The World of Numbers | Exercise 3.2

Question 3

Calculate the following using Brahmagupta's laws:

(i) (−12)×5(-12) \times 5

(ii) (−8)×(−7)(-8) \times (-7)

(iii) 0−(−14)0 - (-14)

(iv) (−20)÷4(-20) \div 4

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Solution
Understand the Question

Brahmagupta introduced foundational rules for arithmetic with positive numbers (Fortune), negative numbers (Debt), and zero:

  • Debt ×\times Fortune = Debt: The product of a negative number and a positive number is negative ((−a)×b=−(a×b)(-a) \times b = -(a \times b)).
  • Debt ×\times Debt = Fortune: The product of two negative numbers is positive ((−a)×(−b)=a×b(-a) \times (-b) = a \times b).
  • Zero −- Debt = Fortune: Subtracting a negative number from zero results in a positive number (0−(−a)=a0 - (-a) = a).
  • Debt ÷\div Fortune = Debt: The quotient of a negative number divided by a positive number is negative ((−a)÷b=−(a÷b)(-a) \div b = -(a \div b)).

(i) (−12)×5(-12) \times 5

Step 1 · Multiply a Negative and a Positive Number

According to Brahmagupta's law, the product of Debt (negative) and Fortune (positive) is Debt (negative).Diagram 1

(−12)×5=−(12×5)=−60\begin{aligned} (-12) \times 5 &= -(12 \times 5) \\ &= -60 \end{aligned}
Answer

(i) −60-60

(ii) (−8)×(−7)(-8) \times (-7)

Step 1 · Multiply Two Negative Numbers

According to Brahmagupta's law, the product of Debt (negative) and Debt (negative) is Fortune (positive).

(−8)×(−7)=(8×7)=56\begin{aligned} (-8) \times (-7) &= (8 \times 7) \\ &= 56 \end{aligned}
Answer

(ii) 5656

(iii) 0−(−14)0 - (-14)

Step 1 · Subtract a Negative Number from Zero

According to Brahmagupta's law, zero minus Debt (negative) is Fortune (positive).

0−(−14)=0+14=14\begin{aligned} 0 - (-14) &= 0 + 14 \\ &= 14 \end{aligned}
Answer

(iii) 1414

(iv) (−20)÷4(-20) \div 4

Step 1 · Divide a Negative Number by a Positive Number

According to Brahmagupta's law, Debt (negative) divided by Fortune (positive) is Debt (negative).

(−20)÷4=−(20÷4)=−5\begin{aligned} (-20) \div 4 &= -(20 \div 4) \\ &= -5 \end{aligned}
Answer

(iv) −5-5

Common Mistakes
  • Sign Rule in Multiplication: Forgetting that multiplying two negative numbers gives a positive product ((−8)×(−7)=+56(-8) \times (-7) = +56, not −56-56).
  • Double Negative Subtraction: Mistaking 0−(−14)0 - (-14) for 0−14=−140 - 14 = -14. Subtracting a negative number is equivalent to addition (0−(−14)=0+14=140 - (-14) = 0 + 14 = 14).
  • Division Sign Confusion: Confusing division of opposite signs with same signs; dividing a negative by a positive always yields a negative result.

More questions in Exercise 3.2

Q1

The temperature in the high-altitude desert of Ladakh is recorded as 4∘C4^\circ\text{C} at noon. By midnight, it drops by 15∘C15^\circ\text{C}. What is the midnight temperature?

Q2

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.

Q3

Calculate the following using Brahmagupta's laws:

(i) (−12)×5(-12) \times 5

(ii) (−8)×(−7)(-8) \times (-7)

(iii) 0−(−14)0 - (-14)

(iv) (−20)÷4(-20) \div 4

Q4

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)=1510 - (-5) = 15).

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