A man travelled a distance of 47 km in 6 hours. He travelled partly on foot at the rate of `6(1)/2` km/h and partly on bicycle at the rate of `8(1)/2` km/h. The distance travelled on foot is:

Options

A. 13 km
B. 15 km
C. 12 km
D. 16 km

Answer

To solve the problem step by step, we will use the information given about the distances traveled on foot and by bicycle, along with their respective speeds.

### Step 1: Define Variables
Let:
- \( x \) = time traveled on foot (in hours)
- \( 6 - x \) = time traveled on bicycle (in hours)

### Step 2: Write Down the Speeds
The speeds are given as:
- Speed on foot = \( 6 \frac{1}{2} \) km/h = \( \frac{13}{2} \) km/h
- Speed on bicycle = \( 8 \frac{1}{2} \) km/h = \( \frac{17}{2} \) km/h

### Step 3: Write the Distance Equations
Using the formula for distance \( \text{Distance} = \text{Speed} \times \text{Time} \):
- Distance traveled on foot = \( \text{Speed on foot} \times \text{Time on foot} = \frac{13}{2} \times x \)
- Distance traveled on bicycle = \( \text{Speed on bicycle} \times \text{Time on bicycle} = \frac{17}{2} \times (6 - x) \)

### Step 4: Set Up the Total Distance Equation
The total distance traveled is 47 km, so we can set up the equation:
\[ \frac{13}{2} x + \frac{17}{2} (6 - x) = 47 \]

### Step 5: Simplify the Equation
Multiply through by 2 to eliminate the fractions:
\[ 13x + 17(6 - x) = 94 \]
Expanding this gives:
\[ 13x + 102 - 17x = 94 \]

### Step 6: Combine Like Terms
Combine the \( x \) terms:
\[ -4x + 102 = 94 \]

### Step 7: Solve for \( x \)
Subtract 102 from both sides:
\[ -4x = 94 - 102 \]
\[ -4x = -8 \]
Dividing both sides by -4 gives:
\[ x = 2 \]

### Step 8: Calculate the Distance Traveled on Foot
Now that we have \( x \), we can find the distance traveled on foot:
\[ \text{Distance on foot} = \text{Speed on foot} \times \text{Time on foot} = \frac{13}{2} \times 2 = 13 \text{ km} \]

### Final Answer
The distance traveled on foot is **13 km**.

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