A person purchases $90$ clocks and sells $40$ clocks at a gain of $10\%$ and $50$ clocks at a gain of $20\%$. If he sold all of them at a uniform profit of $15\%$, then he would have got ₹ $40$ less. The cost price of each clock is

### Concept & Algebraic Profit Modeling When comparing two different sales scenarios for the same inventory, you can express total profit algebraically in terms of the cost price of a single unit. Equating the difference between the two profit scenarios to the given monetary difference allows you to solve for the unit cost price. ### Step-by-Step Solution * Let the cost price (CP) of one clock be $x$. * **Scenario 1 (Split Sales):** * Profit from 40 clocks at 10% gain = $40x \times 10\% = 4x$. * Profit from 50 clocks at 20% gain = $50x \times 20\% = 10x$. * Total Profit in Scenario 1 = $4x + 10x = 14x$. * **Scenario 2 (Uniform Sale):** * Total clocks = $90$. * Profit from all clocks at 15% uniform gain = $90x \times 15\% = 13.5x$. * **Compare the Scenarios:** * The problem states that Scenario 2 yields ₹ 40 less than Scenario 1. * $\text{Profit}_1 - \text{Profit}_2 = 40$ * $14x - 13.5x = 40$ * $0.5x = 40$ * $x = \frac{40}{0.5} = 80$. * The cost price of each clock is ₹ 80. ### Exam Strategy & Shortcut Instead of algebraic variables, use weighted averages on percentages. Scenario 1 net profit % = $\frac{(40 \times 10\%) + (50 \times 20\%)}{90} = \frac{400 + 1000}{90} = \frac{1400}{90} = \frac{140}{9}\%$. Scenario 2 net profit % = $15\% = \frac{135}{9}\%$. Difference in percentages = $\frac{140}{9}\% - \frac{135}{9}\% = \frac{5}{9}\%$. This $\frac{5}{9}\%$ of the total Cost Price equals ₹ 40. $\frac{5}{900} \times \text{Total CP} = 40 \implies \text{Total CP} = \frac{40 \times 900}{5} = 8 \times 900 = 7200$. CP per clock = $\frac{7200}{90} = 80$. ### Common Pitfall A common error is confusing the total cost price of the inventory with the cost price of a single unit. Ensure you divide the total CP by the number of units (90) if you solve for the aggregate first. ### Final Answer Therefore, the correct answer is **₹ 80**.