Question

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 meet each other from the bottom?

• A. \(8^{\text{TH}}\)
• B. \(10^{\text{TH}}\)
• C. \(9^{\text{TH}}\)
• D. \(11^{\text{TH}}\)
• E. \(15^{\text{TH}}\)

Hint:

For "meeting" problems like this, the faster person covers more distance. Since Ram is twice as fast, he covers $2/3$ of the distance and Shyam covers $1/3$. $23 \div 3 \approx 7.66$. Since Shyam is moving up, he will be on the 8th step when they encounter each other.

Answer 1

The Correct Option is A

Concept: This is a relative speed problem. We can determine the meeting point by calculating how many steps each person covers per unit of time and finding where their total distance covered equals the total number of steps.
Step 1: Define the speeds and initial positions.
• Total steps = 23.
• Ram starts from the top (
Step 23) and descends.
• Shyam starts from the bottom (
Step 0) and ascends.
• Ratio of speed (Ram:Shyam) = $2:1$.

Step 2: Set up the meeting equation.
Let the number of steps covered by Shyam be $x$. Since Ram is twice as fast, the number of steps covered by Ram in the same time will be $2x$. When they meet, the total steps between them (23) must be covered. However, since they meet on a step, we account for the fact that they start moving simultaneously. \[ x + 2x = 23 \] \[ 3x = 23 \] \[ x = \frac{23}{3} = 7.66 \]

Step 3: Identify the meeting step from the bottom.
Shyam has covered 7.66 steps from the bottom. This means:
• After 7 intervals: Shyam is at step 7, Ram has descended 14 steps (is at $23 - 14 = 9$). They have not met.
• During the 8th interval: Shyam moves toward
step 8. Ram moves from step 9 toward
step 7.
• They will cross each other at the \(8^{\text{TH}}\) step.