If Abhi travels a certain distance at 6 km/h. he reaches his destination 12 minutes early, but if he travels at 4 km/h, he reaches his destination 10 minutes late. The speed (in km/h) at which he should travel to reach his destination on time is ........

Options

A. `5^(4/4)`
B. `5^(1/8)`
C. `4^(3/7)`
D. `4^(5/7)`

Answer

To solve the problem step by step, we will use the information given about Abhi's travel at different speeds and the time differences.

### Step 1: Understand the problem
Abhi travels a certain distance at two different speeds: 6 km/h and 4 km/h. He arrives 12 minutes early when traveling at 6 km/h and 10 minutes late when traveling at 4 km/h. We need to find the speed at which he should travel to reach his destination on time.

### Step 2: Convert time differences into hours
- 12 minutes early is equivalent to \( \frac{12}{60} = \frac{1}{5} \) hours.
- 10 minutes late is equivalent to \( \frac{10}{60} = \frac{1}{6} \) hours.

### Step 3: Set up equations for distance
Let the distance to the destination be \( D \) km.

Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

1. When traveling at 6 km/h:
\[ \text{Time taken} = \frac{D}{6} \quad \text{(arrives 12 minutes early)} \]
Hence, the actual time to reach on time is:
\[ \frac{D}{6} + \frac{1}{5} \]

2. When traveling at 4 km/h:
\[ \text{Time taken} = \frac{D}{4} \quad \text{(arrives 10 minutes late)} \]
Hence, the actual time to reach on time is:
\[ \frac{D}{4} - \frac{1}{6} \]

### Step 4: Set the two expressions for actual time equal
Since both expressions represent the same actual time to reach the destination on time, we can set them equal to each other:
\[ \frac{D}{6} + \frac{1}{5} = \frac{D}{4} - \frac{1}{6} \]

### Step 5: Clear the fractions
To eliminate the fractions, find a common denominator, which is 60:
\[ 10D + 12 = 15D - 10 \]

### Step 6: Solve for \( D \)
Rearranging the equation:
\[ 10D + 12 + 10 = 15D \]
\[ 22 = 15D - 10D \]
\[ 22 = 5D \]
\[ D = \frac{22}{5} = 4.4 \text{ km} \]

### Step 7: Find the actual time to reach on time
Substituting \( D \) back into either time equation to find the actual time:
Using \( \frac{D}{6} + \frac{1}{5} \):
\[ \text{Time} = \frac{4.4}{6} + \frac{1}{5} = \frac{22}{30} + \frac{6}{30} = \frac{28}{30} = \frac{14}{15} \text{ hours} \]

### Step 8: Calculate the required speed to reach on time
To find the speed \( S \) at which he should travel to reach on time:
\[ S = \frac{D}{\text{Time}} = \frac{4.4}{\frac{14}{15}} = 4.4 \times \frac{15}{14} = \frac{66}{14} = \frac{33}{7} \approx 4.71 \text{ km/h} \]

### Final Answer
The speed at which Abhi should travel to reach his destination on time is approximately \( 4.71 \) km/h.