A person walks `2/5` th of the total distance and covers one-third of the remaining distance by bus. What is the fraction of 2 the remaining distance to be covered?

Options

A. `3/5`
B. `2/5`
C. `7/15`
D. `1/5`

Answer

To solve the problem step by step, let's break it down:

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

### Step 2: Calculate the distance covered by walking
The person walks \( \frac{2}{5} \) of the total distance. Therefore, the distance covered by walking is:
\[ \text{Distance covered by walking} = \frac{2}{5} x \]

### Step 3: Calculate the remaining distance after walking
To find the remaining distance after walking, we subtract the distance covered by walking from the total distance:
\[ \text{Remaining distance} = x - \left(\frac{2}{5} x\right) \]
Calculating this gives:
\[ \text{Remaining distance} = x - \frac{2}{5} x = \frac{5}{5} x - \frac{2}{5} x = \frac{3}{5} x \]

### Step 4: Calculate the distance covered by bus
The person covers one-third of the remaining distance by bus. The remaining distance is \( \frac{3}{5} x \), so the distance covered by bus is:
\[ \text{Distance covered by bus} = \frac{1}{3} \left(\frac{3}{5} x\right) \]
Calculating this gives:
\[ \text{Distance covered by bus} = \frac{3}{15} x = \frac{1}{5} x \]

### Step 5: Calculate the remaining distance after bus travel
Now, we need to find the remaining distance after covering some distance by bus. We subtract the distance covered by bus from the remaining distance:
\[ \text{Remaining distance after bus} = \frac{3}{5} x - \left(\frac{1}{5} x\right) \]
Calculating this gives:
\[ \text{Remaining distance after bus} = \frac{3}{5} x - \frac{1}{5} x = \frac{2}{5} x \]

### Step 6: Find the fraction of the remaining distance to be covered
The remaining distance to be covered is \( \frac{2}{5} x \). To find the fraction of the remaining distance to be covered in relation to the total distance, we express it as:
\[ \text{Fraction of remaining distance} = \frac{\text{Remaining distance after bus}}{\text{Total distance}} = \frac{\frac{2}{5} x}{x} \]
This simplifies to:
\[ \text{Fraction of remaining distance} = \frac{2}{5} \]

### Final Answer
The fraction of the remaining distance to be covered is \( \frac{2}{5} \).

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