Aman travelled `3/5` of his journey by rail `1/4` by a taxi, `1//8` by a bus and the remaining 2 km on Foot. What is the length of his total journey?

Answer

To find the total length of Aman's journey, we can denote the total distance of the journey as \( x \) kilometers. According to the problem, Aman travels different fractions of his journey by different modes of transport, and we know the remaining distance he traveled on foot.

1. **Identify the fractions of the journey:**
- By rail: \( \frac{3}{5}x \)
- By taxi: \( \frac{1}{4}x \)
- By bus: \( \frac{1}{8}x \)
- On foot: \( 2 \) km

2. **Set up the equation:**
The total distance traveled by Aman can be expressed as the sum of the distances traveled by each mode of transport plus the distance traveled on foot. Therefore, we can write:
\[ \frac{3}{5}x + \frac{1}{4}x + \frac{1}{8}x + 2 = x \]

3. **Combine the fractions:**
To combine the fractions, we need a common denominator. The least common multiple (LCM) of \( 5, 4, \) and \( 8 \) is \( 40 \).
- Convert each fraction:
- \( \frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40} \)
- \( \frac{1}{4} = \frac{1 \times 10}{4 \times 10} = \frac{10}{40} \)
- \( \frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40} \)

Now, substitute these values back into the equation:
\[ \frac{24}{40}x + \frac{10}{40}x + \frac{5}{40}x + 2 = x \]

4. **Combine the fractions on the left side:**
\[ \left(\frac{24 + 10 + 5}{40}\right)x + 2 = x \]
\[ \frac{39}{40}x + 2 = x \]

5. **Isolate \( x \):**
To isolate \( x \), subtract \( \frac{39}{40}x \) from both sides:
\[ 2 = x - \frac{39}{40}x \]
\[ 2 = \left(1 - \frac{39}{40}\right)x \]
\[ 2 = \frac{1}{40}x \]

6. **Solve for \( x \):**
Multiply both sides by \( 40 \):
\[ x = 2 \times 40 \]
\[ x = 80 \]

7. **Conclusion:**
The total length of Aman's journey is \( 80 \) km.