A man travels `(3)/(4) th` of the distance of his journey by bus, `(1)/(6)th` by rickshow and 2km on foot. The total distance travelled by the man is

Options

A. 12km
B. 18km
C. 20km
D. 24km

Answer

To solve the problem step by step, we need to find the total distance traveled by the man based on the fractions of the journey he completed by different modes of transport.

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

### Step 2: Express the distances traveled by each mode of transport
According to the question:
- The distance traveled by bus = \( \frac{3}{4} D \)
- The distance traveled by rickshaw = \( \frac{1}{6} D \)
- The distance traveled on foot = 2 km

### Step 3: Set up the equation
The total distance traveled can be expressed as the sum of the distances traveled by bus, rickshaw, and on foot:
\[ \frac{3}{4} D + \frac{1}{6} D + 2 = D \]

### Step 4: Find a common denominator
To combine the fractions, we need a common denominator. The least common multiple of 4 and 6 is 12. Therefore, we can rewrite the fractions:
\[ \frac{3}{4} D = \frac{9}{12} D \]
\[ \frac{1}{6} D = \frac{2}{12} D \]

### Step 5: Substitute back into the equation
Now we substitute these back into the equation:
\[ \frac{9}{12} D + \frac{2}{12} D + 2 = D \]

### Step 6: Combine the fractions
Combine the fractions on the left side:
\[ \frac{11}{12} D + 2 = D \]

### Step 7: Isolate \( D \)
To isolate \( D \), subtract \( \frac{11}{12} D \) from both sides:
\[ 2 = D - \frac{11}{12} D \]
This simplifies to:
\[ 2 = \frac{1}{12} D \]

### Step 8: Solve for \( D \)
Now, multiply both sides by 12 to solve for \( D \):
\[ D = 2 \times 12 = 24 \text{ km} \]

### Conclusion
The total distance traveled by the man is \( 24 \) km.

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