A trader professes to sell his goods at a nominal gain percentage but actually earns $37 \frac{1}{2}\%$ profit by using false weight. If for a kg he uses a weight of $800$ gm, what is the nominal gain percentage at which he claims to be selling his goods?

### Concept & Nominal vs. Actual Profit A trader's nominal (professed) profit is based on the declared selling price and the assumed standard weight. The actual profit depends on the actual money received compared to the cost of the actual physical weight delivered. ### Step-by-Step Solution * Let the cost price (CP) of $1$ gm of goods be Re. $1$. * **Actual Transaction:** The trader uses an $800$ gm weight for a kilogram ($1000$ gm). * The actual CP incurred by the trader is for the goods he physically gives away: CP of $800$ gm = Rs. $800$. * He earns an actual profit of $37 \frac{1}{2}\%$. Convert to fraction: $37.5\% = \frac{3}{8}$. * Actual Profit = $\frac{3}{8} \times \text{Actual CP} = \frac{3}{8} \times 800 = \text{Rs. } 300$. * Actual Selling Price (SP) = $\text{Actual CP} + \text{Actual Profit} = 800 + 300 = \text{Rs. } 1100$. * **Nominal (Professed) Claim:** He claims to be selling $1000$ gm (which costs Rs. $1000$) for the SP of Rs. $1100$. * Nominal Profit = $\text{SP} - \text{Claimed CP} = 1100 - 1000 = \text{Rs. } 100$. * Nominal Gain Percentage = $\left( \frac{\text{Nominal Profit}}{\text{Claimed CP}} \right) \times 100$ * $$ \text{Nominal Gain } \% = \frac{100}{1000} \times 100 = 10\% $$ ### Exam Strategy & Shortcut Let professed profit factor be $x$. The false weight factor is $\frac{1000}{800} = \frac{5}{4}$ (he gets paid for $5$ parts but gives $4$). Total profit factor = $1 + 37.5\% = 1 + \frac{3}{8} = \frac{11}{8}$. Equation: $x \times \frac{5}{4} = \frac{11}{8} \implies x = \frac{11}{8} \times \frac{4}{5} = \frac{11}{10} = 1.1$. A factor of $1.1$ means a $10\%$ nominal gain. ### Common Pitfall It is easy to confuse the base of the percentages. Ensure that actual profit is calculated on the actual weight delivered ($800$g), while nominal profit is calculated on the standard weight ($1000$g). ### Final Answer Therefore, the correct answer is **$10\%$**.