Class 7 Notes

Class 7 Mathematics Notes Chapter 9 (Rational Numbers) – Mathematics Book

Philoid

Philoid

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Mathematics
Detailed Notes with MCQs of Chapter 9: Rational Numbers. This is a fundamental chapter, building upon your understanding of integers and fractions, and it's crucial for various competitive exams. Pay close attention to the definitions and operations.

Chapter 9: Rational Numbers - Detailed Notes

1. What are Rational Numbers?

  • Need: We already know about Natural Numbers (1, 2, 3...), Whole Numbers (0, 1, 2...), Integers (..., -2, -1, 0, 1, 2, ...), and Fractions (like 1/2, 3/4). Rational numbers extend this system to include numbers that can be expressed as a ratio of two integers. Think about representing values like a debt of half a rupee (-1/2) or a temperature of -5.5 degrees Celsius (-55/10).
  • Definition: A number that can be expressed in the form p/q, where 'p' and 'q' are integers and q ≠ 0 (q is not equal to zero), is called a rational number.
    • 'p' is called the Numerator.
    • 'q' is called the Denominator.
  • Examples: 2/3, -5/7, 4/-9, -11/-13, 5 (since 5 = 5/1), -3 (since -3 = -3/1), 0 (since 0 = 0/1 or 0/5 etc.).
  • Key Insight: All integers are rational numbers. All fractions are rational numbers.

2. Numerator and Denominator

  • In p/q, 'p' is the numerator and 'q' is the denominator.
  • The denominator (q) tells us how many equal parts the whole is divided into.
  • The numerator (p) tells us how many of those parts are being considered.

3. Equivalent Rational Numbers

  • Rational numbers that represent the same value are called equivalent rational numbers.
  • We can obtain equivalent rational numbers by multiplying or dividing both the numerator and the denominator of a given rational number by the same non-zero integer.
  • Example:
    • 1/2 = (1 × 2) / (2 × 2) = 2/4
    • 1/2 = (1 × -3) / (2 × -3) = -3/-6
    • -18/24 = (-18 ÷ 6) / (24 ÷ 6) = -3/4
    • -18/24 = (-18 ÷ -2) / (24 ÷ -2) = 9/-12

4. Positive and Negative Rational Numbers

  • Positive Rational Number: A rational number is positive if both its numerator and denominator have the same sign (both positive or both negative).
    • Examples: 3/5, -2/-7 (since -2/-7 = 2/7)
  • Negative Rational Number: A rational number is negative if its numerator and denominator have opposite signs (one positive, one negative).
    • Examples: -3/5, 2/-7
  • Zero (0): The number 0 is a rational number (0/1, 0/-2 etc.), but it is neither positive nor negative.

5. Rational Numbers on a Number Line

  • Rational numbers can be represented on a number line just like integers.
  • To represent p/q:
    1. Divide the unit length (distance between 0 and 1, or -1 and 0, etc.) into 'q' equal parts.
    2. Move 'p' such parts from 0. Move to the right for positive 'p' and to the left for negative 'p'.
  • Example: To represent 3/4, divide the segment between 0 and 1 into 4 equal parts. The third mark from 0 is 3/4.
  • Example: To represent -5/3, divide the segment between 0 and -1, and -1 and -2 into 3 equal parts each. Starting from 0, move 5 parts to the left. This point will be between -1 and -2. (-5/3 = -1 2/3).

6. Standard Form of a Rational Number

  • A rational number p/q is said to be in standard form (or simplest form) if:
    1. Its denominator 'q' is a positive integer.
    2. The numerator 'p' and denominator 'q' have no common factor other than 1 (they are co-prime).
  • How to convert to Standard Form:
    1. If the denominator is negative, make it positive by multiplying both numerator and denominator by -1. (e.g., 3/-5 = -3/5)
    2. Find the Highest Common Factor (HCF) of the absolute values of the numerator and denominator.
    3. Divide both the numerator and the denominator by their HCF.
  • Example: Reduce 36/-24 to standard form.
    1. Make denominator positive: 36/-24 = -36/24
    2. Find HCF of 36 and 24: HCF(36, 24) = 12
    3. Divide by HCF: (-36 ÷ 12) / (24 ÷ 12) = -3/2
    • So, the standard form of 36/-24 is -3/2.

