, and geometric proportions appear in painting, sculpture, and design.

39. Why is mathematical literacy important for global competence?

Answer: It enables one to participate in tech-driven economies, data interpretation, and scientific reasoning.

40. How do mathematics and logic support ethical reasoning?

Answer: They provide structured frameworks to evaluate consistency and validity of arguments.

41. Design an advanced competency question: Propose a scenario showing how mathematics informs decision-making in public policy.

Answer: Use statistics to analyze survey data, apply probability models to forecast outcomes, and present recommendations accordingly.

42. Design a challenging question on abstraction and modeling.

Answer: Model traffic flow in a city using graphs and queuing theory; abstract vehicles and roads into nodes and edges.

43. Formulate a question assessing knowledge of branches. Compare and contrast Algebra, Geometry, and Statistics in terms of real-world uses.

Answer: Algebra is used in finance and computer science; Geometry in architecture; Statistics in health and social research.

44. Create a high‑order thinking question: Analyze how probability and calculus can be combined in modeling epidemics.

Answer: Use differential equations with probabilities to describe infection rates and recovery for SIR models.

45. Reflective essay question: Discuss why mathematics is considered both an art and a science.

Answer: As an art it involves creativity and imagination; as a science it relies on logic and methodical reasoning.

46. Suggest a classroom activity to show mathematics’ importance.

Answer: Have students conduct a mini-survey, collect data, compute statistics, and conclude insights on a local issue.

47. Advanced question on interdisciplinary links: Explain how topology can apply in robotics or biology.

Answer: Topology helps study shape invariants in data or robot motion planning avoiding obstacles.

48. Write a question assessing understanding of mathematical precision. Why does small rounding error in financial calculations matter over time?

Answer: Even tiny errors compound, leading to significant discrepancies in long-term interest or savings.

49. Devise a problem on modeling using abstraction: Model queue lengths in a bank using simple assumptions and explain limits.

Answer: Use Poisson arrivals and exponential service times to estimate average wait times and discuss real-life deviations.

50. Final question: Evaluate why mastery of the concept of mathematics is essential for examination competence worldwide.

Answer: Because it forms a foundation for advanced topics; enhances analytical and critical thinking; bridges to sciences and technology.

Chapter Two: Numbers I - Questions and Answers

Chapter Two: Numbers I - Questions and Detailed Answers

1. Base Ten Numeration

Q1: Write the number 4,827 in expanded form.

Answer:

  • Method 1: 4,827 = 4,000 + 800 + 20 + 7
  • Method 2: (4 × 1000) + (8 × 100) + (2 × 10) + (7 × 1)

Q2: Express 30506 in base ten numeration.

Answer:

  • Method 1: 30,506 = 3×10,000 + 0×1,000 + 5×100 + 0×10 + 6×1
  • Method 2: Expanded = 30000 + 0 + 500 + 0 + 6

2. Place Value of a Digit in a Number

Q3: What is the place value of 7 in 76,234?

Answer:

  • Method 1: 7 is in the ten thousands place, so its place value is 70,000.
  • Method 2: (7 × 10,000) = 70,000

Q4: What is the place value of 3 in 432?

Answer:

  • Method 1: 3 is in the tens place → 30
  • Method 2: (3 × 10) = 30

3. Total Value

Q5: Find the total value of 5 in 25,614.

Answer:

  • Method 1: 5 is in the thousands place = 5,000
  • Method 2: (5 × 1,000) = 5,000

Q6: What is the total value of 2 in 2,391?

Answer:

  • Method 1: 2 is in thousands = 2,000
  • Method 2: (2 × 1,000) = 2,000

4. Writing Numbers in Words and in Numerals

Q7: Write 18,302 in words.

Answer:

  • Method 1: Eighteen thousand three hundred two
  • Method 2: 18,000 + 300 + 2 = Eighteen thousand three hundred and two

Q8: Write "Seventy-five thousand, six hundred twenty" in numerals.

Answer:

  • Method 1: 75,620
  • Method 2: 70,000 + 5,000 + 600 + 20 = 75,620

5. Natural and Whole Numbers

Q9: Define natural numbers and whole numbers. Give examples.

Answer:

  • Method 1: Natural numbers = 1, 2, 3, ... Whole numbers = 0, 1, 2, 3, ...
  • Method 2: Whole numbers include zero; natural numbers do not.

Q10: Identify if 0 and 1 are natural or whole numbers.

