There are 23 steps to reach a temple. On descending from the temple Ram takes two steps in the same time. Shyam ascends one step. If they start to work simultaneously, at which step will they pass each other?

Options

A. `8^(th)`
B. `9^(th)`
C. `10^(th)`
D. `11^(th)`

Answer

To solve the problem, we need to determine at which step Ram and Shyam will pass each other as they move in opposite directions on the steps leading to the temple.

### Step-by-Step Solution:

1. **Understanding the Movement**:
- Ram is descending from the temple and takes 2 steps down in the same time it takes Shyam to take 1 step up.
- There are a total of 23 steps.

2. **Initial Positions**:
- Ram starts at step 23 (the top of the temple).
- Shyam starts at step 0 (the ground).

3. **Movement Representation**:
- Let \( R \) be the number of steps Ram has descended.
- Let \( S \) be the number of steps Shyam has ascended.
- Since Ram descends 2 steps for every step Shyam ascends, we can express their positions as:
- Ram's position after \( t \) seconds: \( 23 - 2t \)
- Shyam's position after \( t \) seconds: \( 0 + t \)

4. **Setting Up the Equation**:
- They will pass each other when their positions are equal:
\[ 23 - 2t = t \]

5. **Solving the Equation**:
- Rearranging the equation:
\[ 23 = 3t \]
- Dividing both sides by 3:
\[ t = \frac{23}{3} \approx 7.67 \]

6. **Finding the Step Number**:
- To find the step at which they pass each other, we substitute \( t \) back into Shyam's position equation:
\[ S = t = \frac{23}{3} \approx 7.67 \]
- Since Shyam can only be on whole steps, we take the integer part which is 7. At this point, Ram will be:
\[ R = 23 - 2 \times 7 = 23 - 14 = 9 \]
- Therefore, they pass each other between step 7 and step 8.

### Conclusion:
They pass each other at step 8.