Let a and b be two non-zero rational numbers such that a + 1b = 0. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.
Answer: a = −1b, so ab = −1b × b = −1. Hence ab is negative.
Step-by-step solution
Idea: Rearrange to get a in terms of b, then multiply by b: the product has a fixed value.
- a + 1b = 0 gives a = −1b (b ≠ 0, so 1b is defined).1 mark
- Multiply both sides by b: ab = −1b × b = −1. Since −1 < 0, ab is negative. (So a and b always have opposite signs.)1 mark
ab = −1, so ab is negative.
Check: Example only to check (not as proof): b = 2, a = −12: a + 1b = 0 and ab = −1 ✓.
Answer to write in the exam
a + 1b = 0 ⇒ a = −1b
ab = −1b × b = −1
∴ ab = −1 < 0; ab is negative.
Common mistakes that cost marks
- Substituting numbers to decide. The question says ‘without assigning values’; an example is not a proof.
- Writing a = 1b (losing the minus sign) and concluding ab = 1.
- Multiplying only one side by b.
How this can come in the exam
MCQ (1 mark)
If p − 1q = 0 for non-zero rationals p, q, then pq =
- −1
- 0
- 1
- cannot be found
Show answer
(C) 1
p = 1q, so pq = 1.
Assertion–Reason (1 mark)
Assertion (A): If a + 1b = 0, then a and b have opposite signs.
Reason (R): ab = −1.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
ab = −1 < 0 means one of a, b is negative and the other positive; R explains A.
Try one yourself
If m and n are non-zero rationals with mn + 1 = 0, is mn positive or negative?
Show answer
m = −n, so mn = −n2 < 0: negative.
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