Answer:

  • 0: Whole number only
  • 1: Both natural and whole number

6. Even, Odd, and Prime Numbers

Q11: List the even, odd, and prime numbers from 1 to 20.

Answer:

  • Even: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
  • Odd: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
  • Prime: 2, 3, 5, 7, 11, 13, 17, 19

Q12: Is 9 a prime number? Why?

Answer:

  • Method 1: No, because it has more than 2 factors: 1, 3, 9
  • Method 2: 9 is divisible by 3 → not a prime

7. Operations on Whole Numbers

Q13: Add: 4,529 + 3,781

Answer: 8,310

  • Method 1: Add column-wise
  • Method 2: Use calculator for check

Q14: Multiply: 123 × 47

Answer:

  • Method 1: Long multiplication = 5781
  • Method 2: 123×40 + 123×7 = 4920 + 861 = 5781

8. BODMAS

Q15: Evaluate: 5 + 2 × (3 + 1)² ÷ 2

Answer:

  • Step 1: Inside bracket = 4
  • Step 2: (4)² = 16
  • Step 3: 2 × 16 = 32 → 32 ÷ 2 = 16
  • Final: 5 + 16 = 21

Q16: Simplify: (8 + 2) × 3 - 4²

Answer:

  • Method 1: (10 × 3) - 16 = 30 - 16 = 14
  • Method 2: Use BODMAS sequence to validate

9. Factors and Multiples of Numbers

Q17: List all factors and multiples of 12 up to 100.

Answer:

  • Factors: 1, 2, 3, 4, 6, 12
  • Multiples: 12, 24, 36, 48, 60, 72, 84, 96

Q18: Find LCM and HCF of 18 and 24.

Answer:

  • LCM = 72
  • HCF = 6
  • Method: Prime factorization

10. Operations on Integers

Q19: Evaluate: -5 + 7 - (-3) + (-8)

Answer:

  • Step-by-step: -5 + 7 = 2 → 2 + 3 = 5 → 5 + (-8) = -3
  • Final Answer: -3

Q20: Evaluate: (-4) × 3 + 6 ÷ (-2)

Answer:

  • Step 1: -4 × 3 = -12
  • Step 2: 6 ÷ -2 = -3
  • Final: -12 + (-3) = -15
Chapter Two: Numbers I - Questions and Answers

Chapter Two: Numbers I - Questions and Detailed Answers

Base Ten Numeration

Q1: Write the number 8,472 in expanded form.

Answer:

  • Method 1: 8,000 + 400 + 70 + 2
  • Method 2: (8 × 1000) + (4 × 100) + (7 × 10) + (2 × 1)

Q2: Convert 50,806 to base ten expanded form.

Answer:

  • Method 1: 50,000 + 800 + 6
  • Method 2: (5 × 10,000) + (0 × 1,000) + (8 × 100) + (0 × 10) + (6 × 1)

Q3: Which digit is in the ten-thousands place in 76,294?

Answer:

  • Method 1: From left, the 7 is in the ten-thousands place.
  • Method 2: 76,294 → 7 × 10,000 = 70,000

Q4: What is the value of each digit in 98,731?

Answer:

  • 9 → 90,000
  • 8 → 8,000
  • 7 → 700
  • 3 → 30
  • 1 → 1

Q5: Fill in the blank: 3 × 1,000 + 5 × 100 + 6 × 10 + 9 × 1 = ______

Answer:

  • Method 1: 3,000 + 500 + 60 + 9 = 3,569
  • Method 2: Evaluating each term: 3000 + 500 + 60 + 9 = 3569

Q6: Arrange the following in ascending order: 9,084; 9,048; 9,804; 9,840

Answer:

  • Step-by-step: Compare thousands → hundreds → tens → units
  • Order: 9,048; 9,084; 9,804; 9,840

Q7: Identify the number with 6 in thousands place, 4 in hundreds place, 0 in tens place, and 9 in ones place.

Answer:

  • Method 1: Form number = 6,409
  • Method 2: Expanded = 6000 + 400 + 0 + 9 = 6409

Q8: Write in base ten numeration: (9 × 1000) + (3 × 100) + (2 × 10) + (5 × 1)

Answer:

  • Method 1: 9000 + 300 + 20 + 5 = 9,325
  • Method 2: Read as a single number = 9325

Q9: Which digit has the greatest place value in 124,789?

