Q1.
Which of these are in inverse proportion? (i) x: 40, 80, 25, 16 and y: 20, 10, 32, 50. (ii) x: 40, 80, 25, 16 and y: 20, 10, 12.5, 8. (iii) x: 30, 90, 150, 10 and y: 15, 5, 3, 45.
Answer
Test each column by multiplying x by y. If every product is the same constant, the pair is in inverse proportion.
| Table | Products x × y | Inverse proportion? |
|---|---|---|
| (i) | 40×20 = 800, 80×10 = 800, 25×32 = 800, 16×50 = 800 | Yes, k = 800 |
| (ii) | 40×20 = 800, 80×10 = 800, 25×12.5 = 312.5, 16×8 = 128 | No |
| (iii) | 30×15 = 450, 90×5 = 450, 150×3 = 450, 10×45 = 450 | Yes, k = 450 |
(i) and (iii) are in inverse proportion; (ii) is not.
Why (ii) fails: Its first two columns do keep the product 800, so the first pairs look promising. But in the last two columns the numbers behave quite differently — 25 → 12.5 and 16 → 8 are simply halvings, with x and y falling together. A single table cannot be inverse in one part and direct in another, so (ii) is neither. Checking only two columns would have led you astray; the constant product must hold for every column.
Tip: For a direct proportion you divide (x/y must be constant); for an inverse proportion you multiply (xy must be constant). In (i), 40/20 = 2 but 25/32 is not 2 — so it is certainly not direct.