A man covers `6(1)/(4)%` distance via bus at 80 km/hr, 25% of the distance via car at 120 km/hr., 30% distance via bicycle at 32 km/hr. and remaining distance via train at 62 km/hr. If total distance covered by man is 640km, then find the total time taken man during the entire journey.

Options

A. A)`(65)/(6)` hours
B. B)13 hours
C. C)`(44)/(3)` hours
D. D)`(71)/(6)` hours

Answer

To solve the problem step by step, we need to calculate the distance covered by the man using different modes of transportation and then find the time taken for each segment of the journey.

### Step 1: Calculate the total distance covered by each mode of transport.

Given the total distance is 640 km:

1. **Distance via Bus**:
- The man covers \(6\frac{1}{4}\%\) of the distance via bus.
- Convert \(6\frac{1}{4}\%\) to a fraction:
\[ 6\frac{1}{4}\% = \frac{25}{4}\% = \frac{25}{4 \times 100} = \frac{1}{16} \]
- Distance via bus:
\[ \text{Distance}_{\text{bus}} = 640 \times \frac{1}{16} = 40 \text{ km} \]

2. **Distance via Car**:
- The man covers \(25\%\) of the distance via car.
- Distance via car:
\[ \text{Distance}_{\text{car}} = 640 \times \frac{25}{100} = 160 \text{ km} \]

3. **Distance via Bicycle**:
- The man covers \(30\%\) of the distance via bicycle.
- Distance via bicycle:
\[ \text{Distance}_{\text{bicycle}} = 640 \times \frac{30}{100} = 192 \text{ km} \]

4. **Remaining Distance via Train**:
- Total distance covered so far:
\[ 40 + 160 + 192 = 392 \text{ km} \]
- Remaining distance:
\[ \text{Distance}_{\text{train}} = 640 - 392 = 248 \text{ km} \]

### Step 2: Calculate the time taken for each segment of the journey.

1. **Time via Bus**:
- Speed of bus = 80 km/hr
- Time taken via bus:
\[ \text{Time}_{\text{bus}} = \frac{\text{Distance}_{\text{bus}}}{\text{Speed}_{\text{bus}}} = \frac{40}{80} = \frac{1}{2} \text{ hours} \]

2. **Time via Car**:
- Speed of car = 120 km/hr
- Time taken via car:
\[ \text{Time}_{\text{car}} = \frac{\text{Distance}_{\text{car}}}{\text{Speed}_{\text{car}}} = \frac{160}{120} = \frac{4}{3} \text{ hours} \]

3. **Time via Bicycle**:
- Speed of bicycle = 32 km/hr
- Time taken via bicycle:
\[ \text{Time}_{\text{bicycle}} = \frac{\text{Distance}_{\text{bicycle}}}{\text{Speed}_{\text{bicycle}}} = \frac{192}{32} = 6 \text{ hours} \]

4. **Time via Train**:
- Speed of train = 62 km/hr
- Time taken via train:
\[ \text{Time}_{\text{train}} = \frac{\text{Distance}_{\text{train}}}{\text{Speed}_{\text{train}}} = \frac{248}{62} = 4 \text{ hours} \]

### Step 3: Calculate the total time taken for the entire journey.

Now, we sum up all the times:
\[ \text{Total Time} = \text{Time}_{\text{bus}} + \text{Time}_{\text{car}} + \text{Time}_{\text{bicycle}} + \text{Time}_{\text{train}} \]
\[ \text{Total Time} = \frac{1}{2} + \frac{4}{3} + 6 + 4 \]

To add these fractions, we need a common denominator, which is 6:
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{4}{3} = \frac{8}{6}, \quad 6 = \frac{36}{6}, \quad 4 = \frac{24}{6} \]
Now, adding them together:
\[ \text{Total Time} = \frac{3 + 8 + 36 + 24}{6} = \frac{71}{6} \text{ hours} \]

### Final Answer:
The total time taken by the man during the entire journey is \(\frac{71}{6}\) hours.

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