Left pan of a faulty balance weighs $100$ grams more than its right pan. A shopkeeper keeps the weight measure in the left pan while buying goods but keeps it in the right pan while selling his goods. He uses only $1$ kg weight measure. If he sells his goods at the listed cost price, what is his gain? (Civil Services, 2005; Hotel Mgmt, 2007)

### Concept & Faulty Balances When a balance is faulty such that the left pan is heavier, the empty weights relate as $L = R + 100$. Placing standard weights in different pans changes the effective mass of goods required to balance the scales. ### Step-by-Step Solution Let the empty right pan weigh $w$ grams. The empty left pan weighs $w + 100$ grams. Buying Phase: * The shopkeeper puts a $1000$g weight in the left pan and goods in the right pan. * Total weight on left = $(w + 100) + 1000 = w + 1100$. * Total weight on right = $w +$ goods. * Equating the pans: $w + 1100 = w + \text{goods} \implies \text{goods} = 1100$g. * He pays for $1000$g but receives $1100$g. Selling Phase: * He puts a $1000$g weight in the right pan and goods in the left pan. * Total weight on right = $w + 1000$. * Total weight on left = $(w + 100) + \text{goods}$. * Equating the pans: $w + 100 + \text{goods} = w + 1000 \implies \text{goods} = 900$g. * He sells $900$g for the cost price of $1000$g. Profit Calculation: * Let the cost price per gram be $1$ unit. * Total cost for $1100$g bought = $1000$ units. (CP per gram = $1000/1100$) * Total selling price for $900$g sold = $1000$ units. (SP per gram = $1000/900$) * For an equivalent quantity (e.g., $9900$g), total CP = $9000$ units, total SP = $11000$ units. * Gain = $11000 - 9000 = 2000$ units. * Gain % = $\frac{2000}{9000} \times 100 = \frac{200}{9}\%$. ### Exam Strategy & Shortcut Instead of finding SP and CP of $1$g, use the ratio of quantity bought to quantity sold for the same monetary amount. He effectively gets $1100$g for $P$ amount and gives $900$g for $P$ amount. $$ \text{Profit } \% = \frac{\text{Goods Bought} - \text{Goods Sold}}{\text{Goods Sold}} \times 100 $$ $$ \text{Profit } \% = \frac{1100 - 900}{900} \times 100 = \frac{200}{9}\% $$ ### Common Pitfall A common mistake is assuming he gains $100$g while buying and loses $100$g while selling on a flat $1000$g base, and simply adding percentages. It is crucial to trace the actual physical grams exchanged. ### Final Answer Therefore, the correct answer is **$\frac{200}{9}\%$**.