To solve the problem, we can break it down step by step.

1. Initial Price and Markup: Let the cost price (CP) of the article be $x$. After marking up by 100%, the marked price (MP) becomes: $MP = x + 100\% \text{ of } x = 2x$

2. Discount Calculation: The shopkeeper offers three successive discounts of $a\%$, $b\%$, and $c\%$ such that $a + b + c = 30$. The selling price (SP) after these discounts can be calculated using the formula for successive discounts: $SP = MP \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100})$

3. Profit Calculation: The profit can be calculated as: $Profit = SP - CP$ To express it as a percentage of the cost price: $Profit \% = \frac{SP - CP}{CP} \times 100 = \frac{SP - x}{x} \times 100$

4. Combining the Equations: Substitute the SP: $Profit \% = \frac{(2x \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100})) - x}{x} \times 100$ Simplifying this: $Profit \% = (2 \times (1 - \frac{a}{100}) \times (1 - \frac{b}{100}) \times (1 - \frac{c}{100}) - 1) \times 100$

5. Expressing $a, b, c$: Given $a + b + c = 30$, we can try different combinations to maximize the profit.

To maximize profit, the discounts should be structured as follows:

• $a = b = 10\%$ and $c = 10\%$ or
• $a = 12\%, b = 12\%, c = 6\%$

Calculation of Profits for Different Combinations

Let's calculate for $a = 10, b = 10, c = 10$: $SP = 2x \times (1 - 0.1) \times (1 - 0.1) \times (1 - 0.1) = 2x \times 0.9^3 = 2x \times 0.729 = 1.458x$ Thus, $Profit = 1.458x - x = 0.458x$ Profit percentage: $Profit \% = \frac{0.458x}{x} \times 100 = 45.8\%$

Conclusion:

Thus, the maximum profit percentage he can earn is 45.8%.

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. How does changing the discount percentages affect the overall profit?
2. What would be the profit percentage if only one discount was applied?
3. How do successive discounts differ from a single discount?
4. Can the shopkeeper still profit if the cost price increases?
5. What is the impact of the initial markup on profit margins?

Tip: Always calculate your selling price after discounts to determine the actual profit before making pricing decisions.