A 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

Let's break down the problem to find the percentage of profit or loss.

### Step 1: Set up the initial profit scenario

Let the cost price (C.P.) of the product be \( C \).

The retailer intends to earn a 40% profit even after offering a \( 4a\% \) discount. Therefore, the marked price (M.P.) is set to achieve this.

The selling price after discount should still yield a 40% profit:

\[
\text{Selling Price (S.P.) after } 4a\% \text{ discount} = C \times (1 + 0.40) = 1.4C
\]

Given that a \( 4a\% \) discount is offered on the marked price (M.P.), the selling price can also be expressed as:

\[
\text{Selling Price} = \text{M.P.} \times \left(1 - \frac{4a}{100}\right)
\]

Equating the two expressions for the selling price:

\[
\text{M.P.} \times \left(1 - \frac{4a}{100}\right) = 1.4C
\]

Solving for M.P.:

\[
\text{M.P.} = \frac{1.4C}{1 - \frac{4a}{100}} = \frac{1.4C \times 100}{100 - 4a}
\]

### Step 2: Calculate the new selling price with the new discount

Now, a discount of \( (3a + 25)\% \) is offered instead. The new selling price (S.P.') is:

\[
\text{S.P.'} = \text{M.P.} \times \left(1 - \frac{3a + 25}{100}\right)
\]

Substituting the value of M.P. from the previous step:

\[
\text{S.P.'} = \frac{1.4C \times 100}{100 - 4a} \times \left(1 - \frac{3a + 25}{100}\right)
\]

Simplifying further:

\[
\text{S.P.'} = \frac{1.4C \times 100 \times (100 - 3a - 25)}{(100 - 4a) \times 100}
\]

\[
\text{S.P.'} = \frac{1.4C \times (75 - 3a)}{100 - 4a}
\]

### Step 3: Determine the percentage of profit or loss

To find the percentage profit or loss:

\[
\text{Profit or Loss Percentage} = \frac{\text{S.P.'} - \text{C.P.}}{\text{C.P.}} \times 100
\]

Substitute \( \text{S.P.'} = \frac{1.4C \times (75 - 3a)}{100 - 4a} \) into the equation:

\[
\text{Profit or Loss Percentage} = \frac{\frac{1.4C \times (75 - 3a)}{100 - 4a} - C}{C} \times 100
\]

Simplifying further:

\[
\text{Profit or Loss Percentage} = \frac{1.4(75 - 3a) - (100 - 4a)}{100 - 4a} \times 100
\]

Expanding and simplifying:

\[
\text{Profit or Loss Percentage} = \frac{105 - 4.2a - 100 + 4a}{100 - 4a} \times 100
\]

\[
\text{Profit or Loss Percentage} = \frac{5 - 0.2a}{100 - 4a} \times 100
\]

This is the final expression that gives the percentage of profit or loss based on the value of \( a \).

### Conclusion
- If \( 5 - 0.2a > 0 \), there will be a profit.
- If \( 5 - 0.2a < 0 \), there will be a loss.
- If \( 5 - 0.2a = 0 \), there will be neither profit nor loss.

Would you like to explore specific values for \( a \) or any other details?

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Here are some related questions:

1. What happens to the profit or loss if \( a = 10\% \)?
2. How does increasing the value of \( a \) affect the profit or loss?
3. What is the profit percentage when \( a = 5\% \)?
4. Can there be a scenario where the retailer neither makes a profit nor incurs a loss?
5. How would the calculations change if the cost price is doubled?

**Tip:** Always double-check your profit and discount percentages to ensure you're setting the correct marked price for the desired profit margin.

A 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?

Determine the percentage of profit or loss when a retailer changes the discount percentage from 4a% to (3a + 25)%. Learn how to solve this complex problem using algebraic expressions, percentage calculations, and profit-loss formulas. Suitable for high school students in Grades 10-12.

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