A person drops a ball from an 80 m tall building and each time the ball bounces, it rebounds to p% of its previous height. If the ball travels a total distance of 320 m, then the value of p is

Explanation

To solve the problem, we need to determine the value of \( p \) given that a ball is dropped from a height of 80 meters and travels a total distance of 320 meters after bouncing back to \( p\% \) of its previous height.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The ball is dropped from a height of 80 meters.
- After each bounce, it rebounds to \( p\% \) of the height from which it fell.

2. **Distance Calculation**:
- The initial drop distance is 80 meters.
- After the first bounce, the ball rises to \( \frac{p}{100} \times 80 \) meters.
- It then falls the same distance, so the distance for the first bounce (up and down) is \( 2 \times \frac{p}{100} \times 80 \).

3. **Total Distance**:
- The total distance traveled by the ball can be expressed as:
\[ \text{Total Distance} = 80 + 2 \left( \frac{p}{100} \times 80 \right) + 2 \left( \frac{p^2}{100^2} \times 80 \right) + 2 \left( \frac{p^3}{100^3} \times 80 \right) + \ldots \]
- This can be simplified to:
\[ \text{Total Distance} = 80 + 160 \left( \frac{p}{100} \right) + 160 \left( \frac{p^2}{100^2} \right) + 160 \left( \frac{p^3}{100^3} \right) + \ldots \]

4. **Recognizing the Series**:
- The series \( 160 \left( \frac{p}{100} \right) + 160 \left( \frac{p^2}{100^2} \right) + 160 \left( \frac{p^3}{100^3} \right) + \ldots \) is a geometric series with:
- First term \( a = 160 \left( \frac{p}{100} \right) \)
- Common ratio \( r = \frac{p}{100} \)

5. **Sum of the Geometric Series**:
- The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
- Therefore, the total distance becomes:
\[ \text{Total Distance} = 80 + \frac{160 \left( \frac{p}{100} \right)}{1 - \left( \frac{p}{100} \right)} \]

6. **Setting Up the Equation**:
- Given that the total distance is 320 meters, we can set up the equation:
\[ 320 = 80 + \frac{160 \left( \frac{p}{100} \right)}{1 - \left( \frac{p}{100} \right)} \]
- Simplifying this gives:
\[ 240 = \frac{160 \left( \frac{p}{100} \right)}{1 - \left( \frac{p}{100} \right)} \]

7. **Cross Multiplying**:
- Cross multiplying gives:
\[ 240 \left( 1 - \frac{p}{100} \right) = 160 \left( \frac{p}{100} \right) \]
- Expanding this results in:
\[ 240 - \frac{240p}{100} = \frac{160p}{100} \]

8. **Combining Like Terms**:
- Rearranging gives:
\[ 240 = \frac{240p}{100} + \frac{160p}{100} \]
- This simplifies to:
\[ 240 = \frac{400p}{100} \]

9. **Solving for \( p \)**:
- Multiplying both sides by 100 gives:
\[ 24000 = 400p \]
- Dividing by 400 results in:
\[ p = 60 \]

### Final Answer:
The value of \( p \) is \( 60\% \).