Two Boys begin together to write out a booklet containing 817 lines.The first boy starts with first line, writing at the rate of 200 lines an hour and the second boy start with the last line. He writes line 817 and so on backward proeccding at the rate of 150 lines an hour. At what line will they meet?

Options

A. 466th
B. 465th
C. 467th
D. 468th

Answer

To solve the problem, we need to find the line number at which both boys meet while writing the booklet. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- The first boy starts writing from line 1 and writes at a speed of 200 lines per hour.
- The second boy starts writing from line 817 and writes backward at a speed of 150 lines per hour.
- We need to find the line number where they meet.

2. **Setting Up the Equation:**
- Let \( n \) be the line number where they meet.
- The first boy will write \( n \) lines, and the time taken by him can be expressed as:
\[ \text{Time taken by first boy} = \frac{n}{200} \text{ hours} \]
- The second boy will write \( 817 - n \) lines, and the time taken by him can be expressed as:
\[ \text{Time taken by second boy} = \frac{817 - n}{150} \text{ hours} \]

3. **Setting the Times Equal:**
- Since both boys start at the same time and meet at the same moment, we can set their times equal to each other:
\[ \frac{n}{200} = \frac{817 - n}{150} \]

4. **Cross-Multiplying to Solve for \( n \):**
- Cross-multiply to eliminate the fractions:
\[ 150n = 200(817 - n) \]
- Expanding the right side:
\[ 150n = 163400 - 200n \]

5. **Combining Like Terms:**
- Add \( 200n \) to both sides:
\[ 150n + 200n = 163400 \]
\[ 350n = 163400 \]

6. **Solving for \( n \):**
- Divide both sides by 350:
\[ n = \frac{163400}{350} \]
- Simplifying the fraction:
\[ n = 466.8571 \]

7. **Rounding to the Nearest Whole Number:**
- Since \( n \) must be a whole number (as it represents a line), we round \( n \) to the nearest whole number:
\[ n \approx 467 \]

### Conclusion:
The boys will meet at line number **467**.