Class 9 Maths Chapter 1: Numbers Systems – Detailed Solutions to All Exercises

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Introduction

Chapter 1 of Class 9 Maths, titled Number Systems, explores the different types of numbers that we use in our daily lives and mathematical operations. In this chapter, we cover real numbers, including natural numbers, whole numbers, integers, fractions, and irrational numbers, as well as their properties and operations.

Let’s dive into each exercise and solve them step by step.

Exercise 1.1

Q1: Represent the following numbers on the number line:

  • (a) 0.4
  • (b) 2.8
  • (c) -1.5
  • (d) -2.2

Solution:

  1. 0.4 – To represent 0.4, divide the segment between 0 and 1 into 10 equal parts. Mark 4th part, which is 0.4.
  2. 2.8 – Divide the segment between 2 and 3 into 10 equal parts and mark the 8th part.
  3. -1.5 – Divide the segment between -1 and -2 into 10 equal parts and mark the 5th part.
  4. -2.2 – Divide the segment between -2 and -3 into 10 equal parts and mark the 2nd part.

Exercise 1.2

Q2: Which of the following are rational numbers?

  • (a) 5
  • (b) 0.75
  • (c) √2
  • (d) -3.4
  • (e) 0.333…

Solution:

  1. 5 is a rational number because it can be written as 5/1.
  2. 0.75 is a rational number because it can be written as 3/4.
  3. √2 is an irrational number because it cannot be expressed as a fraction of two integers.
  4. -3.4 is a rational number because it can be written as -17/5.
  5. 0.333… is a rational number because it can be expressed as 1/3.

Conclusion: Rational numbers are numbers that can be written as the ratio of two integers.

Exercise 1.3

Q3: Simplify the following:

  • (a) 5/3 + 2/3
  • (b) 3/5 – 1/4
  • (c) (3/7) × (2/5)
  • (d) (4/9) ÷ (2/3)

Solution:

  1. 5/3 + 2/3 = (5 + 2) / 3 = 7/3
  2. 3/5 – 1/4 = (12 – 5) / 20 = 7/20
  3. (3/7) × (2/5) = (3 × 2) / (7 × 5) = 6/35
  4. (4/9) ÷ (2/3) = (4/9) × (3/2) = (4 × 3) / (9 × 2) = 12/18 = 2/3

Exercise 1.4

Q4: Write the following decimals in the form of p/q (where p and q are integers):

  • (a) 0.6
  • (b) 0.75
  • (c) 0.333…
  • (d) 1.25

Solution:

  1. 0.6 = 6/10 = 3/5 (after simplifying).
  2. 0.75 = 75/100 = 3/4 (after simplifying).
  3. 0.333… = 1/3 (as it is a repeating decimal).
  4. 1.25 = 125/100 = 5/4 (after simplifying).

Exercise 1.5

Q5: Represent the following numbers on the number line:

  • (a) √3
  • (b) 1.4
  • (c) -√2

Solution:

  1. √3 is approximately 1.732. To represent √3, find a number slightly more than 1.7 and mark it.
  2. 1.4 – Mark a point between 1 and 2 but slightly closer to 1.
  3. -√2 is approximately -1.414. To represent -√2, find a point slightly more than -1.4 but less than -1.5.

Exercise 1.6

Q6: Find the HCF and LCM of 36 and 60.

Solution:

  • Prime Factorization of 36: 36 = 2² × 3²
  • Prime Factorization of 60: 60 = 2² × 3 × 5

HCF = Product of the lowest powers of common factors = 2² × 3 = 12
LCM = Product of the highest powers of all prime factors = 2² × 3² × 5 = 180

So, HCF = 12 and LCM = 180.

