NCERT Solutions for Class 9 Maths Chapter 3 — Exercise Set 3.2#
Integers $\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$ add zero and the negative numbers to the natural numbers. Brahmagupta described positive numbers as fortunes (dhana) and negative numbers as debts (ṛiṇa), and gave the rules for working with them:
| Operation | Rule |
|---|---|
| Fortune × fortune, or debt × debt | fortune (positive) |
| Fortune × debt, or debt × fortune | debt (negative) |
| Subtracting a debt | the same as adding a fortune |
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
A drop of 15 °C means we subtract 15 from the noon temperature:
Midnight temperature $= 4 - 15$
Midnight temperature $= -11$
∴ The midnight temperature is −11 °C (11 °C below zero).
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
A debt or loss is a negative integer and a profit is a positive integer.
The loan is −850, the profit is +1,200 and the loss is −450, so:
$$\text{Final standing} = (-850) + 1200 + (-450)$$We work out the final standing:
$$= 350 + (-450)$$$$= -100$$
∴ His final standing is −₹100, that is, he is still in debt of ₹100.
3. Calculate the following using Brahmagupta’s laws: #
(i) $(-12) \times 5$
A debt multiplied by a fortune is a debt:
$$(-12) \times 5 = -60$$(ii) $(-8) \times (-7)$
A debt multiplied by a debt is a fortune:
$$(-8) \times (-7) = 56$$(iii) $0 - (-14)$
Subtracting a debt is the same as adding a fortune:
$$0 - (-14) = 0 + 14$$$$0 - (-14) = 14$$
(iv) $(-20) \div 4$
A debt divided by a fortune is a debt:
$$(-20) \div 4 = -5$$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
Suppose you have ₹15 in your pocket, but you owe a friend ₹5. Your net worth (what you really own) is:
$$\text{Net worth} = 15 + (-5)$$$$\text{Net worth} = 10$$
Now your friend says, “Forget the ₹5 you owe me.” The debt of ₹5 is taken away from your accounts, so we subtract $-5$ from your net worth:
$$\text{New net worth} = 10 - (-5)$$You still have the ₹15 in your pocket and owe nothing, so:
$$\text{New net worth} = 15$$So $10 - (-5) = 15$. Removing a debt of ₹5 made you ₹5 richer, exactly as if someone had given you ₹5:
$$10 - (-5) = 10 + 5$$∴ Subtracting a negative number (taking away a debt) is the same as adding the positive number (receiving a fortune).