Direct and Inverse Proportions (NCERT Ch. 13)

• Distinguish direct proportion (both quantities increase/decrease together at a constant ratio) from inverse proportion (one increases as the other decreases, product stays constant).
• Solve real-life problems using direct and inverse proportion.

1. Two quantities x and y are in direct proportion if x/y = k (a constant) — as one increases, the other increases proportionally.
2. Two quantities x and y are in inverse proportion if x × y = k (a constant) — as one increases, the other decreases proportionally.
3. In direct proportion problems: x₁/y₁ = x₂/y₂.
4. In inverse proportion problems: x₁y₁ = x₂y₂.

If 5 workers can build a wall in 12 days, how many days will 10 workers take (inverse proportion, assuming same work rate)?

More workers -> fewer days needed (inverse proportion).
x1*y1 = x2*y2
5 x 12 = 10 x y2
60 = 10 x y2
y2 = 6 days

• Applying direct proportion logic to an inverse proportion situation (or vice versa) — always ask "does more of one mean more or less of the other?"
• Forgetting that speed and time are inversely proportional for a fixed distance, while distance and time are directly proportional for fixed speed.
• Setting up the wrong equation (cross-multiplication for direct, product-equality for inverse).

• If 3 pens cost ₹45, is this direct or inverse proportion, and what would 5 pens cost?
• If 4 workers finish a job in 9 days, how many days would 6 workers take (inverse proportion)?

Open the Practice tab for graded questions on Direct and Inverse Proportions (NCERT Ch. 13).