Direct & Inverse Proportions — Class 8

Clear notes on direct and inverse proportion with unitary method, solution strategies, solved word problems and practice — mobile-friendly & SEO-optimized.

1. Direct Proportion

Two quantities x and y are said to be in direct proportion if they increase or decrease together at the same rate. We write y ∝ x or y = kx where k is the constant of proportionality.

Property: If y₁/x₁ = y₂/x₂, then y₁ : x₁ = y₂ : x₂ (ratios equal). For multiple quantities: y₁/x₁ = y₂/x₂ = y₃/x₃.

2. Inverse Proportion

Two quantities x and y are in inverse proportion if one increases while the other decreases so that their product is constant. We write y ∝ 1/x or xy = k where k is constant.

Property: If x₁y₁ = x₂y₂, then x₁ : x₂ = y₂ : y₁. For example, if 3 workers take 10 days, then 6 workers (twice) will take 5 days (half).

3. Unitary Method (Quick strategy)

Find value for one unit, then multiply to get required units. Useful for direct proportion problems.

Example: If 5 pens cost ₹75, cost of 1 pen = ₹75/5 = ₹15. Cost of 8 pens = 8×15 = ₹120.

4. How to Solve Proportion Problems

  1. Identify whether the quantities are directly or inversely proportional.
  2. Use the relation y = kx (direct) or xy = k (inverse) to find the constant k.
  3. Apply the constant to find unknown value or set up ratio equations and use cross-multiplication.
  4. Use unitary method for straightforward money, speed, time, work problems.

Tip: For combined proportionality (e.g., y ∝ x and y ∝ 1/z), form y = k x / z and use given values to find k.

5. Solved Examples

Example 1 (Direct): If 4 kg of rice costs ₹320, find cost of 10 kg.
Cost per kg = 320/4 = 80 → Cost of 10 kg = 10×80 = ₹800.
Example 2 (Inverse): 6 machines take 12 days to finish a job. How many days will 8 machines take (same efficiency)?
Since machines ∝ 1/days, 6×12 = 8×d → d = (6×12)/8 = 9 days.
Example 3 (Combined): If y ∝ x and y ∝ 1/z, and y = 12 when x = 3, z = 2, find y when x = 6, z = 1.
y = k x / z. From data: 12 = k×3/2 → k = 8. For x=6, z=1 → y = 8×6/1 = 48.

6. Practice Questions (with answers)

  1. If 7 workers complete a task in 20 days, how many days will 14 workers take?
    Answer
    Inverse proportion: 7×20 = 14×d → d = 10 days.
  2. If y ∝ x and y = 24 when x = 6, find y when x = 15.
    Answer
    k = 24/6 = 4 → y = 4×15 = 60.
  3. A car travels at 40 km/h and takes 3 hours. How long at 60 km/h?
    Answer
    Distance = 120 km. Time = Distance/Speed = 120/60 = 2 hours (inverse relation between speed and time).
  4. y ∝ x². If y = 18 when x = 3, find y when x = 6.
    Answer
    k = 18/9 = 2 → y = 2×36 = 72.

Always state whether direct/inverse and show intermediate calculation for full marks.