Chapter 1, Integers (पूर्णांक), is fundamental to Class 7 Maths and covers the operations and properties of positive and negative whole numbers, including zero.
Table of Contents
The solutions below follow the structure of the latest rationalized syllabus for the Rajasthan Board (RBSE), which aligns with the NCERT textbook.
Exercise 1.1: Introduction to Integers
This exercise deals with representing integers on a number line, calculating temperature differences, and applying addition/subtraction.
| Q. No. | Type | Example (Step-by-Step Solution) | Key Concept |
| 1 | Number Line | Find the difference between the hottest (Bengaluru, 22°C) and coldest (Lahulspiti, -8°C) places. Difference = $22 – (-8)$ $= 22 + 8 = \mathbf{30°C}$ | Subtraction of negative integers ($a – (-b) = a + b$) |
| 2 | Word Problem | If Team A scores -40, 10, 0 and Team B scores 10, 0, -40, which team scored more? Team A Total: $-40 + 10 + 0 = \mathbf{-30}$ Team B Total: $10 + 0 + (-40) = \mathbf{-30}$ Both teams scored the same. | Addition of Integers (Commutative Property) |
| 3 | Temperature Change | If Srinagar’s temperature was -5°C on Monday and dropped by 2°C on Tuesday, what was the temperature on Tuesday? Tuesday’s Temp = $-5 – 2 = \mathbf{-7°C}$ | Adding/Subtracting on a Number Line (Moving left for subtraction) |
| 4 | Pair of Integers | Write a pair of negative integers whose difference is 8. Example: $(-5) – (-13)$ $= -5 + 13 = \mathbf{8}$ | Creating pairs based on a required sum or difference. |
Exercise 1.2: Properties of Integer Multiplication
This exercise focuses on the multiplication of integers and related properties like Commutativity, Associativity, and Distributivity.
| Q. No. | Type | Example (Step-by-Step Solution) | Key Concept |
| 1 | Product | Find the product: (a) $3 \times (-1)$ $= \mathbf{-3}$ (b) $(-21) \times (-30)$ $= 21 \times 30 = \mathbf{630}$ | Rules of Multiplication: $(+) \times (-) = (-)$ and $(-) \times (-) = (+)$ |
| 2 | Verification | Verify $a \times (b + c) = a \times b + a \times c$ for $a=10, b=-2, c=5$. LHS: $10 \times (-2 + 5) = 10 \times 3 = 30$ RHS: $(10 \times -2) + (10 \times 5) = -20 + 50 = 30$ LHS = RHS. | Distributive Property of multiplication over addition. |
| 3 | Identity | What is $(-1) \times a$? Any integer multiplied by $-1$ gives the Additive Inverse of that integer. Result: $\mathbf{-a}$ | Multiplicative Identity (1) and its variation. |
| 4 | Closure Property | Fill in the blanks: $-5 \times (-10) = -10 \times \_\_$ Answer: $\mathbf{-5}$ | Commutative Property for Multiplication ($a \times b = b \times a$). |
Exercise 1.3: Division of Integers
This exercise explores the division of integers and its properties.
| Q. No. | Type | Example (Step-by-Step Solution) | Key Concept |
| 1 | Quotient | Find the quotient: (a) $-30 \div 10$ $= \mathbf{-3}$ (b) $-36 \div (-9)$ $= \mathbf{4}$ | Rules of Division: $(-) \div (+) = (-)$ and $(-) \div (-) = (+)$ |
| 2 | Division Identity | Find the value of $20 \div (-1)$ $= \mathbf{-20}$ | Division by -1 gives the additive inverse. |
| 3 | Word Problem | In a test, +5 marks are given for every correct answer and -2 for every incorrect answer. Mohan got 4 correct and 6 incorrect answers. What is his score? Correct Marks: $4 \times 5 = 20$ Incorrect Marks: $6 \times (-2) = -12$ Total Score: $20 + (-12) = \mathbf{8}$ | Application of integers in scoring and real-life scenarios. |
📌 Important Definitions
- Integers: The set of all whole numbers and their negatives $(\dots, -3, -2, -1, 0, 1, 2, 3, \dots)$.
- Additive Inverse: For any integer $a$, its additive inverse is $-a$, such that $a + (-a) = 0$.
- Commutative Property (Addition/Multiplication): The order doesn’t matter: $a + b = b + a$ and $a \times b = b \times a$.
- Distributive Property: $a \times (b + c) = a \times b + a \times c$.