Along a road lies an odd number of stones placed at intervals of 10m. These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried out the job starting with the stone in the middle, carrying stones in succession, thereby covering a distance of 4.8 km. Then the number of stones is

Options

A. 15
B. 29
C. 31
D. 35

Answer

To solve the problem, we need to determine the number of stones placed along the road based on the distance covered by a person carrying them. Let's break down the solution step by step.

### Step 1: Define the Variables
Let the total number of stones be \( x \). Since the number of stones is odd, we can express the middle stone as the \( \frac{x + 1}{2} \)th stone.

### Step 2: Determine the Number of Stones on Each Side
Since there are \( x \) stones, there will be \( \frac{x - 1}{2} \) stones on each side of the middle stone. Thus, the left side has \( \frac{x - 1}{2} \) stones and the right side also has \( \frac{x - 1}{2} \) stones.

### Step 3: Calculate the Distance for Each Stone
Each stone is placed at intervals of 10 meters. The distance to the stones from the middle stone is as follows:
- The first stone on either side is at a distance of 10 meters.
- The second stone is at a distance of 20 meters.
- The \( n \)th stone is at a distance of \( 10n \) meters.

For the left side, the distances to the stones are:
- 10 meters (1st stone) + 10 meters (return) = 20 meters
- 20 meters (2nd stone) + 20 meters (return) = 40 meters
- ...
- \( 10 \cdot n \) meters (nth stone) + \( 10 \cdot n \) meters (return) = \( 20n \) meters

### Step 4: Total Distance for All Stones
The total distance covered for the stones on one side (left or right) can be calculated as:
\[ \text{Total distance for one side} = 20 \times \left(1 + 2 + 3 + \ldots + \frac{x - 1}{2}\right) \]
Using the formula for the sum of the first \( n \) natural numbers:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, the total distance for one side becomes:
\[ 20 \times \frac{\frac{x - 1}{2} \left(\frac{x - 1}{2} + 1\right)}{2} = 5 \times (x - 1) \times \left(\frac{x + 1}{2}\right) \]

### Step 5: Total Distance for Both Sides
Since the person carries stones from both sides, the total distance covered is:
\[ \text{Total distance} = 2 \times 5 \times (x - 1) \times \left(\frac{x + 1}{2}\right) = 5(x - 1)(x + 1) \]

### Step 6: Set Up the Equation
The total distance covered by the person is given as 4.8 km, which is 4800 meters:
\[ 5(x - 1)(x + 1) = 4800 \]

### Step 7: Simplify the Equation
Dividing both sides by 5:
\[ (x - 1)(x + 1) = 960 \]
This simplifies to:
\[ x^2 - 1 = 960 \]
Thus:
\[ x^2 = 961 \]

### Step 8: Solve for \( x \)
Taking the square root of both sides:
\[ x = 31 \quad (\text{since } x \text{ must be positive}) \]

### Conclusion
The total number of stones is \( \boxed{31} \).

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