3.2 Direct Proportion

Two quantities are in direct proportion if an increase in one causes a proportional increase in the other (their ratio remains constant). If x and y are directly proportional, x/y = constant, or x₁/y₁ = x₂/y₂.

Example 17: If 5 pens cost ₹75, find the cost of 8 pens (cost is directly proportional to number of pens). 75/5 = x/8 → x = 75 × 8/5 = ₹120.

Example 18: A car covers 180 km using 15 litres of petrol. How much distance will it cover using 22 litres (distance directly proportional to petrol, at constant efficiency)? 180/15 = x/22 → x = 12 × 22 = 264 km.

Example 18A: If the cost of 18 chairs is ₹27,000, find the cost of 25 similar chairs. Cost per chair = 27000/18 = 1500. Cost of 25 chairs = 25 × 1500 = ₹37,500. (Direct proportion — more chairs, proportionally more cost.)

A quick way to identify direct proportion in a word problem: ask “if one quantity doubles, does the other also double (all else constant)?” If yes, it’s direct. Typical direct-proportion contexts: cost vs quantity, distance vs time (at constant speed), work done vs number of workers (for a fixed number of days).

Example 18B (additional): The weight of 12 identical books is 9.6 kg. Find the weight of 25 such books. Weight per book = 9.6/12 = 0.8 kg. Weight of 25 books = 25 × 0.8 = 20 kg.

Example 18C (additional): A recipe requires 250 g of sugar for 4 servings. How much sugar is needed for 10 servings, assuming the same proportion? Sugar per serving = 250/4 = 62.5 g. Sugar for 10 servings = 10 × 62.5 = 625 g.