Answer:

  • Method 1: 1 is in hundred-thousands = 100,000
  • Method 2: Compare each digit’s place value → 1 × 100000 is largest

Q10: Choose the correct value: In 703,182 the digit 1 stands for:

Answer:

  • Method 1: 1 is in hundreds place = 100
  • Method 2: (1 × 100) = 100
Chapter Two: Numbers I - Questions and Answers

Chapter Two: Numbers I - Questions and Detailed Answers

Additional 20 Questions: Place and Total Value

Q21: What is the place value of 4 in 4,589?

Answer:

  • Method 1: 4 is in the thousands place = 4,000
  • Method 2: 4 × 1,000 = 4,000

Q22: What is the total value of 2 in 1,243?

Answer:

  • Method 1: 2 is in hundreds = 200
  • Method 2: 2 × 100 = 200

Q23: What is the place value of 0 in 3,048?

Answer:

  • Method 1: 0 is in hundreds = 0
  • Method 2: 0 × 100 = 0

Q24: What is the total value of 7 in 7,002?

Answer:

  • Method 1: 7 is in thousands = 7,000
  • Method 2: 7 × 1,000 = 7,000

Q25: Identify the place value and total value of 3 in 3,120

Answer:

  • Place Value: Thousands
  • Total Value: 3 × 1,000 = 3,000

Q26: What is the place value of 9 in 892?

Answer:

  • Place: Tens
  • Total: 9 × 10 = 90

Q27: Find place value of 5 in 5,432.

Answer:

  • Place: Thousands
  • Total Value: 5,000

Q28: What is the total value of 6 in 6,601?

Answer:

  • Method 1: 6 in thousands = 6,000
  • Method 2: 6 × 1,000 = 6,000

Q29: What is the place value of 8 in 2,847?

Answer:

  • Place: Hundreds
  • Total: 800

Q30: Total value of 1 in 1,013?

Answer:

  • Place: Thousands
  • Value: 1,000

Q31: What is the place value of 2 in 462?

Answer:

  • Place: Units
  • Total Value: 2

Q32: What is the total value of 9 in 9,000?

Answer:

  • Place: Thousands
  • Value: 9,000

Q33: Write the place and total value of 6 in 6,654.

Answer:

  • 1st 6: Thousands → 6,000
  • 2nd 6: Hundreds → 600

Q34: Identify place value of 7 in 4,372

Answer:

  • Place: Tens
  • Value: 70

Q35: Total value of 5 in 5,005?

Answer:

  • First 5: Thousands = 5,000
  • Last 5: Units = 5

Q36: Place and total value of 0 in 2048?

Answer:

  • Place: Hundreds
  • Total: 0 × 100 = 0

Q37: Value of 8 in 8,080?

Answer:

  • Thousands place: 8,000
  • Tens place: 80

Q38: Write the place value of 3 in 3,003

Answer:

  • First 3: Thousands = 3,000
  • Second 3: Units = 3

Q39: Value of 1 in 1,101?

Answer:

  • Thousands: 1,000
  • Hundreds: 100
  • Units: 1

Q40: What is the total value of each digit in 2,345?

Answer:

  • 2 = 2,000
  • 3 = 300
  • 4 = 40
  • 5 = 5
Writing Numbers in Words and Numerals - Questions and Answers

Writing Numbers in Words and Numerals - Questions and Detailed Answers

1. Write the number 56,238 in words.

Answer:

  • Method 1: Fifty-six thousand two hundred thirty-eight
  • Method 2: 56,000 + 200 + 30 + 8 = Fifty-six thousand two hundred and thirty-eight

2. Write \"Ninety-four thousand, five hundred and twelve\" in numerals.

Answer:

  • Method 1: 94,512
  • Method 2: 90,000 + 4,000 + 500 + 10 + 2 = 94,512

3. Write the number 7,050 in words.

Answer:

  • Method 1: Seven thousand fifty
  • Method 2: 7,000 + 50 = Seven thousand and fifty

4. Convert \"Thirty-two thousand, one hundred and four\" into numerals.

Answer:

  • Method 1: 32,104
  • Method 2: 30,000 + 2,000 + 100 + 4 = 32,104

5. Write the number 120,409 in words.

Answer:

  • Method 1: One hundred twenty thousand four hundred nine
  • Method 2: 120,000 + 400 + 9 = One hundred twenty thousand four hundred and nine

6. Write \"Two hundred seventy-five thousand, six hundred and eighty-two\" in numerals.

Answer:

  • Method 1: 275,682
  • Method 2: 200,000 + 70,000 + 5,000 + 600 + 80 + 2 = 275,682

