Calculate the following using Brahmagupta’s laws:
- (i) (−12) × 5
- (ii) (−8) × (−7)
- (iii) 0 − (−14)
- (iv) (−20) ÷ 4
Answer: (i) −60 (ii) 56 (iii) 14 (iv) −5
Step-by-step solution
Idea: Brahmagupta’s laws: a debt times a fortune is a debt (negative × positive = negative); a debt times a debt is a fortune (negative × negative = positive); removing a debt adds to your worth; division follows the same sign rules as multiplication.
(i) (−12) × 5
- Debt × fortune = debt: 5 debts of 12 make a debt of 60. So (−12) × 5 = −60.½ mark
−60
(ii) (−8) × (−7)
- Debt × debt = fortune: 8 × 7 = 56 and the sign is positive. So (−8) × (−7) = 56.½ mark
56
(iii) 0 − (−14)
- Taking away a debt of 14 from nothing leaves you 14 better off: 0 − (−14) = 0 + 14 = 14.½ mark
14
(iv) (−20) ÷ 4
- A debt of 20 shared equally among 4 is a debt of 5 each. So (−20) ÷ 4 = −5.½ mark
−5
(i) −60 (ii) 56 (iii) 14 (iv) −5
Check: (iv) (−5) × 4 = −20 ✓. (iii) 14 + (−14) = 0 ✓.
Answer to write in the exam
(i)
(−12) × 5 = −60
(ii)
(−8) × (−7) = 56
(iii)
0 − (−14) = 0 + 14 = 14
(iv)
(−20) ÷ 4 = −5
Common mistakes that cost marks
- Writing (−8) × (−7) = −56. Two negatives multiply to a positive.
- Writing 0 − (−14) = −14. Subtracting a negative is the same as adding the positive.
- Giving (−20) ÷ 4 = 5: the signs differ, so the quotient is negative.
How this can come in the exam
MCQ (1 mark)
Which of these has a positive value?
- (−9) × 4
- (−36) ÷ 6
- (−3) × (−11)
- 0 − 15
Show answer
(C) (−3) × (−11)
(−3) × (−11) = 33. The others are −36, −6 and −15.
Short answer (2 marks)
Evaluate [(−6) × (−9)] − [(−45) ÷ 5].
Show answer
(−6) × (−9) = 54 and (−45) ÷ 5 = −9 (1 mark). 54 − (−9) = 54 + 9 = 63 (1 mark).Try one yourself
Calculate (a) (−15) × (−4) (b) (−72) ÷ (−8) (c) 3 − (−17).
Show answer
(a) 60 (b) 9 (c) 20
More questions like this
- 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).
- As society grew more complex, measuring became just as important as counting. If a farmer divides a field of wheat among his three children, how much does each get? If a recipe calls for half a cup of ghee, how do we represent that mathematically?
- Can you explain why we need q ≠ 0 in the definition of a rational number?
- 1. While adding or subtracting two rational numbers having different denominators, how will you make the denominators equal?
2. Verify the distributive law for rational numbers. - Prove that the following rational numbers are equal: