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Let's solve this problem step by step:

### Given Information:

1. **Selling price ratios of the 3 shopkeepers**: 8:9:5
2. **Profit percentages ratios**: 8:7:14
3. **First shopkeeper's profit**: \( \frac{1}{7} \) or \( 14.28\% \)
4. **Second shopkeeper's cost price**: ₹400
5. We need to find the **overall profit percentage**.

---

### Step 1: Define Variables

- Let the selling prices be 8x, 9x, and 5x for the three shopkeepers.
- Let the cost prices be \(C_1\), \(C_2 = 400\), and \(C_3\).

### Step 2: Profit Calculation

- For the first shopkeeper, profit is \( \frac{1}{7} \). This implies the profit percentage is 14.28%.
  \[
  \text{Profit} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100
  \]
  For the first shopkeeper:
  \[
  \frac{8x - C_1}{C_1} = \frac{1}{7} \quad \Rightarrow \quad 8x - C_1 = \frac{C_1}{7}
  \]
  Solving for \(C_1\):
  \[
  8x = \frac{8C_1}{7} \quad \Rightarrow \quad C_1 = 7x
  \]
  
- For the second shopkeeper, the profit percentage is 7%. Given the cost price of ₹400, we can find the selling price:
  \[
  \frac{9x - 400}{400} \times 100 = 7 \quad \Rightarrow \quad 9x - 400 = 28 \quad \Rightarrow \quad 9x = 428 \quad \Rightarrow \quad x = \frac{428}{9} = 47.56
  \]

- For the third shopkeeper, profit percentage is 14%.
  \[
  \frac{5x - C_3}{C_3} \times 100 = 14 \quad \Rightarrow \quad 5x - C_3 = \frac{14C_3}{100}
  \]
  Solving for \(C_3\):
  \[
  5x = \frac{114C_3}{100} \quad \Rightarrow \quad C_3 = \frac{500x}{114}
  \]
  Substituting \(x = 47.56\), we get:
  \[
  C_3 = \frac{500 \times 47.56}{114} = 208.52
  \]

### Step 3: Calculate Total Cost and Selling Price

- Total selling price = \( 8x + 9x + 5x = 22x \)
  \[
  \text{Total Selling Price} = 22 \times 47.56 = 1046.32
  \]
  
- Total cost price = \( 7x + 400 + 208.52 \)
  \[
  \text{Total Cost Price} = 7 \times 47.56 + 400 + 208.52 = 333.92 + 400 + 208.52 = 942.44
  \]

### Step 4: Find Overall Profit Percentage

\[
\text{Overall Profit Percentage} = \frac{\text{Total Selling Price} - \text{Total Cost Price}}{\text{Total Cost Price}} \times 100
\]
\[
= \frac{1046.32 - 942.44}{942.44} \times 100 = \frac{103.88}{942.44} \times 100 = 11.02\%
\]

### Conclusion:

None of the given options (14.87%, 14.28%, 16.67%, 15.78%) match the calculated profit of 11.02%. Therefore, the correct answer is:

**e) None of these**

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Do you need further details or clarifications?

### Related Questions:

1. How do you calculate profit percentages when both cost price and selling price are given?
2. Can profit ratios directly determine overall profit percentages?
3. What role does the selling price ratio play in this calculation?
4. How would the answer change if the profit percentages were different?
5. What happens if one shopkeeper incurs a loss instead of profit?

**Tip:** Always ensure to correctly match the cost price and selling price relationships before calculating the profit percentage.

3 shopkeepers sell a bike at different prices. The selling prices of the 3 are in the ratio 8:9:5 and they earn a profit percentage in the ratio of 8:7:14 respectively. If the first shopkeeper earned 1/7th profit on the cost price and the second shopkeeper had a cost price of ₹400/-, find the overall percentage of the 3 combined.

This problem involves calculating the overall profit percentage of 3 shopkeepers selling bikes at different prices. Learn how to use profit percentage, selling price ratios, and cost price calculations to determine the combined profit. Suitable for Grades 10-12.

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