7. Write the number 9,001 in words.

Answer:

  • Method 1: Nine thousand one
  • Method 2: 9,000 + 1 = Nine thousand and one

8. Convert \"Fifteen thousand, eight\" into numerals.

Answer:

  • Method 1: 15,008
  • Method 2: 15,000 + 8 = 15,008

9. Write 60,370 in words.

Answer:

  • Method 1: Sixty thousand three hundred seventy
  • Method 2: 60,000 + 300 + 70 = Sixty thousand three hundred and seventy

10. Write \"One hundred five thousand, two hundred and thirty\" in numerals.

Answer:

  • Method 1: 105,230
  • Method 2: 100,000 + 5,000 + 200 + 30 = 105,230

11. Write the number 801,609 in words.

Answer:

  • Method 1: Eight hundred one thousand six hundred nine
  • Method 2: 800,000 + 1,000 + 600 + 9 = Eight hundred one thousand six hundred and nine

12. Write \"Two thousand, seven hundred and four\" in numerals.

Answer:

  • Method 1: 2,704
  • Method 2: 2,000 + 700 + 4 = 2,704

13. Write the number 45,020 in words.

Answer:

  • Method 1: Forty-five thousand twenty
  • Method 2: 40,000 + 5,000 + 20 = Forty-five thousand and twenty

14. Convert \"Seventy-one thousand, three hundred and ninety-nine\" into numerals.

Answer:

  • Method 1: 71,399
  • Method 2: 70,000 + 1,000 + 300 + 90 + 9 = 71,399

15. Write the number 100,005 in words.

Answer:

  • Method 1: One hundred thousand five
  • Method 2: 100,000 + 5 = One hundred thousand and five

16. Write \"Eighty-two thousand, six hundred and one\" in numerals.

Answer:

  • Method 1: 82,601
  • Method 2: 80,000 + 2,000 + 600 + 1 = 82,601

17. Write the number 37,804 in words.

Answer:

  • Method 1: Thirty-seven thousand eight hundred four
  • Method 2: 30,000 + 7,000 + 800 + 4 = Thirty-seven thousand eight hundred and four

18. Convert \"Nine hundred ninety-nine thousand, nine hundred ninety-nine\" into numerals.

Answer:

  • Method 1: 999,999
  • Method 2: 900,000 + 90,000 + 9,000 + 900 + 90 + 9 = 999,999

19. Write the number 2,450 in words.

Answer:

  • Method 1: Two thousand four hundred fifty
  • Method 2: 2,000 + 400 + 50 = Two thousand four hundred and fifty

20. Write \"Six hundred twelve thousand, five hundred and seventy-one\" in numerals.

Answer:

  • Method 1: 612,571
  • Method 2: 600,000 + 12,000 + 500 + 70 + 1 = 612,571
Natural and Whole Numbers - Questions and Answers

Natural and Whole Numbers - Questions and Detailed Answers

1. Define natural numbers.

Answer: Natural numbers are positive integers starting from 1, 2, 3, ... used for counting.

2. Define whole numbers.

Answer: Whole numbers are natural numbers including zero: 0, 1, 2, 3, ...

3. Is 0 a natural number? Explain.

Answer: No. Natural numbers start at 1, so zero is not a natural number but a whole number.

4. Write down the first 10 natural numbers.

Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

5. Write down the first 10 whole numbers.

Answer: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

6. Find the sum of the first 20 natural numbers.

Answer:

  • Method 1: Use formula \( S_n = \frac{n(n+1)}{2} \) → \( \frac{20 \times 21}{2} = 210 \)
  • Method 2: Add numbers 1+2+3+...+20 manually or with calculator = 210

7. Find the difference between the sum of the first 10 whole numbers and the first 10 natural numbers.

Answer:

  • Sum of first 10 whole numbers = 0 + 1 + 2 + ... + 9 = 45
  • Sum of first 10 natural numbers = 1 + 2 + ... + 10 = 55
  • Difference = 55 - 45 = 10

8. List five natural numbers greater than 100.

Answer: 101, 102, 103, 104, 105

9. List five whole numbers less than 10.

Answer: 0, 1, 2, 3, 4

10. Are negative numbers whole numbers? Justify.

Answer: No. Whole numbers include zero and positive integers only, no negatives.

11. Find the next three natural numbers after 57.

Answer: 58, 59, 60

12. What is the smallest natural number?

Answer: 1

13. What is the smallest whole number?

Answer: 0

14. If \(n\) is a natural number, express the next two natural numbers in terms of \(n\).

Answer: \(n+1\) and \(n+2\)

15. Which set does the number 25 belong to: natural numbers, whole numbers, or both?

Answer: Both natural and whole numbers (since 25 is positive).

