Instead of a metre scale, a cloth merchant uses a $120$ cm scale while buying but uses an $80$ cm scale while selling the same cloth. If he offers a discount of $20\%$ on cash payment what is his overall profit percentage?

### Concept & Double Dishonesty in Trading When a merchant cheats during both buying and selling, we must track the actual cost price (CP) incurred for the physical goods obtained and the selling price (SP) realized for the physical goods delivered. Calculating the CP and SP per single unit of measurement normalizes the transaction. ### Step-by-Step Solution * Let the true cost price of $1$ cm of cloth be Re. $1$. * **Buying Phase:** He uses a $120$ cm scale instead of $100$ cm. This means he pays for $100$ cm but receives $120$ cm. * Total cost paid = Rs. $100$ for $120$ cm. * Actual CP per cm = $\frac{100}{120} = \frac{5}{6}$ Rupees. * **Selling Phase:** He uses an $80$ cm scale instead of $100$ cm. This means he charges the customer for $100$ cm but only delivers $80$ cm. * The marked price for the $80$ cm he delivers is Rs. $100$. * He offers a $20\%$ discount on cash payment. * Actual SP for the $80$ cm = $100 - 20\%$ of $100 = \text{Rs. } 80$. * Actual SP per cm = $\frac{80}{80} = 1$ Rupee. * **Profit Calculation:** * Profit per cm = $\text{SP} - \text{CP} = 1 - \frac{5}{6} = \frac{1}{6}$ Rupees. * Profit Percentage = $\frac{\text{Profit}}{\text{CP}} \times 100$ * $$ \text{Profit } \% = \frac{\frac{1}{6}}{\frac{5}{6}} \times 100 = \frac{1}{5} \times 100 = 20\% $$ ### Exam Strategy & Shortcut Use the multiplier method. Multiplier while buying: gets $120$ for price of $100 \implies \frac{120}{100} = 1.2$ Multiplier while selling: gives $80$ for price of $100 \implies \frac{100}{80} = 1.25$ Discount multiplier: $20\%$ discount $\implies 0.8$ Overall multiplier = $1.2 \times 1.25 \times 0.8 = 1.2$ A final multiplier of $1.2$ corresponds to a $20\%$ profit. ### Common Pitfall A common mistake is simply adding up the percentage gains from buying ($20\%$), selling ($25\%$), and subtracting the discount ($20\%$). Successive percentage changes are multiplicative, not additive. ### Final Answer Therefore, the correct answer is **$20\%$**.