7. Comparison of Rational Numbers

  • Method 1: Using Number Line: Numbers on the right are always greater than numbers on the left. Positive numbers are always greater than negative numbers and zero. Negative numbers are always less than zero.
  • Method 2: Making Denominators Equal (Using LCM):
    1. Express both rational numbers with a positive denominator (if not already).
    2. Find the Least Common Multiple (LCM) of the denominators.
    3. Convert each rational number into an equivalent rational number with the LCM as the common denominator.
    4. Compare the numerators. The rational number with the greater numerator is greater.
  • Example: Compare 5/6 and 7/8.
    1. Denominators are positive (6 and 8).
    2. LCM(6, 8) = 24.
    3. 5/6 = (5 × 4) / (6 × 4) = 20/24
    4. 7/8 = (7 × 3) / (8 × 3) = 21/24
    5. Since 21 > 20, we have 21/24 > 20/24. Therefore, 7/8 > 5/6.
  • Example: Compare -3/4 and -5/6.
    1. Denominators are positive (4 and 6).
    2. LCM(4, 6) = 12.
    3. -3/4 = (-3 × 3) / (4 × 3) = -9/12
    4. -5/6 = (-5 × 2) / (6 × 2) = -10/12
    5. Since -9 > -10 (on the number line, -9 is to the right of -10), we have -9/12 > -10/12. Therefore, -3/4 > -5/6.

8. Rational Numbers Between Two Rational Numbers

  • Unlike integers, between any two distinct rational numbers, there lie infinitely many rational numbers.
  • Method: To find rational numbers between two given rational numbers, say a/b and c/d:
    1. Convert them to equivalent rational numbers with the same denominator (using LCM). Let them be p/q and r/q.
    2. If p and r are consecutive integers, find further equivalent rational numbers by multiplying the numerator and denominator by a suitable number (like 2, 10, etc.) to create a gap between the numerators.
    3. The rational numbers with the same denominator 'q' and numerators lying between 'p' and 'r' are the required rational numbers.
  • Example: Find 3 rational numbers between 1/3 and 1/2.
    1. LCM(3, 2) = 6.
    2. 1/3 = 2/6; 1/2 = 3/6. Numerators (2, 3) are consecutive.
    3. Multiply by 4 (to get at least 3 numbers in between):
      • 2/6 = (2 × 4) / (6 × 4) = 8/24
      • 3/6 = (3 × 4) / (6 × 4) = 12/24
    4. Numerators between 8 and 12 are 9, 10, 11.
    5. Required rational numbers are 9/24, 10/24, 11/24. (These can be simplified to 3/8, 5/12, 11/24).

9. Operations on Rational Numbers

  • a) Addition:
    • Same Denominator: Add the numerators and keep the common denominator. p/q + r/q = (p+r)/q.
      • Example: 3/7 + 2/7 = (3+2)/7 = 5/7.
      • Example: -5/8 + 3/8 = (-5+3)/8 = -2/8 = -1/4.
    • Different Denominators: Find the LCM of the denominators. Convert each rational number to an equivalent rational number with the LCM as the denominator. Then add as above.
      • Example: 2/3 + 4/5. LCM(3, 5) = 15.
      • (2×5)/(3×5) + (4×3)/(5×3) = 10/15 + 12/15 = (10+12)/15 = 22/15.
  • b) Additive Inverse: For any rational number p/q, its additive inverse is -(p/q) or -p/q, such that p/q + (-p/q) = 0.
    • Example: Additive inverse of 4/9 is -4/9. Additive inverse of -5/7 is -(-5/7) = 5/7.
  • c) Subtraction: Subtracting a rational number is the same as adding its additive inverse. a/b - c/d = a/b + (-c/d).
    • Same Denominator: p/q - r/q = (p-r)/q.
      • Example: 7/9 - 2/9 = (7-2)/9 = 5/9.
    • Different Denominators: Find LCM, convert to equivalent fractions, then subtract numerators.
      • Example: 5/6 - 3/4. LCM(6, 4) = 12.
      • (5×2)/(6×2) - (3×3)/(4×3) = 10/12 - 9/12 = (10-9)/12 = 1/12.
  • d) Multiplication: Multiply the numerators together and the denominators together. (a/b) × (c/d) = (a × c) / (b × d).
    • Example: (3/5) × (-2/7) = (3 × -2) / (5 × 7) = -6/35.
    • Example: (-4/9) × (-5/8) = (-4 × -5) / (9 × 8) = 20/72 = 5/18 (Always simplify).
  • e) Multiplicative Inverse (Reciprocal): For a non-zero rational number p/q, its multiplicative inverse or reciprocal is q/p, such that (p/q) × (q/p) = 1.
    • Example: Reciprocal of 3/7 is 7/3. Reciprocal of -5/8 is 8/-5 or -8/5. Reciprocal of -1 is -1. Reciprocal of 1 is 1. Zero (0) has no reciprocal.
  • f) Division: Dividing by a rational number is the same as multiplying by its reciprocal. (a/b) ÷ (c/d) = (a/b) × (d/c). (where c/d ≠ 0).
    • Example: (2/3) ÷ (5/7) = (2/3) × (7/5) = (2 × 7) / (3 × 5) = 14/15.
    • Example: (-4/9) ÷ (2/3) = (-4/9) × (3/2) = (-4 × 3) / (9 × 2) = -12/18 = -2/3.

