A shopkeeper advertises for selling cloth at $4\%$ loss. However, by using a false metre scale he actually gains $20\%$. What is the actual length of the scale? (R.R.B., 2006)

### Concept & Reverse Percentage Calculation When the overall gain percentage and the professed price percentage are given, we can trace backward to find the actual physical quantity delivered by equating the final selling price to the marked up actual cost. ### Step-by-Step Solution * Let the true cost price of $100$ cm ($1$ metre) of cloth be Rs. $100$. * The shopkeeper advertises a $4\%$ loss. * Selling Price (SP) for a promised $100$ cm = $100 - 4 = \text{Rs. } 96$. * Let the actual length of the false scale be $x$ cm. * The actual Cost Price (CP) for the cloth given is Rs. $x$. * The problem states he makes an actual gain of $20\%$ on his actual CP. * Therefore, $\text{SP} = \text{Actual CP} + 20\%$ of Actual CP. * $$ 96 = x + 0.20x $$ * $$ 96 = 1.2x $$ * $$ x = \frac{96}{1.2} = 80 $$ * The actual length of the scale used is $80$ cm. ### Exam Strategy & Shortcut Let the false scale length be $x$. He charges for $96$ cm (due to $4\%$ loss on $100$ cm). His profit is $20\%$, meaning the amount he charges represents $120\%$ of what he actually gives. $120\% \text{ of } x = 96$ $1.2x = 96 \implies x = 80$. ### Common Pitfall A frequent error is taking $20\%$ of $96$ and subtracting it, which incorrectly assumes the $20\%$ margin is calculated on the selling price instead of the cost price. Profit is always based on the CP. ### Final Answer Therefore, the correct answer is **$80$ cm**.