16. Are fractions whole numbers or natural numbers? Explain.

Answer: No. Whole and natural numbers are integers without fractions or decimals.

17. Calculate \(5 + 0 + 7\) and state whether the sum is a natural number or whole number.

Answer:

  • Sum = 12
  • 12 is both a natural and whole number

18. Find the product of the first 5 natural numbers.

Answer:

  • Product = \(1 \times 2 \times 3 \times 4 \times 5 = 120\)
  • This is a natural number

19. Express the number 0 in terms of natural and whole numbers.

Answer: 0 is a whole number but NOT a natural number.

20. State whether the following statements are True or False:

  • (a) All whole numbers are natural numbers. False
  • (b) Zero is a natural number. False
  • (c) Natural numbers start from 1. True
  • (d) Negative numbers are whole numbers. False
Natural and Whole Numbers - Questions and Answers

Natural and Whole Numbers: Questions and Detailed Answers

1. Define natural numbers and whole numbers with examples.

Solution 1: Natural numbers are the counting numbers starting from 1, 2, 3, ... Whole numbers include all natural numbers plus zero: 0, 1, 2, 3, ...

Solution 2: Natural numbers = {1, 2, 3, 4, ...}; Whole numbers = {0, 1, 2, 3, 4, ...}

Solution 3: Whole numbers extend natural numbers by including zero, useful for representing quantities that can be zero or more.

2. Is zero a natural number? Justify your answer.

Solution 1: No, zero is not a natural number because natural numbers start from 1.

Solution 2: Zero is a whole number but not a natural number because natural numbers are positive integers used for counting.

Solution 3: Zero represents 'nothing' or 'absence' and is included in whole numbers for completeness but excluded in natural numbers.

3. List all natural numbers less than 15 and all whole numbers less than or equal to 15.

Solution 1: Natural numbers less than 15: 1, 2, 3, ..., 14

Solution 2: Whole numbers less than or equal to 15: 0, 1, 2, ..., 15

Solution 3: Natural numbers exclude zero, whole numbers include zero up to the number given.

4. Represent the natural numbers from 1 to 7 on a number line.

Solution 1: The natural numbers from 1 to 7 are points on the number line starting at 1 and ending at 7, equally spaced.

1
2
3
4
5
6
7

Solution 2: Draw a horizontal line, mark equal intervals, label 1 through 7 sequentially.

Solution 3: This visual representation helps understand counting and order of natural numbers.

5. What is the smallest natural number? What is the smallest whole number?

Solution 1: Smallest natural number is 1.

Solution 2: Smallest whole number is 0.

Solution 3: Natural numbers start counting at 1; whole numbers include zero to represent empty sets.

6. If a number belongs to both natural numbers and whole numbers, what can it be?

Solution 1: Any natural number (1, 2, 3, ...) is also a whole number.

Solution 2: Whole numbers include natural numbers, so intersection excludes zero.

Solution 3: The set of natural numbers is a subset of whole numbers, so every natural number belongs to both sets.

7. Are negative numbers natural or whole numbers? Explain.

Solution 1: Negative numbers are neither natural nor whole numbers since both sets only include zero or positive integers.

Solution 2: Natural and whole numbers represent counts and quantities, which cannot be negative.

Solution 3: Negative numbers belong to integers but not to natural or whole numbers.

8. Find the sum of the first 10 natural numbers using two different methods.

Solution 1 (Formula method): Sum = n(n + 1)/2 = 10 × 11 / 2 = 55

Solution 2 (Pairing method): Pair 1+10=11, 2+9=11, ..., 5+6=11; 5 pairs × 11 = 55

Solution 3 (Iterative addition): Add numbers 1+2+3+...+10 step by step, resulting in 55.

9. Write the first 5 whole numbers and their squares.

Solution 1: Whole numbers: 0,1,2,3,4. Squares: 0,1,4,9,16.

Solution 2: Calculate by multiplying each number by itself.

Solution 3: Use a table:

NumberSquare
00
11
24
39
416
10. Explain why zero is important in whole numbers but not included in natural numbers.

Solution 1: Zero represents 'nothing' or the absence of quantity, so it’s essential for whole numbers to represent empty sets.

Solution 2: Natural numbers are for counting tangible objects, so zero is excluded.

