The cost of carpeting a room is ₹ 120. If the width had been 4 metres less, the cost of the carpet would have been ₹ 20 less. The width of the room is

### Concept & Proportionality of Area and Cost Assuming the rate of carpeting per square meter is uniform, the total cost of carpeting is directly proportional to the area of the room. A reduction in cost corresponds proportionally to a reduction in area. ### Step-by-Step Solution * **Given:** Initial total cost = ₹ 120. * Let the length of the room be $l$, original width be $w$, and the cost per square meter be $R$. * Original Cost Equation: $R \times (l \times w) = 120$. (Equation 1) * If the width is reduced by 4 metres (new width = $w - 4$), the new cost is ₹ 20 less, meaning $120 - 20 = 100$. * New Cost Equation: $R \times (l \times (w - 4)) = 100$. (Equation 2) * To eliminate the unknown variables $R$ and $l$, divide Equation 1 by Equation 2: * $\frac{R \times l \times w}{R \times l \times (w - 4)} = \frac{120}{100}$ * $\frac{w}{w - 4} = \frac{6}{5}$ * Cross-multiply to solve for $w$: * $5w = 6(w - 4)$ * $5w = 6w - 24$ * $w = 24$. * The original width of the room is 24 metres. ### Exam Strategy & Shortcut Think in terms of direct proportion. A ₹ 20 reduction in cost is caused by a 4m reduction in width (keeping length constant). So, ₹ 20 cost corresponds to a width of 4m. The total cost is ₹ 120, which is $6 \times 20$. Therefore, the total original width must be $6 \times 4\text{m} = 24\text{m}$. This logical deduction takes less than 10 seconds. ### Common Pitfall A common mistake is trying to assume a value for the rate $R$ or length $l$, creating complex systems of equations, whereas simply observing the ratio immediately isolates the width variable. ### Final Answer Therefore, the correct answer is **24 m**.