Question
If 5 workers can build a wall in 12 days, how many days will 10 workers take to build the same wall? Is this direct or inverse proportion? Also, if 3 metres of cloth costs Rs 210, find the cost of 7 metres.
Solution — Step by Step
More workers → less time needed. As one quantity increases, the other decreases. This is inverse proportion.
More cloth → more cost. As one quantity increases, the other also increases. This is direct proportion.
Why This Works
graph TD
A["How to identify proportion type?"] --> B["When one increases, does the other increase too?"]
B -->|Yes| C["DIRECT proportion: x/y = constant"]
B -->|No| D["When one increases, does the other decrease?"]
D -->|Yes| E["INVERSE proportion: x × y = constant"]
D -->|No| F["Not a proportion relationship"]In direct proportion, the ratio stays constant. Double , double . Triple , triple .
In inverse proportion, the product stays constant. Double , halve . Triple , reduce to one-third.
Real-life examples of direct proportion: distance and petrol consumed, items bought and total cost. Inverse proportion: workers and time, speed and time for a fixed distance.
Alternative Method
The unitary method always works for both types:
For the cloth problem: 3 metres costs Rs 210. So 1 metre costs Rs 70. Therefore 7 metres costs Rs 490.
For the workers problem: 5 workers take 12 days. So 1 worker takes days (more workers gone, more time). Therefore 10 workers take days.
The unitary method avoids the confusion of which formula to use — just find the value for 1 unit first.
Common Mistake
Applying direct proportion when it should be inverse, or vice versa. The most common error: “5 workers take 12 days, so 10 workers take 24 days” (applying direct proportion). But more workers should mean less time, not more. Always do a quick sanity check: does your answer make physical sense? If doubling workers increases time, something is wrong.