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Class 7 Mathematics: Rational Numbers

Standard form of rational numbers, representation on number line, rational density, addition, subtraction, multiplication, and reciprocal division.

Class 7 Mathematics 5 Generated Practice Exercises NCERT Exercise Solutions

⚡ Key Formulas & Core Relations for Rational Numbers

Standard form: Denominator q > 0 and HCF(|p|, q) = 1
Additive Inverse of p/q is -p/q
Multiplicative Inverse (reciprocal) of p/q is q/p (p ≠ 0)
Density: Rational number between a and b = (a + b) / 2

NCERT Textbook Exercise Solutions

1 Solutions
Exercise 9.1 - Q1

Textbook Exercise Problem

List five rational numbers between -1 and 0.

Step-by-Step Verified Solution:

  1. Step 1: Write -1 and 0 with common denominator (e.g. multiply by 6/6): -1 = -6/6, 0 = 0/6.
  2. Step 2: List integers strictly between -6 and 0: -5, -4, -3, -2, -1.
  3. Step 3: Form rational numbers: -5/6, -4/6, -3/6, -2/6, -1/6.
  4. Step 4: Simplify fractions: -5/6, -2/3, -1/2, -1/3, -1/6.
Final Answer: -5/6, -2/3, -1/2, -1/3, -1/6
Key Concept / Exam Strategy: Infinitely many rational numbers lie between any two real integers.

5 AI-Generated Chapter Exercises (Foundation to HOTS)

5 Problems with Full Steps
Generated Exercise #1 Foundation
Question: Reduce -45/30 to its standard form.
Core Concept Tested: HCF reduction to irreducible standard form
Mentor Hint: Find HCF of 45 and 30 (which is 15).

Step-by-Step Solution:

  1. Step 1: HCF of 45 and 30 is 15.
  2. Step 2: Divide numerator and denominator by 15: (-45 ÷ 15) / (30 ÷ 15) = -3/2.
Verified Answer: -3/2
Generated Exercise #2 Intermediate
Question: Find the product: (-9/2) × (-7/4) and express in lowest terms.
Core Concept Tested: Multiplication of signed rational fractions
Mentor Hint: Negative times negative is positive.

Step-by-Step Solution:

  1. Step 1: Multiply numerators: (-9) × (-7) = +63.
  2. Step 2: Multiply denominators: 2 × 4 = 8.
  3. Step 3: Result = 63/8 (or 7 7/8).
Verified Answer: 63/8
Generated Exercise #3 Intermediate
Question: Divide: -7/12 ÷ (-2/13).
Core Concept Tested: Division using reciprocal inversion
Mentor Hint: Change division to multiplication by inverting -2/13 to -13/2.

Step-by-Step Solution:

  1. Step 1: Reciprocal of -2/13 is -13/2.
  2. Step 2: Multiply: (-7/12) × (-13/2).
  3. Step 3: (-7) × (-13) = +91; 12 × 2 = 24.
  4. Step 4: Result = 91/24.
Verified Answer: 91/24
Generated Exercise #4 JEE-NEET Foundation
Question: If the product of two rational numbers is -14/27 and one of the numbers is 7/9, find the other number.
Core Concept Tested: Formulation and solution of rational equation
Mentor Hint: Other number = Product ÷ Given number.

Step-by-Step Solution:

  1. Step 1: Let other number = x. Then (7/9) × x = -14/27.
  2. Step 2: x = (-14/27) ÷ (7/9) = (-14/27) × (9/7).
  3. Step 3: Cross cancel: -14/7 = -2; 9/27 = 1/3.
  4. Step 4: x = -2/3.
Verified Answer: -2/3
Generated Exercise #5 Advanced HOTS
Question: Find three rational numbers strictly between 1/4 and 1/3.
Core Concept Tested: Decimal expansion or common denominator interpolation
Mentor Hint: Convert to denominator with 48 or 100: 1/4 = 0.25, 1/3 ≈ 0.333.

Step-by-Step Solution:

  1. Step 1: 1/4 = 30/120 and 1/3 = 40/120.
  2. Step 2: Choose numerators between 30 and 40: 31/120, 35/120, 37/120.
  3. Step 3: Simplifying 35/120 gives 7/24.
Verified Answer: 31/120, 7/24, 37/120