Ratio, Proportion & Commercial Maths (VI–VIII) • Topic 5 of 5

Direct & Inverse Proportion

Two quantities are in DIRECT proportion when they increase or decrease together so that their RATIO stays constant: x/y = k. More petrol, more distance; more articles, more cost. They are in INVERSE proportion when one increases as the other decreases so that their PRODUCT stays constant: x × y = k. More workers, fewer days; more speed, less time. The whole skill is deciding which relationship applies before computing — and the test is simply asking 'if one quantity goes up, does the other go up (direct) or down (inverse)?'. For direct proportion you equate ratios (x1/y1 = x2/y2); for inverse you equate products (x1 × y1 = x2 × y2). PEDAGOGY: the topic grows directly out of the unitary method and ratio, so it should be taught with hands-on tables children fill in (1 pen ₹8, 2 pens ₹16, 3 pens ₹24) and the question 'what stays the same?' — the ratio for direct, the product for inverse. COMMON MISCONCEPTIONS: the dominant error is treating every situation as direct ('more is always more'), so children multiply when they should divide in a workers-and-days problem; they also struggle to articulate what quantity stays constant. HOW TESTED: identify direct vs inverse from a described situation, a direct-proportion 'cost of more items' computation, and an inverse-proportion 'men and days' or 'speed and time' calculation.

✅ Solved examples

1. State whether the number of workers and the days taken to finish a fixed job are in direct or inverse proportion.
Inverse proportion. More workers finish the SAME job in fewer days, so as one increases the other decreases; their product (workers × days) stays constant.
2. If 6 books cost ₹240, what will 9 books cost? (direct proportion)
Cost varies directly with number of books, so cost/books is constant. 240/6 = 40 per book, so 9 books cost 9 × 40 = ₹360. (Or 6/240 = 9/x ⇒ x = 360.)
3. 8 men can build a wall in 6 days. How many days will 12 men take? (inverse proportion)
Men and days are in inverse proportion: men × days is constant = 8 × 6 = 48. For 12 men, days = 48 ÷ 12 = 4 days.
4. A car covers 150 km in 3 hours at a steady speed. How far will it go in 5 hours at the same speed?
Distance is in direct proportion to time at constant speed. Speed = 150/3 = 50 km/h, so in 5 hours distance = 50 × 5 = 250 km.

✏️ Practice — try these, take hints as needed

1. The speed of a car and the time it takes to cover a fixed distance are in ____ proportion.
Faster means less time.
One up, the other down.
Their product (or speed × time = distance) is fixed.
Inverse proportion
2. If 4 kg of rice cost ₹200, what will 7 kg cost?
Direct proportion — find the cost of 1 kg.
200 ÷ 4.
₹350 (₹50 per kg)
3. 15 workers can finish a task in 8 days. How many days will 10 workers take for the same task?
Inverse: workers × days is constant.
15 × 8 = 120.
12 days
4. A tap fills a tank in 20 minutes. If two identical taps are opened together, how long will it take?
More taps → less time (inverse).
Total work = 1 tap × 20 min.
10 minutes

📝 Topic test — 8 questions

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