A person started his journey in the morning. At 11 a.m. he covered `(3)/(8)` of the journey and on the same day at 4.30p.m. he covered `(5)/(6)` of the journey. He started his journey at

Options

A. 6.00 a.m
B. 3.30 a.m
C. 7.00 a.m
D. 6.30 a.m

Answer

To solve the problem step by step, we need to find out the time the person started his journey based on the information given about the distances covered by certain times.

### Step 1: Understand the distances covered
The person covered \( \frac{3}{8} \) of the journey by 11 a.m. and \( \frac{5}{6} \) of the journey by 4:30 p.m.

### Step 2: Determine the total journey
Let the total distance of the journey be \( D \).

### Step 3: Calculate the distance covered by 11 a.m. and 4:30 p.m.
- By 11 a.m., the distance covered is:
\[ \text{Distance at 11 a.m.} = \frac{3}{8}D \]
- By 4:30 p.m., the distance covered is:
\[ \text{Distance at 4:30 p.m.} = \frac{5}{6}D \]

### Step 4: Calculate the distance covered between 11 a.m. and 4:30 p.m.
To find the distance covered between these two times, we subtract the distance at 11 a.m. from the distance at 4:30 p.m.:
\[ \text{Distance covered between 11 a.m. and 4:30 p.m.} = \frac{5}{6}D - \frac{3}{8}D \]

### Step 5: Find a common denominator to simplify
The least common multiple of 6 and 8 is 24. We convert the fractions:
\[ \frac{5}{6}D = \frac{20}{24}D \quad \text{and} \quad \frac{3}{8}D = \frac{9}{24}D \]
Now, substituting back:
\[ \text{Distance covered} = \frac{20}{24}D - \frac{9}{24}D = \frac{11}{24}D \]

### Step 6: Calculate the time taken between 11 a.m. and 4:30 p.m.
The time between 11 a.m. and 4:30 p.m. is 5.5 hours (from 11 a.m. to 4:30 p.m. is 5 hours and 30 minutes).

### Step 7: Calculate the speed
Using the distance covered and the time taken, we can find the speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{11}{24}D}{5.5} = \frac{11D}{24 \times 5.5} \]
Converting 5.5 hours to a fraction:
\[ 5.5 = \frac{11}{2} \implies \text{Speed} = \frac{11D}{24 \times \frac{11}{2}} = \frac{D}{\frac{24}{2}} = \frac{D}{12} \]

### Step 8: Calculate the total time for the journey
If the speed is \( \frac{D}{12} \), then the total time for the journey is:
\[ \text{Total Time} = \frac{D}{\frac{D}{12}} = 12 \text{ hours} \]

### Step 9: Determine the starting time
If the person finished the journey in 12 hours and reached the final distance at 4:30 p.m., we can find the starting time:
\[ \text{Ending time} - \text{Total time} = 4:30 \text{ p.m.} - 12 \text{ hours} = 4:30 \text{ a.m.} \]

Thus, the person started his journey at **4:30 a.m.**.