A man walks half of the journey at 4 km/h by cycle does one third of journey at 12 km/h and rides the remainder journey in a horse cart at 9 km/h, thus completing the whole journey in 6 hours and 12 minutes. The length of the journey is

Options

A. 36 km
B. `(1332)/(67)km`
C. 40 km
D. 28 km

Answer

To solve the problem step by step, we will break down the journey into parts based on the given speeds and the total time taken.

### Step 1: Define the total distance
Let the total distance of the journey be \( D \) kilometers.

### Step 2: Break down the journey
- The man walks half of the journey at 4 km/h:
\[ \text{Distance}_1 = \frac{D}{2} \]
- He cycles one third of the journey at 12 km/h:
\[ \text{Distance}_2 = \frac{D}{3} \]
- He rides the remainder of the journey in a horse cart at 9 km/h. The remainder can be calculated as:
\[ \text{Distance}_3 = D - \left(\frac{D}{2} + \frac{D}{3}\right) \]
To simplify this, we first find a common denominator for \(\frac{D}{2}\) and \(\frac{D}{3}\):
\[ \frac{D}{2} + \frac{D}{3} = \frac{3D}{6} + \frac{2D}{6} = \frac{5D}{6} \]
Thus,
\[ \text{Distance}_3 = D - \frac{5D}{6} = \frac{D}{6} \]

### Step 3: Calculate the time taken for each part of the journey
- Time taken for the first part (walking):
\[ T_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{\frac{D}{2}}{4} = \frac{D}{8} \text{ hours} \]
- Time taken for the second part (cycling):
\[ T_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{\frac{D}{3}}{12} = \frac{D}{36} \text{ hours} \]
- Time taken for the third part (horse cart):
\[ T_3 = \frac{\text{Distance}_3}{\text{Speed}_3} = \frac{\frac{D}{6}}{9} = \frac{D}{54} \text{ hours} \]

### Step 4: Total time taken for the journey
The total time for the journey is given as 6 hours and 12 minutes. We convert this into hours:
\[ 6 \text{ hours } 12 \text{ minutes} = 6 + \frac{12}{60} = 6.2 \text{ hours} = \frac{31}{5} \text{ hours} \]

### Step 5: Set up the equation
Now, we can set up the equation based on the total time:
\[ T_1 + T_2 + T_3 = \frac{D}{8} + \frac{D}{36} + \frac{D}{54} = \frac{31}{5} \]

### Step 6: Find a common denominator
The least common multiple of 8, 36, and 54 is 216. We will convert each term:
- \(\frac{D}{8} = \frac{27D}{216}\)
- \(\frac{D}{36} = \frac{6D}{216}\)
- \(\frac{D}{54} = \frac{4D}{216}\)

Combining these gives:
\[ \frac{27D + 6D + 4D}{216} = \frac{37D}{216} \]

### Step 7: Solve for D
Setting the equation:
\[ \frac{37D}{216} = \frac{31}{5} \]
Cross-multiplying gives:
\[ 37D \cdot 5 = 31 \cdot 216 \]
\[ 185D = 6696 \]
\[ D = \frac{6696}{185} \approx 36.2 \text{ kilometers} \]

### Step 8: Conclusion
Since we need to find the length of the journey, we round it to the nearest option available. The closest option is:
\[ \text{Length of the journey} = 36 \text{ kilometers} \]