Class 9 Maths Chapter 3 Exercise 3.2 Solutions Ganita Manjari NCERT
Looking for reliable and easy-to-understand answers for NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.2? You have landed at the right place! Here at Sid Classes, we break down every problem step by step to help you master integer operations. This specific exercise dives deep into real-life integer scenarios—such as tracking sub-zero mountain temperatures and calculating loans or profits—while exploring the classic arithmetic laws framed by ancient Indian mathematician Brahmagupta.
🔑 Key Concepts to Remember:
- Brahmagupta’s Rules on Fortune and Debt: A fortune (positive quantity) added to a fortune is a fortune; a debt (negative quantity) added to a debt is a debt.
- Multiplication/Division Signs: Multiplying or dividing two numbers with like signs gives a positive result; unlike signs give a negative result.
- Subtracting a Negative Number: Subtracting a negative number is equivalent to adding its corresponding positive counterpart (e.g., \(a - (-b) = a + b\)).
NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.2 Solutions
Question 1
The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?
Solution:
- Temperature at noon: \(+4\text{ }^\circ\text{C}\)
- Temperature drop by midnight: \(15\text{ }^\circ\text{C}\)
To find the midnight temperature, we subtract the drop from the noon temperature:
$$\text{Midnight Temperature} = 4 - 15$$
$$\text{Midnight Temperature} = -11\text{ }^\circ\text{C}$$
Final Answer: The midnight temperature in Ladakh is \(\mathbf{-11\text{ }^\circ\text{C}}\).
Question 2
A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing.
Solution:
Let's represent debts/losses as negative integers and profits/fortunes as positive integers:
- Loan (Debt): \(-850\)
- Profit (Fortune): \(+1200\)
- Loss: \(-450\)
Writing the Sequence as an Equation:
$$\text{Final Standing} = -850 + 1200 - 450$$
Calculating Step-by-Step:
1. Combine the first two terms: \(-850 + 1200 = 350\)
2. Subtract the loss: \(350 - 450 = -100\)
Final Answer: His final financial standing is a net debt/loss of \(\mathbf{-₹100}\) (or an overall deficit of ₹100).
Question 3
Calculate the following using Brahmagupta’s laws:
Solution:
(i) \((-12) \times 5\)
$$\text{Product} = -(12 \times 5) = \mathbf{-60}$$
(ii) \((-8) \times (-7)\)
$$\text{Product} = +(8 \times 7) = \mathbf{56}$$
(iii) \(0 - (-14)\)
$$\text{Expression} = 0 + 14 = \mathbf{14}$$
(iv) \((-20) \div 4\)
$$\text{Quotient} = -(20 \div 4) = \mathbf{-5}$$
Question 4
Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 – (–5) = 15).
Solution:
Imagine you have ₹10 in your hand, but you also owe someone a debt of ₹5 (which can be represented as having \(-5\) relative to your net worth).
Now, suppose a financial institution or a friend decides to cancel, forgive, or take away your debt of ₹5. Subtracting that debt (\(-(-5)\)) means removing a negative obligation from your account. When a debt is removed, your net financial standing increases because you no longer owe that money.
Mathematically and logically:
$$\text{Starting Money} - (\text{Removal of Debt}) = 10 - (-5) = 10 + 5 = 15$$
Final Answer: Removing a negative entity (debt) increases your overall value, making subtracting a negative number equivalent to adding a positive number (\(\mathbf{10 + 5 = 15}\)).
Frequently Asked Questions (FAQs)
Q1: What are Brahmagupta’s rules regarding integers?
A: Ancient Indian mathematician Brahmagupta formulated rules for operations using fortunes (positive numbers) and debts (negative numbers), stating foundational properties like a debt added to a debt becomes a larger debt, and the product of two debts is a fortune.
Q2: Why does multiplying two negative numbers result in a positive number?
A: Conceptually, reversing a negative direction twice (multiplying a negative quantity by a negative multiplier) turns the result back into a positive direction.
Q3: What textbook does Exercise 3.2 belong to?
A: These answers strictly follow the new Ganita Manjari Part I textbook issued for NCERT Grade 9 Mathematics.
Q4: How do you handle consecutive negative signs in integer subtraction?
A: When you subtract a negative number, the two negative signs combine to form a positive sign (e.g., \(-(-a) = +a\)).
Q5: Can temperature values drop below zero?
A: Yes, temperatures below freezing point are represented as negative integers (like \(-11\text{ }^\circ\text{C}\) in Ladakh during midnight).
Q6: Are loans represented as positive or negative numbers in equations?
A: Loans represent money owed, which is a financial deficit or debt, so they are expressed as negative integers.
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