Short Intro

Exercise 3.2 of “The World of Numbers” focuses on integers, negative numbers, and Brahmagupta’s arithmetic laws. Students learn how to solve real-life problems involving temperature changes, debts, profits, and operations with negative numbers. These step-by-step solutions are written in simple English for easy understanding and exam preparation.

Quick Information Box

Topic	Details
Chapter	The World of Numbers
Exercise	3.2
Subject	Mathematics
Class	Grade 9
Main Concept	Integers & Brahmagupta’s Laws
Difficulty Level	Easy
Useful For	School Exams & Practice

Concepts Used (Topics Covered)

• Positive and Negative Numbers
• Integers
• Brahmagupta’s Laws
• Multiplication of Integers
• Division of Integers
• Real-life Applications of Integers
• Temperature Problems
• Profit and Loss Using Integers

Important Formulas

Integer Set

$\mathbb{Z}=\{\dots,-3,-2,-1,0,1,2,3,\dots\}$

Product Rules of Integers

$(-a)\times(+b)=-ab$

$(-a)\times(-b)=+ab$

Zero Property

$a+0=a,\ a-0=a,\ a\times0=0$

Questions & Step-by-Step Solutions

Question 1

The temperature in Ladakh is 4°C at noon. By midnight, it drops by 15°C. Find the midnight temperature.

Solution

Initial temperature = 4°C

Temperature drop = 15°C

So,$$4 – 15$$$$= -11$$

Answer

The midnight temperature is:$$\boxed{-11^\circ C}$$

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Question 2

A spice trader takes a loan of ₹850. The next day he earns a profit of ₹1200. Later he incurs a loss of ₹450. Find his final financial standing.

Solution

Loan = −850

Profit = +1200

Loss = −450

Equation:$$-850 + 1200 – 450$$

Step 1:$$-850 + 1200 = 350$$

Step 2:$$350 – 450 = -100$$

Answer

Final financial standing:$$\boxed{-₹100}$$

This means the trader still has a debt of ₹100.

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Question 3

Calculate the following using Brahmagupta’s laws:

(i) $(-12) \times 5$

Using integer multiplication rule:

Negative × Positive = Negative$$(-12)\times5=-60$$

Answer

$$\boxed{-60}$$

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(ii) $(-8) \times (-7)$(−8)×(−7)

Negative × Negative = Positive$$(-8)\times(-7)=56$$

Answer

$$\boxed{56}$$

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(iii) $0 – (-14)$

Subtracting a negative means adding a positive.$$0-(-14)=0+14$$$$=14$$

Answer

$$\boxed{14}$$

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(iv) $(-20) \div 4$

Negative ÷ Positive = Negative$$(-20)\div4=-5$$

Answer

$$\boxed{-5}$$

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Question 4

Explain using a real-life example why subtracting a negative number is the same as adding a positive number.

Solution

Suppose you owe your friend ₹5.

Your current money = ₹10

Mathematically:$$10-(-5)$$

Subtracting a negative means removing debt.

If your ₹5 debt is removed, your financial condition improves by ₹5.

So,$$10+5=15$$

Answer

$$10-(-5)=15$$

Therefore, subtracting a negative number is the same as adding a positive number.

Common Mistakes

• Forgetting sign rules while multiplying integers
• Confusing subtraction of negative numbers
• Writing positive answers for negative operations
• Ignoring real-life meaning of debts and fortunes
• Division sign mistakes with integers

Exam Tips

• Remember:
  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
• Practice temperature and money-based questions
• Always write signs carefully
• Learn Brahmagupta’s laws properly
• Use step-by-step calculation in exams

Practice MCQs

1. What is the value of:

$$(-6)\times(-4)$$(−6)×(−4)

A) −24
B) 24
C) −10
D) 10

Answer

B) 24

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2. What is:

$$0-(-9)$$0−(−9)

A) −9
B) 0
C) 9
D) −18

Answer

C) 9

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3. Which of the following is an integer?

A) $\frac12$21​
B) $\sqrt2$2​
C) −7
D) 3.5

Answer

C) −7

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4. The result of:

$$(-15)\div3$$(−15)÷3

is:

A) 5
B) −5
C) 45
D) −45

Answer

B) −5

FAQ Section

Q1. What are integers?

Integers are positive numbers, negative numbers, and zero.

Q2. Who introduced negative numbers formally?

Brahmagupta formally introduced negative numbers.

Q3. Why does negative × negative become positive?

Because removing a debt improves financial value.

Q4. Is zero a positive number?

No, zero is neither positive nor negative.

Q5. Can integers be represented on a number line?

Yes, integers are represented on a number line.

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