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Let's break down the problem and solve it step by step:

### Given:
- The dimensions of the rectangular field are 50 m by 40 m.
- A flower bed is prepared inside the field, leaving a uniform gravel path around it.
- The cost of laying the flower bed is ₹30 per square meter, and for gravelling the path, it is ₹20 per square meter.
- The total cost for both the flower bed and the gravel path is ₹52,000.

### To find:
- The width of the gravel path.

### Solution:

1. **Dimensions of the flower bed**:
   Let the width of the gravel path be \( x \) meters. Therefore, the dimensions of the flower bed will be reduced by \( 2x \) meters in both length and width, since the path is around the flower bed on all sides.
   
   Thus, the dimensions of the flower bed will be:
   - Length = \( 50 - 2x \) meters
   - Width = \( 40 - 2x \) meters

2. **Area of the flower bed**:
   The area of the flower bed is:
   \[
   \text{Area of the flower bed} = (50 - 2x)(40 - 2x)
   \]

3. **Area of the entire field**:
   The area of the field (without the path) is:
   \[
   \text{Area of the field} = 50 \times 40 = 2000 \, \text{m}^2
   \]

4. **Area of the gravel path**:
   The area of the gravel path will be the total area of the field minus the area of the flower bed:
   \[
   \text{Area of the gravel path} = 2000 - (50 - 2x)(40 - 2x)
   \]

5. **Cost of laying the flower bed**:
   The cost to lay the flower bed is \( 30 \, \text{₹/m}^2 \), so:
   \[
   \text{Cost of the flower bed} = 30 \times (50 - 2x)(40 - 2x)
   \]

6. **Cost of gravelling the path**:
   The cost to gravel the path is \( 20 \, \text{₹/m}^2 \), so:
   \[
   \text{Cost of the gravel path} = 20 \times (2000 - (50 - 2x)(40 - 2x))
   \]

7. **Total cost**:
   The total cost for both the flower bed and the gravel path is given as ₹52,000. Therefore, we can write the equation:
   \[
   30 \times (50 - 2x)(40 - 2x) + 20 \times (2000 - (50 - 2x)(40 - 2x)) = 52000
   \]

We can now solve this equation to find \( x \), the width of the gravel path. I'll calculate this step by step.The solutions for the width of the gravel path are \( x = 5 \) meters and \( x = 40 \) meters.

However, the value \( x = 40 \) meters would make the flower bed's dimensions negative, which isn't possible. Therefore, the valid solution is:

\[
x = 5 \, \text{meters}
\]

Thus, the width of the gravel path is 5 meters.

Would you like more details, or do you have any other questions?

Here are 5 related questions for further practice:
1. How would the total cost change if the cost of laying the flower bed increased to ₹40 per square meter?
2. What would the width of the gravel path be if the field's dimensions were 60 m by 45 m with the same costs?
3. How can you calculate the area of the gravel path directly if the flower bed dimensions are known?
4. If the gravel path were only on two opposite sides of the flower bed, how would the total cost be affected?
5. What if the cost of the gravel was ₹25 per square meter? How would that change the width of the path?

**Tip:** Always verify whether a solution is physically realistic by considering the context, especially when solving quadratic equations!

The dimensions of a rectangular field are 50 m by 40 m. A flower bed is prepared inside this field, leaving a gravel path of uniform width all around the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 30 and ₹ 20 per square metre, respectively, is ₹ 52,000. Find the width of the gravel path.

This problem involves calculating the width of a gravel path surrounding a flower bed in a rectangular field. Using quadratic equations and area calculations, the solution demonstrates how to find the width based on given costs and dimensions.

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