shopkeeper bought certain number of apples for Rs 3,600 He sold one-fifth of them at a loss of 10%, one-fourth of the remaining apples at a loss of 5%, and two-third of the rest at a  profit of 15%. At what price (in Rs.) should he sell the remaining apples to earn a profit of 27% overall?

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

### Step 1: Let the number of apples be \( x \).

### Step 2: Calculate the cost price (CP) per apple.
The total cost is Rs. 3,600, so the cost price per apple is:
\[
\text{CP per apple} = \frac{3600}{x}
\]

### Step 3: Calculate the number of apples sold in each stage and their corresponding selling prices.

#### 1. Selling one-fifth of the apples at a loss of 10%:
\[
\text{Number of apples sold} = \frac{x}{5}
\]
\[
\text{Selling price per apple} = \text{CP per apple} \times (1 - 0.10) = \frac{3600}{x} \times 0.90
\]
\[
\text{Total selling price for these apples} = \frac{x}{5} \times \frac{3600}{x} \times 0.90 = 720 \times 0.90 = 648 \text{ Rs.}
\]

#### 2. Selling one-fourth of the remaining apples at a loss of 5%:
\[
\text{Remaining apples after the first sale} = x - \frac{x}{5} = \frac{4x}{5}
\]
\[
\text{Number of apples sold in this stage} = \frac{1}{4} \times \frac{4x}{5} = \frac{x}{5}
\]
\[
\text{Selling price per apple} = \text{CP per apple} \times (1 - 0.05) = \frac{3600}{x} \times 0.95
\]
\[
\text{Total selling price for these apples} = \frac{x}{5} \times \frac{3600}{x} \times 0.95 = 720 \times 0.95 = 684 \text{ Rs.}
\]

#### 3. Selling two-thirds of the remaining apples at a profit of 15%:
\[
\text{Remaining apples after the second sale} = \frac{4x}{5} - \frac{x}{5} = \frac{3x}{5}
\]
\[
\text{Number of apples sold in this stage} = \frac{2}{3} \times \frac{3x}{5} = \frac{2x}{5}
\]
\[
\text{Selling price per apple} = \text{CP per apple} \times (1 + 0.15) = \frac{3600}{x} \times 1.15
\]
\[
\text{Total selling price for these apples} = \frac{2x}{5} \times \frac{3600}{x} \times 1.15 = 1440 \times 1.15 = 1656 \text{ Rs.}
\]

### Step 4: Find the number of remaining apples.
\[
\text{Remaining apples} = \frac{3x}{5} - \frac{2x}{5} = \frac{x}{5}
\]

### Step 5: Calculate the total selling price needed for the remaining apples to achieve an overall profit of 27%.

#### Total cost:
\[
\text{Total cost} = 3600 \text{ Rs.}
\]

#### Desired selling price for a 27% profit:
\[
\text{Desired total selling price} = 3600 \times 1.27 = 4572 \text{ Rs.}
\]

#### Selling price already obtained:
\[
\text{Selling price obtained from the first three sales} = 648 + 684 + 1656 = 2988 \text{ Rs.}
\]

#### Required selling price for the remaining apples:
\[
\text{Required selling price for the remaining apples} = 4572 - 2988 = 1584 \text{ Rs.}
\]

### Step 6: Calculate the selling price per apple for the remaining apples:
\[
\text{Required selling price per apple} = \frac{1584}{\frac{x}{5}} = 1584 \times \frac{5}{x} = \frac{7920}{x}
\]

### Final Answer:
The shopkeeper should sell the remaining apples at a rate of Rs. \( \frac{7920}{x} \) per apple to achieve an overall profit of 27%.

Would you like to go through any part of this calculation in more detail, or do you have any questions?

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Here are 5 related questions you might find interesting:
1. How would the solution change if the overall profit target was 30% instead of 27%?
2. What would be the effect on the selling price if the number of apples were doubled?
3. How does the percentage loss or profit in each stage affect the final required selling price?
4. What would happen if the shopkeeper had made a loss in the final sale instead of a profit?
5. Can you find a formula for the required selling price based on different percentages of profit or loss?

**Tip:** Always double-check your calculations by verifying each step, especially when dealing with percentages and multiple stages of sales.

Learn how to calculate the selling price of apples to achieve a 27% profit after selling a portion at varying losses and profits. This problem involves profit and loss calculations, suitable for Grades 7-9 students.

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