If the speed of a train is increased by `30%` it takes 36 minutes less to cover the same distance . What is the time taken (in hours) to cover the same distance with the original speed ?

Options

A. `3.2`
B. `2.4`
C. `3.5`
D. `2.6`

Answer

To solve the problem step-by-step, we can follow these instructions:

### Step 1: Understand the relationship between speed, time, and distance.
The formula for distance is given by:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Since the distance is the same in both cases (original speed and increased speed), we can set up a relationship between the times taken at the two speeds.

### Step 2: Define the variables.
Let:
- \( S \) = original speed of the train
- \( T \) = time taken to cover the distance with the original speed (in hours)
- The new speed after a 30% increase is:
\[ S' = S + 0.3S = 1.3S \]

### Step 3: Set up the equations for time.
The time taken to cover the same distance with the new speed is:
\[ T' = \frac{\text{Distance}}{S'} = \frac{D}{1.3S} \]
Using the original speed, the time taken is:
\[ T = \frac{D}{S} \]

### Step 4: Relate the times.
According to the problem, the new time \( T' \) is 36 minutes less than the original time \( T \). We need to convert 36 minutes into hours:
\[ 36 \text{ minutes} = \frac{36}{60} \text{ hours} = 0.6 \text{ hours} \]
Thus, we have:
\[ T - T' = 0.6 \]

### Step 5: Substitute the expressions for \( T \) and \( T' \).
Substituting the expressions for \( T \) and \( T' \):
\[ T - \frac{D}{1.3S} = 0.6 \]
Now, substituting \( D = S \times T \) into the equation:
\[ T - \frac{S \times T}{1.3S} = 0.6 \]
This simplifies to:
\[ T - \frac{T}{1.3} = 0.6 \]

### Step 6: Solve for \( T \).
To solve for \( T \), we can find a common denominator:
\[ T \left(1 - \frac{1}{1.3}\right) = 0.6 \]
Calculating \( 1 - \frac{1}{1.3} \):
\[ 1 - \frac{1}{1.3} = \frac{1.3 - 1}{1.3} = \frac{0.3}{1.3} \]
So we have:
\[ T \cdot \frac{0.3}{1.3} = 0.6 \]
Now, solving for \( T \):
\[ T = 0.6 \cdot \frac{1.3}{0.3} \]
\[ T = 0.6 \cdot \frac{13}{3} \]
\[ T = \frac{7.8}{3} = 2.6 \text{ hours} \]

### Final Answer:
The time taken to cover the same distance with the original speed is:
\[ \boxed{2.6} \text{ hours} \]