Multiple Choice Questions (MCQs)

  1. Which of the following is NOT a rational number?
    (a) 5/1
    (b) -3/4
    (c) 0/8
    (d) 6/0

  2. The standard form of the rational number -48/60 is:
    (a) 48/60
    (b) -24/30
    (c) -12/15
    (d) -4/5

  3. Which rational number is equivalent to 5/8?
    (a) 10/18
    (b) 15/24
    (c) 20/30
    (d) 25/32

  4. Which of the following rational numbers is positive?
    (a) -3/5
    (b) 4/-7
    (c) -5/-9
    (d) 0/-2

  5. Which symbol should be placed in the box: -3/7 ☐ -4/9 ?
    (a) >
    (b) <
    (c) =
    (d) Cannot be determined

  6. What is the sum of 2/5 and -4/5?
    (a) 6/5
    (b) -6/5
    (c) 2/5
    (d) -2/5

  7. What is the value of (-3/4) × (8/-9)?
    (a) -2/3
    (b) 2/3
    (c) -32/27
    (d) 1/3

  8. The additive inverse of -7/11 is:
    (a) 11/7
    (b) -11/7
    (c) 7/11
    (d) -7/11

  9. The reciprocal (multiplicative inverse) of -3 is:
    (a) 3
    (b) 1/3
    (c) -1/3
    (d) Does not exist

  10. What is the result of (-5/6) ÷ (2/3)?
    (a) -10/18
    (b) -5/4
    (c) -4/5
    (d) 5/4

Answer Key for MCQs:

  1. (d) - Division by zero is undefined.
  2. (d) - HCF(48, 60) = 12. (-48 ÷ 12) / (60 ÷ 12) = -4/5.
  3. (b) - (5 × 3) / (8 × 3) = 15/24.
  4. (c) - Both numerator and denominator are negative, making the number positive (-5/-9 = 5/9).
  5. (b) - Convert to common denominator 63: -3/7 = -27/63; -4/9 = -28/63. Since -27 > -28, -3/7 > -4/9. Oops, let me recheck the comparison. -27 is indeed greater than -28. So -3/7 > -4/9. The correct option should be (a). Let me correct the answer key.
    Correction: Comparing -3/7 and -4/9. LCM is 63. -3/7 = -27/63. -4/9 = -28/63. Since -27 > -28, -3/7 > -4/9. So the answer is (a).
  6. (d) - (2 + (-4))/5 = -2/5.
  7. (b) - (-3 × 8) / (4 × -9) = -24 / -36 = 24/36 = 2/3.
  8. (c) - Additive inverse changes the sign. -(-7/11) = 7/11.
  9. (c) - Reciprocal of -3 (which is -3/1) is 1/-3 = -1/3.
  10. (b) - (-5/6) × (3/2) = (-5 × 3) / (6 × 2) = -15/12 = -5/4.

Corrected Answer Key:

  1. (d)
  2. (d)
  3. (b)
  4. (c)
  5. (a)
  6. (d)
  7. (b)
  8. (c)
  9. (c)
  10. (b)

Make sure you understand each concept, especially standard form, comparison, and all the operations. Practice regularly! Let me know if any part needs further clarification.

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