A man misses a train by 1 h if he travels at a speed of 4 km/h . If he increases his speed to 5 km/h, he still misses the train by 24 min . At what speed should he travel so that he reaches the station exactly on time ?

Options

A. 12 kmph
B. 8 kmph
C. 6 kmph
D. 10 kmph

Answer

To solve the problem, we need to find the speed at which the man should travel to reach the station exactly on time. Let's break down the information given and derive the solution step by step.

### Step 1: Define Variables
Let:
- \( d \) = distance to the station (in km)
- \( t \) = time taken to reach the station on time (in hours)

### Step 2: Set Up Equations Based on Given Information
1. **At 4 km/h**: The man misses the train by 1 hour.
- Time taken at 4 km/h = \( \frac{d}{4} \)
- Therefore, the equation is:
\[ \frac{d}{4} = t + 1 \]

2. **At 5 km/h**: The man misses the train by 24 minutes (which is \( \frac{24}{60} = 0.4 \) hours).
- Time taken at 5 km/h = \( \frac{d}{5} \)
- Therefore, the equation is:
\[ \frac{d}{5} = t + 0.4 \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( \frac{d}{4} = t + 1 \) (Equation 1)
2. \( \frac{d}{5} = t + 0.4 \) (Equation 2)

We can express \( t \) from both equations:
- From Equation 1:
\[ t = \frac{d}{4} - 1 \]

- From Equation 2:
\[ t = \frac{d}{5} - 0.4 \]

### Step 4: Set the Two Expressions for \( t \) Equal
Now, we can set the two expressions for \( t \) equal to each other:
\[ \frac{d}{4} - 1 = \frac{d}{5} - 0.4 \]

### Step 5: Solve for \( d \)
To eliminate the fractions, we can multiply through by 20 (the least common multiple of 4 and 5):
\[ 20 \left(\frac{d}{4}\right) - 20(1) = 20 \left(\frac{d}{5}\right) - 20(0.4) \]
This simplifies to:
\[ 5d - 20 = 4d - 8 \]
Now, rearranging gives:
\[ 5d - 4d = 20 - 8 \]
\[ d = 12 \text{ km} \]

### Step 6: Substitute \( d \) Back to Find \( t \)
Now that we have \( d \), we can substitute it back into either equation to find \( t \). Using Equation 1:
\[ t = \frac{12}{4} - 1 = 3 - 1 = 2 \text{ hours} \]

### Step 7: Calculate the Required Speed
Now we can find the speed at which he should travel to reach the station on time:
\[ \text{Required Speed} = \frac{d}{t} = \frac{12}{2} = 6 \text{ km/h} \]

### Final Answer
The speed at which the man should travel to reach the station exactly on time is **6 km/h**.

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