To solve this problem, let's break it down step-by-step:

Step 1: Understanding the Problem

1. Let $C$ be the cost price of the product for the retailer.
2. The retailer aims for a 40% profit even after offering a discount of $4a\%$.
3. The selling price (SP) after giving a $4a\%$ discount should provide a 40% profit on the cost price.

Step 2: Calculate the Marked Price

1. Let $M$ be the marked price of the product.

2. After a discount of $4a\%$, the selling price $SP$ is: $SP = M \times \left(1 - \frac{4a}{100}\right)$

3. Since the retailer aims for a 40% profit after the discount, the selling price $SP$ must also equal: $SP = C \times \left(1 + \frac{40}{100}\right) = 1.4C$

4. Equating the two expressions for $SP$: $M \times \left(1 - \frac{4a}{100}\right) = 1.4C$

5. Rearranging for the marked price $M$: $M = \frac{1.4C}{1 - \frac{4a}{100}}$

Step 3: Calculate the New Selling Price with the Increased Discount

1. The new discount offered is $(3a + 25)\%$.

2. The new selling price $SP_{new}$ after this discount is: $SP_{new} = M \times \left(1 - \frac{3a + 25}{100}\right)$

3. Substitute the value of $M$ from the previous equation: $SP_{new} = \frac{1.4C}{1 - \frac{4a}{100}} \times \left(1 - \frac{3a + 25}{100}\right)$

Step 4: Find the Profit or Loss Percentage

1. The profit or loss is determined by comparing $SP_{new}$ to the cost price $C$.

2. The profit or loss percentage is: $\text{Profit or Loss \%} = \left(\frac{SP_{new} - C}{C}\right) \times 100$

3. Substitute the expression for $SP_{new}$: $\text{Profit or Loss \%} = \left(\frac{\frac{1.4C}{1 - \frac{4a}{100}} \times \left(1 - \frac{3a + 25}{100}\right) - C}{C}\right) \times 100$

4. Simplify the expression: $\text{Profit or Loss \%} = \left(\frac{1.4 \left(1 - \frac{3a + 25}{100}\right)}{1 - \frac{4a}{100}} - 1\right) \times 100$

Step 5: Final Expression for Profit or Loss

To determine the exact value of the profit or loss percentage, substitute the value of $a$ if known or keep it in terms of $a$.

Would you like to proceed with a specific value for $a$, or keep the result in terms of $a$? Feel free to ask any further questions or request additional clarifications!

Related Questions:

1. What would be the profit percentage if $a = 10\%$?
2. How would the profit percentage change if the usual discount were increased to $5a\%$?
3. What is the new selling price formula in terms of $a$ only?
4. How does the retailer's strategy change if the profit target is increased to 50%?
5. What is the effect of combining discounts on overall profit margins?

Tip:

Always break down profit and discount problems into smaller steps to find the relationship between cost, selling price, and profit percentage effectively.