Solution 3: Zero allows arithmetic completeness, especially for addition and subtraction, in whole numbers.

11. Identify whether the following numbers belong to natural numbers, whole numbers, or neither: -3, 0, 7, 15.5, 20

Solution 1: -3: Neither; 0: Whole; 7: Both; 15.5: Neither; 20: Both.

Solution 2: Natural numbers are positive integers starting from 1; whole numbers include 0 and positives; decimals and negatives do not belong.

Solution 3: Only integers ≥0 belong to whole numbers; natural numbers exclude zero.

12. Find the difference between the largest natural number less than 100 and the smallest whole number.

Solution 1: Largest natural number < 100 is 99; smallest whole number is 0; difference = 99 - 0 = 99.

Solution 2: Using subtraction directly: 99 - 0 = 99.

Solution 3: Conceptually, the difference equals the largest natural number itself.

13. Write down the next 5 whole numbers after 12.

Solution 1: 13, 14, 15, 16, 17

Solution 2: Add 1 repeatedly starting from 12.

Solution 3: Recognize that whole numbers increase by 1 each step.

14. Are natural numbers closed under subtraction? Justify with an example.

Solution 1: No. Example: 3 - 5 = -2 (not natural).

Solution 2: Subtracting larger natural numbers can produce negative numbers, which are outside natural numbers.

Solution 3: Natural numbers are closed under addition and multiplication but not subtraction.

15. Find the sum of the whole numbers between 5 and 10 inclusive.

Solution 1: Sum = 5 + 6 + 7 + 8 + 9 + 10 = 45

Solution 2: Use formula for consecutive integers: sum = (n/2)(first + last), here n=6 → (6/2)(5+10)=3×15=45

Solution 3: Add manually or use calculator for verification.

16. Show that whole numbers are closed under addition.

Solution 1: Take any two whole numbers, say 3 and 5; 3 + 5 = 8 which is whole.

Solution 2: Adding zero or any natural number to a whole number results in a whole number.

Solution 3: This can be generalized: sum of any whole numbers is always whole.

17. If x and y are whole numbers, find the value of x + y if x = 0 and y = 9.

Solution 1: x + y = 0 + 9 = 9

Solution 2: Adding zero to any number results in the same number.

Solution 3: Use the identity property of addition.

18. List five natural numbers that are greater than 50 but less than 60.

Solution 1: 51, 52, 53, 54, 55

Solution 2: Count upward from 50 excluding 50 itself.

Solution 3: Write numbers between 50 and 60 excluding 60.

19. Express the natural number 25 in terms of whole numbers.

Solution 1: 25 is both a natural and whole number.

Solution 2: Since whole numbers include natural numbers, 25 belongs to both sets.

Solution 3: No difference for positive integers in natural and whole numbers.

20. Identify all whole numbers less than 5.

Solution 1: 0, 1, 2, 3, 4

Solution 2: Whole numbers start at zero, so all integers from 0 up to 4 inclusive.

Solution 3: Write out all numbers starting from zero below 5.

21. If n is a natural number, show that n + 1 is a whole number.

Solution 1: Since natural numbers start at 1, adding 1 makes n + 1 ≥ 2, which is a whole number.

Solution 2: Whole numbers include natural numbers, so n+1 ∈ whole numbers.

Solution 3: Example: If n=3, n+1=4 which is whole number.

22. Are all whole numbers natural numbers? Explain.

Solution 1: No, because whole numbers include zero which is not natural.

Solution 2: Natural numbers are a subset of whole numbers excluding zero.

Solution 3: Example: 0 ∈ whole numbers but 0 ∉ natural numbers.

23. Write the number that comes before 100 in natural numbers.

Solution 1: 99

Solution 2: Natural numbers increase by 1, so number before 100 is 100 - 1 = 99

Solution 3: Counting backwards from 100 gives 99.

24. Explain the difference between the sets {0,1,2,3} and {1,2,3}.

Solution 1: First set is whole numbers 0 to 3; second set is natural numbers 1 to 3.

Solution 2: Difference is the number zero included only in the first set.

Solution 3: Zero represents absence in whole numbers, excluded in natural numbers.

25. Solve: Using a number line, find the difference between natural numbers 7 and 3.

Solution 1: Difference = 7 - 3 = 4

0
1
2
3
4
5
6
7

Solution 2: Count steps on number line from 3 to 7 → 4 steps

Solution 3: Use subtraction directly: 7 - 3 = 4

Reference Book: N/A

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