Class 8 Maths Direct and Inverse Proportions Notes & MCQs

Direct and Inverse Proportions – Quick Notes (Class 8 NCERT)

1. Direct Proportion

• Definition: Two quantities are in direct proportion if one increases, the other increases, and one decreases, the other decreases, keeping the ratio constant.

$$\text{If } x \propto y, \text{ then } \frac{x}{y} = k$$If x∝y, then yx​=k

• k = constant of proportionality

Example:

• Cost of apples ∝ Number of apples
• If 5 apples cost ₹50 → 10 apples cost ₹100

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2. Inverse Proportion

• Definition: Two quantities are in inverse proportion if one increases, the other decreases so that the product remains constant.

$$x \propto \frac{1}{y}, \quad xy = k$$x∝y1​,xy=k

Example:

• Time taken ∝ 1 / Speed
• If a car takes 5 hours at 60 km/h → time at 30 km/h = 10 hours

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3. Important Notes / Tips

• Direct proportion → constant ratio ($x/y = k$x/y=k)
• Inverse proportion → constant product ($xy = k$xy=k)
• Always check what varies: increase/decrease → identify type

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Direct and Inverse Proportions – MCQ Q&A

1. Q: If 4 pens cost ₹20, how much do 6 pens cost? (Direct proportion)
   A: 6 × 5 = ₹30
2. Q: x ∝ y → if x = 10, y = 5, find y when x = 20
   A: x/y = k → 10/5 = 20/y → y = 10
3. Q: Time ∝ 1/Speed → A car takes 5 h at 60 km/h. Time at 30 km/h?
   A: 5 × (60/30) = 10 h
4. Q: 3 workers complete a job in 12 days. Inverse proportion: 6 workers?
   A: 3 × 12 = 6 × x → x = 6 days
5. Q: Direct proportion → product or ratio constant?
   A: Ratio constant ✅
6. Q: Inverse proportion → product or ratio constant?
   A: Product constant ✅
7. Q: If 8 kg of sugar costs ₹400, cost of 5 kg? (Direct)
   A: 400/8 × 5 = ₹250
8. Q: 5 men can do a work in 20 days. 10 men? (Inverse)
   A: 5 × 20 = 10 × x → x = 10 days
9. Q: Direct proportion → x = 6, y = 9. Find k
   A: k = x/y = 6/9 = 2/3
10. Q: Inverse proportion → x = 4, y = 12. Find k
    A: xy = k → 4 × 12 = 48 → k = 48
11. Q: If quantity of flour ∝ number of cakes → 4 kg → 20 cakes. 6 kg → ?
    A: 6/4 × 20 = 30 cakes
12. Q: 12 workers → 24 days. 8 workers → ? (Inverse)
    A: 12 × 24 = 8 × x → x = 36 days
13. Q: x ∝ y → x = 15, y = 3 → x = 45 → y = ?
    A: x/y = k → 15/3 = 45/y → y = 9
14. Q: Time ∝ 1/Speed → 100 km at 50 km/h → ? h at 25 km/h
    A: 2 × (50/25) = 4 h
15. Q: Direct proportion → x = 10, y = 20 → x = 15 → y = ?
    A: y = 20/10 × 15 = 30
16. Q: Inverse proportion → x = 3, y = 12 → x = 4 → y = ?
    A: xy = 36 → 4 × y = 36 → y = 9
17. Q: Cost of 7 pens = ₹35 → 10 pens?
    A: 35/7 × 10 = ₹50
18. Q: 8 men → 16 days. 4 men → ? days
    A: 8 × 16 = 4 × x → x = 32 days
19. Q: Direct proportion → 2 kg apples → ₹30 → 5 kg → ?
    A: 30/2 × 5 = ₹75
20. Q: Inverse proportion → 5 machines → 10 days → 10 machines → ?
    A: 5 × 10 = 10 × x → x = 5 days