Exercise 1.7

Q7: Solve the following equations:

  • (a) 5x + 3 = 23
  • (b) 2y – 4 = 8

Solution:

  1. 5x + 3 = 23
    Subtract 3 from both sides:
    5x = 20
    Divide by 5:
    x = 4
  2. 2y – 4 = 8
    Add 4 to both sides:
    2y = 12
    Divide by 2:
    y = 6

Additional Questions

1. Multiple Choice Questions (MCQs):

  1. Which of the following is a rational number?
    • (c) 2/3
  2. What is the decimal expansion of 7/8?
    • (b) 0.875
  3. Which of the following is an irrational number?
    • (c) √5
  4. Which of the following numbers is a whole number but not a natural number?
    • (a) 0
  5. What is the result of the addition of two irrational numbers?
    • (c) Sometimes rational, sometimes irrational

2. Short Answer Type Questions:

  1. Represent the number -3.25 on a number line.
    • Mark a point to the left of 0 at -3.25, between -3 and -4.
  2. Simplify: 3/7 + 5/14.
    • 3/7 + 5/14 = (6/14) + (5/14) = 11/14
  3. Subtract 3/5 from 4/5.
    • 4/5 – 3/5 = 1/5
  4. Multiply 3/4 by 2/5.
    • (3/4) × (2/5) = 6/20 = 3/10
  5. Divide 5/8 by 3/4.
    • (5/8) ÷ (3/4) = (5/8) × (4/3) = 20/24 = 5/6
  6. Express 0.4 in the form p/q.
    • 0.4 = 4/10 = 2/5
  7. Express 1.25 in the form of p/q.
    • 1.25 = 125/100 = 5/4
  8. Convert 3.142 into a fraction.
    • 3.142 = 3142/1000 = 1571/500
  9. Convert 1.333… into a fraction.
    • 1.333… = 4/3
  10. Convert 5/9 into a decimal.
    • 5/9 = 0.555… (repeating)

3. Long Answer Type Questions:

  1. Find the HCF and LCM of 24 and 36 using prime factorization.
    • 24 = 2³ × 3
    • 36 = 2² × 3²
    • HCF = 2² × 3 = 12
    • LCM = 2³ × 3² = 72
  2. Find the value of x if 4x – 5 = 15.
    • 4x = 15 + 5 = 20
    • x = 20/4 = 5
  3. Find the value of y if 5y + 9 = 24.
    • 5y = 24 – 9 = 15
    • y = 15/5 = 3
  4. Find the sum of the rational numbers 3/7 and 5/6.
    • (3/7) + (5/6) = (18/42) + (35/42) = 53/42
  5. Simplify: (2/5) × (5/8) × (3/4).
    • (2/5) × (5/8) × (3/4) = (2 × 5 × 3) / (5 × 8 × 4) = 30/160 = 3/16
  6. Solve for z: 3z + 4 = 19.
    • 3z = 19 – 4 = 15
    • z = 15/3 = 5
  7. Add the rational numbers: -1/5 and 3/7.
    • (-1/5) + (3/7) = (-7/35) + (15/35) = 8/35
  8. Subtract 7/9 from 5/6.
    • (5/6) – (7/9) = (15/18) – (14/18) = 1/18
  9. Multiply: 2.5 × 0.4.
    • 2.5 × 0.4 = 1.0
  10. Divide: 6.4 ÷ 0.8.
    • 6.4 ÷ 0.8 = 8

4. Word Problems:

  1. A shopkeeper bought 10 packets of sugar, each weighing 1.75 kg. What is the total weight of the sugar?
    • Total weight = 10 × 1.75 = 17.5 kg
  2. A rectangular plot has length 9.5 meters and width 5.6 meters. Find the area of the plot.
    • Area = length × width = 9.5 × 5.6 = 53.2 m²
  3. A cyclist travels 0.75 km in 1 minute. How far will he travel in 15 minutes?
    • Distance = speed × time = 0.75 × 15 = 11.25 km
  4. If the cost of 2 meters of cloth is ₹36, what will be the cost of 8 meters of cloth?
    • Cost of 1 meter = 36/2 = ₹18
    • Cost of 8 meters = 8 × 18 = ₹144
  5. A car travels 70 km in 2 hours. What is the average speed of the car?
    • Speed = Distance / Time = 70 / 2 = 35 km/h

