While covering a distance of 51 km, a man noticed that after walking for one hour and 40 minutes, the distance covered by him was `5/12` of the remaining distance. What was his speed?

Options

A. 7 km/h
B. 10 km/h
C. 8 km/h
D. 9 km/h

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
The total distance to be covered is 51 km. After walking for 1 hour and 40 minutes, the man covered a distance that is \( \frac{5}{12} \) of the remaining distance.

### Step 2: Convert Time to Hours
1 hour and 40 minutes can be converted into hours:
- 1 hour = 1 hour
- 40 minutes = \( \frac{40}{60} = \frac{2}{3} \) hours
- Total time = \( 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \) hours

### Step 3: Set Up the Equation
Let the distance covered by the man in 1 hour and 40 minutes be \( d \).
The remaining distance after covering \( d \) is \( 51 - d \).

According to the problem, we have:
\[ d = \frac{5}{12} \times (51 - d) \]

### Step 4: Solve for \( d \)
Multiply both sides by 12 to eliminate the fraction:
\[ 12d = 5(51 - d) \]
Expanding the right side:
\[ 12d = 255 - 5d \]
Now, combine like terms:
\[ 12d + 5d = 255 \]
\[ 17d = 255 \]
Now, divide both sides by 17:
\[ d = \frac{255}{17} = 15 \text{ km} \]

### Step 5: Find the Remaining Distance
Now, calculate the remaining distance:
\[ \text{Remaining distance} = 51 - d = 51 - 15 = 36 \text{ km} \]

### Step 6: Calculate Speed
Speed is defined as distance divided by time. We have:
- Distance covered = 15 km
- Time taken = \( \frac{5}{3} \) hours

Using the speed formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{15 \text{ km}}{\frac{5}{3} \text{ hours}} = 15 \times \frac{3}{5} = 9 \text{ km/h} \]

### Final Answer
The speed of the man is **9 km/h**.

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