Exercise 3.2 Practice
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Overview
This page provides comprehensive Ch 3: Coordinate Geometry - Exercise 3.2 Practice. Practice working with negative integers in temperature drops, financial equations of debt and fortune, Brahmagupta's integer laws, and negative number subtraction visualizations. Free step-by-step NCERT solutions and explanations.
Arithmetic of Negative Integers & Historical Math Laws
Q1: Ladakh Temperature Drop
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?
Let the initial temperature at noon be represented by a positive integer:
$$T_{\text{noon}} = 4\text{ °C}$$
The temperature drops by 15 °C. We represent a drop or decrease by subtraction:
$$T_{\text{midnight}} = T_{\text{noon}} - 15$$
Calculate the difference:
$$T_{\text{midnight}} = 4 - 15 = -11\text{ °C}$$
$-11\text{ °C}$
Q2: Spice Trader Financial Standing
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.
Represent each financial event as an integer:
- Loan (debt): $-850$
- Profit (fortune): $+1200$
- Loss: $-450$
Let the final financial standing be $S$. We write the sequence as a summation equation:
$$S = -850 + 1200 + (-450)$$
$$S = -850 + 1200 - 450$$
Calculate the sum step by step:
First, combine the debts (negative integers): $$-850 - 450 = -1300$$
Now, add the fortune: $$S = 1200 - 1300 = -100$$
First, combine the debts (negative integers): $$-850 - 450 = -1300$$
Now, add the fortune: $$S = 1200 - 1300 = -100$$
Since the final result is negative, the trader has a net debt of ₹100.
Equation: $S = -850 + 1200 - 450$; Final standing: Debt of ₹100 (represented by $-100$)
Q3: Brahmagupta's Laws of Integers
Calculate the following using Brahmagupta’s laws:
(i) $(–12) \times 5$
(ii) $(–8) \times (–7)$
(iii) $0 - (–14)$
(iv) $(–20) \div 4$
(i) $(–12) \times 5$
(ii) $(–8) \times (–7)$
(iii) $0 - (–14)$
(iv) $(–20) \div 4$
Brahmagupta's Core Laws:
- "The product of a debt and a fortune is a debt." (Negative $\times$ Positive = Negative)
- "The product of two debts is a fortune." (Negative $\times$ Negative = Positive)
- "Debt subtracted from zero is fortune." ($0 - $ Negative = Positive)
- "Debt divided by fortune is debt." (Negative $\div$ Positive = Negative)
- "The product of a debt and a fortune is a debt." (Negative $\times$ Positive = Negative)
- "The product of two debts is a fortune." (Negative $\times$ Negative = Positive)
- "Debt subtracted from zero is fortune." ($0 - $ Negative = Positive)
- "Debt divided by fortune is debt." (Negative $\div$ Positive = Negative)
(i) $(–12) \times 5$:
Product of debt $(-12)$ and fortune $(5)$ is debt: $$(–12) \times 5 = -60$$
Product of debt $(-12)$ and fortune $(5)$ is debt: $$(–12) \times 5 = -60$$
(ii) $(–8) \times (–7)$:
Product of two debts $(-8)$ and $(-7)$ is fortune: $$(–8) \times (–7) = 56$$
Product of two debts $(-8)$ and $(-7)$ is fortune: $$(–8) \times (–7) = 56$$
(iii) $0 - (–14)$:
Subtracting debt $(-14)$ from zero is fortune: $$0 - (–14) = 14$$
Subtracting debt $(-14)$ from zero is fortune: $$0 - (–14) = 14$$
(iv) $(–20) \div 4$:
Dividing debt $(-20)$ by fortune $(4)$ is debt: $$(–20) \div 4 = -5$$
Dividing debt $(-20)$ by fortune $(4)$ is debt: $$(–20) \div 4 = -5$$
(i) $-60$, (ii) $56$, (iii) $14$, (iv) $-5$
Q4: Visualizing Debt Subtraction
Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., $10 - (-5) = 15$).
Real-World Scenario:
Suppose you have ₹10 in cash, but you also owe your friend ₹5. We can represent this debt as a negative number: $-5$.
Your net financial worth is ₹5: $$10 + (-5) = 5$$
Suppose you have ₹10 in cash, but you also owe your friend ₹5. We can represent this debt as a negative number: $-5$.
Your net financial worth is ₹5: $$10 + (-5) = 5$$
Subtracting the Negative:
Now, suppose your friend decides to be generous and **forgives (removes)** your debt of ₹5.
Removing/subtracting a debt (negative number) from your ledger is represented mathematically as: $$10 - (-5)$$
Now, suppose your friend decides to be generous and **forgives (removes)** your debt of ₹5.
Removing/subtracting a debt (negative number) from your ledger is represented mathematically as: $$10 - (-5)$$
The Result:
Since you no longer owe ₹5, you are free to keep your full ₹10 cash. The removal of the ₹5 obligation has effectively increased your net worth from ₹5 to ₹15.
This has the exact same effect as if someone had handed you (added) an extra ₹5 in cash: $$10 + 5 = 15$$ Therefore, subtracting a debt (negative number) is functionally identical to adding a positive asset.
Since you no longer owe ₹5, you are free to keep your full ₹10 cash. The removal of the ₹5 obligation has effectively increased your net worth from ₹5 to ₹15.
This has the exact same effect as if someone had handed you (added) an extra ₹5 in cash: $$10 + 5 = 15$$ Therefore, subtracting a debt (negative number) is functionally identical to adding a positive asset.
Subtracting a debt removes a financial obligation, which increases net worth the same as adding cash.