Calculate the Sales, Capital, Operating Expenses, Net Profit, and Rate of Return on Capital for Applio Traders, given a 25% margin, Cost of Sales of 75,000, and a Sales to Capital ratio of 1:4, with O

Given Margin is 25% and Cost of Sales (COS) is 75,000.
We know that Margin = $\frac{\text{Gross Profit}}{\text{Sales}}$ and Gross Profit = Sales - COS.
Let Sales be $S$.

$$0.25 = \frac{S - 75,000}{S}$$ $$0.25S = S - 75,000$$ $$75,000 = S - 0.25S$$ $$75,000 = 0.75S$$ $$S = \frac{75,000}{0.75}$$ $$S = 100,000$$

So, Sales = 100,000.
Now, calculate Gross Profit:

$$\text{Gross Profit} = \text{Sales} - \text{Cost of Sales}$$ $$\text{Gross Profit} = 100,000 - 75,000$$ $$\text{Gross Profit} = 25,000$$

We are given that Sales and Capital are in the ratio 1:4.

$$\frac{\text{Sales}}{\text{Capital}} = \frac{1}{4}$$ $$\frac{100,000}{\text{Capital}} = \frac{1}{4}$$ $$\text{Capital} = 100,000 \times 4$$ $$\text{Capital} = 400,000$$

Operating Expenses are 2% of Capital.

$$\text{Operating Expenses} = 0.02 \times \text{Capital}$$ $$\text{Operating Expenses} = 0.02 \times 400,000$$ $$\text{Operating Expenses} = 8,000$$

Net Profit is Gross Profit minus Operating Expenses.

$$\text{Net Profit} = \text{Gross Profit} - \text{Operating Expenses}$$ $$\text{Net Profit} = 25,000 - 8,000$$ $$\text{Net Profit} = 17,000$$

This ratio measures the profitability relative to the capital invested.

$$\text{Rate of Return on Capital} = \frac{\text{Net Profit}}{\text{Capital}} \times 100\%$$ $$\text{Rate of Return on Capital} = \frac{17,000}{400,000} \times 100\%$$ $$\text{Rate of Return on Capital} = 0.0425 \times 100\%$$ $$\text{Rate of Return on Capital} = 4.25\%$$

The ratio is $\boxed{\text{4.25\%}}$.

We use the Cost of Sales formula:

$$\text{Cost of Sales} = \text{Opening Stock} + \text{Purchases} - \text{Closing Stock}$$

Substitute the known values:

$$75,000 = 10,000 + \text{Purchases} - 5,000$$ $$75,000 = 5,000 + \text{Purchases}$$ $$\text{Purchases} = 75,000 - 5,000$$ $$\text{Purchases} = 70,000$$

The purchase amount is $\boxed{\text{70,000}}$.