Shard covers two third of a certain distance at 4 km/hr and the remaining at 5km/hr if he takes 42 minutes in all.The distance is

Options

A. A)2.5km
B. B)4.6km
C. C)4km
D. D)3km

Answer

To solve the problem step by step, we will follow the given information and calculate the total distance covered by Sharad.

### Step 1: Define the total distance
Let the total distance be represented as \( D \). According to the problem, Sharad covers two-thirds of this distance at 4 km/hr and the remaining one-third at 5 km/hr.

### Step 2: Express the distances
The distance covered in two-thirds of the journey is:
\[ \text{Distance from A to C} = \frac{2}{3}D \]
The distance covered in the remaining one-third of the journey is:
\[ \text{Distance from C to B} = \frac{1}{3}D \]

### Step 3: Calculate the time taken for each segment
The time taken to cover the first part of the journey (from A to C) can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
For the first part:
\[ \text{Time from A to C} = \frac{\frac{2}{3}D}{4} = \frac{2D}{12} = \frac{D}{6} \text{ hours} \]

For the second part:
\[ \text{Time from C to B} = \frac{\frac{1}{3}D}{5} = \frac{1D}{15} \text{ hours} \]

### Step 4: Total time taken
The total time taken for the journey is given as 42 minutes. We need to convert this into hours:
\[ 42 \text{ minutes} = \frac{42}{60} = \frac{7}{10} \text{ hours} \]

### Step 5: Set up the equation
Now, we can set up the equation for the total time:
\[ \frac{D}{6} + \frac{D}{15} = \frac{7}{10} \]

### Step 6: Find a common denominator
The least common multiple (LCM) of 6 and 15 is 30. We will convert each term to have a denominator of 30:
\[ \frac{D}{6} = \frac{5D}{30}, \quad \frac{D}{15} = \frac{2D}{30} \]
Thus, the equation becomes:
\[ \frac{5D}{30} + \frac{2D}{30} = \frac{7}{10} \]
Combining the fractions:
\[ \frac{7D}{30} = \frac{7}{10} \]

### Step 7: Cross-multiply to solve for D
Cross-multiplying gives:
\[ 7D \cdot 10 = 7 \cdot 30 \]
\[ 70D = 210 \]
\[ D = \frac{210}{70} = 3 \]

### Step 8: Calculate the total distance
Since \( D \) represents one-third of the total distance, the total distance \( 3D \) is:
\[ 3D = 3 \times 3 = 9 \text{ km} \]

### Final Answer
The total distance covered by Sharad is **9 km**.

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