5. True or False:

  1. The sum of two irrational numbers is always irrational.
    • False
  2. 7/2 is a rational number.
    • True
  3. -√7 is an irrational number.
    • True
  4. All whole numbers are natural numbers.
    • False
  5. The number 0 is a rational number.
    • True

6. More Challenging Questions:

  1. Find the smallest number that is divisible by both 12 and 18.
    • LCM of 12 and 18 = 36
  2. If √3 + √2 = x, find the value of x².
    • x² = (√3 + √2)² = 3 + 2 + 2√6 = 5 + 2√6
  3. Rationalize the denominator of the expression: 5 / (√2 + √3).
    • Multiply numerator and denominator by (√2 – √3):
    • = [5 × (√2 – √3)] / [(√2 + √3)(√2 – √3)]
    • = 5(√2 – √3) / (2 – 3) = 5(√2 – √3) / (-1)
    • = -5(√2 – √3) = -5√2 + 5√3
  4. Simplify the expression: (1/3) ÷ (2/5).
    • (1/3) ÷ (2/5) = (1/3) × (5/2) = 5/6
  5. If x = √2 + √3, find the value of x².
    • x² = (√2 + √3)² = 2 + 3 + 2√6 = 5 + 2√6

These answers cover the 40 questions from Chapter 1: Number Systems. Let me know if you need any further explanations or clarifications!

Also solve these if you have any doubt:

1. Multiple Choice Questions (MCQs):

  1. Which of the following is a non-terminating, non-repeating decimal?
    • (a) 2/3
    • (b) √2
    • (c) 0.7
    • (d) 5/8
  2. The product of two irrational numbers is always:
    • (a) Rational
    • (b) Irrational
    • (c) Can be either rational or irrational
    • (d) None of the above
  3. Which of the following numbers is an integer but not a natural number?
    • (a) -1
    • (b) 0
    • (c) 1
    • (d) 2
  4. Which of the following is true about the number √25?
    • (a) It is a rational number
    • (b) It is an irrational number
    • (c) It is a whole number
    • (d) Both (a) and (c)
  5. Which of the following is an example of a terminating decimal?
    • (a) 1/7
    • (b) 3/5
    • (c) √3
    • (d) π

2. Short Answer Type Questions:

  1. Represent 3/4 on a number line.
  2. Express the number 0.2 in the form p/q where p and q are integers.
  3. Find the difference between 8/9 and 5/9.
  4. Convert the repeating decimal 0.636363… into a fraction.
  5. Find the product of -2/5 and 4/7.

3. Long Answer Type Questions:

  1. Solve for x: 3x – 7 = 11.
  2. Solve for y: 5y + 6 = 36.
  3. Find the sum of the rational numbers 4/5 and 7/10.
  4. Simplify: (7/9) × (6/8) × (3/4).
  5. Find the HCF and LCM of 42 and 56 using prime factorization.

4. Word Problems:

  1. A student scores 25.5 out of 30 in a test. What is the percentage score?
  2. If the cost of 4 pens is ₹80, find the cost of 10 pens.
  3. A car travels 120 km in 3 hours. What is the average speed of the car in km/h?
  4. A person buys 6 shirts, each costing ₹300. What is the total cost of the shirts?
  5. A field has dimensions 8.5 meters by 6.3 meters. Find the area of the field.

These questions cover various aspects of Number Systems, including rational and irrational numbers, fractions, HCF/LCM, basic arithmetic operations, and word problems. Let me know if you need solutions or further explanation for any of these!

In conclusion, Chapter 1: Number Systems provides a thorough understanding of rational, irrational, and real numbers, along with operations on fractions, decimals, and their conversions. The chapter also focuses on HCF, LCM, and their applications, enhancing problem-solving skills in various mathematical contexts.

Also Read : Class 8 Mathematics – Chapter 5: Understanding Quadrilaterals | Complete Guide with Solutions

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