retailer prices his products to earn a 40% profit even after offering a 4a% discount. If on a particular day, instead of the usual discount, the retailer ends up giving a (3a + 25)% discount to a customer, what will be his percentage of profit or loss?

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.

Solve a complex profit and loss problem where a retailer offers different discounts and calculates the percentage of profit or loss. The solution involves using percentage formulas for profit, discount, and selling price, suitable for students in Grades 10-12.

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