John gets on the elevator at the 14th floor of a building and rides up at the rate of 84 floors per minute.At the same time, Vinod gets an on elevator at the 58th floor of the same building and rides down at the rate of 92 floors per minute. If they continue travelling at these rates, then at which floor will their paths cross?

Options

A. 38
B. 36
C. 32
D. 35

Answer

To solve the problem, we need to determine the floor at which John and Vinod will meet as they travel in opposite directions in the building.

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( t \) be the time in minutes after they start moving.
- John's starting floor: 14th floor.
- Vinod's starting floor: 58th floor.

2. **Calculate John's Position:**
- John moves up at a rate of 84 floors per minute.
- After \( t \) minutes, John's position can be expressed as:
\[ \text{John's position} = 14 + 84t \]

3. **Calculate Vinod's Position:**
- Vinod moves down at a rate of 92 floors per minute.
- After \( t \) minutes, Vinod's position can be expressed as:
\[ \text{Vinod's position} = 58 - 92t \]

4. **Set Up the Equation:**
- To find the floor where their paths cross, we set John's position equal to Vinod's position:
\[ 14 + 84t = 58 - 92t \]

5. **Solve for \( t \):**
- Rearranging the equation gives:
\[ 84t + 92t = 58 - 14 \]
\[ 176t = 44 \]
\[ t = \frac{44}{176} = \frac{1}{4} \text{ minutes} \]

6. **Find the Crossing Floor:**
- Now substitute \( t = \frac{1}{4} \) back into John's position equation:
\[ \text{John's position} = 14 + 84 \left(\frac{1}{4}\right) = 14 + 21 = 35 \]

### Conclusion:
John and Vinod will cross paths at the **35th floor**.