Vishal travelled equal distances at speeds of 10 km/hr., 30 km/In., and 8 km/hr., and took a total of 15.5 minutes to complete. Find the total distance he travelled, in km.

Options

A. 1
B. 3
C. 4
D. 2

Answer

To solve the problem, we need to find the total distance Vishal traveled given that he traveled equal distances at three different speeds and took a total of 15.5 minutes. Let's break it down step by step.

### Step 1: Define the Variables
Let the distance traveled at each speed be \( x \) km.

### Step 2: Calculate Time for Each Segment
Using the formula for time, which is \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can express the time taken for each segment of the journey:
- For the speed of 10 km/hr:
\[ t_1 = \frac{x}{10} \text{ hours} \]
- For the speed of 30 km/hr:
\[ t_2 = \frac{x}{30} \text{ hours} \]
- For the speed of 8 km/hr:
\[ t_3 = \frac{x}{8} \text{ hours} \]

### Step 3: Total Time Equation
The total time taken for the journey is given as 15.5 minutes. We need to convert this into hours:
\[ 15.5 \text{ minutes} = \frac{15.5}{60} \text{ hours} = \frac{31}{120} \text{ hours} \]
Thus, we can set up the equation:
\[ t_1 + t_2 + t_3 = \frac{31}{120} \]
Substituting the expressions for \( t_1, t_2, \) and \( t_3 \):
\[ \frac{x}{10} + \frac{x}{30} + \frac{x}{8} = \frac{31}{120} \]

### Step 4: Finding a Common Denominator
To solve the equation, we need a common denominator for the left side. The least common multiple (LCM) of 10, 30, and 8 is 120. We can rewrite each term:
\[ \frac{x}{10} = \frac{12x}{120}, \quad \frac{x}{30} = \frac{4x}{120}, \quad \frac{x}{8} = \frac{15x}{120} \]
Now, substituting these back into the equation gives:
\[ \frac{12x + 4x + 15x}{120} = \frac{31}{120} \]
This simplifies to:
\[ \frac{31x}{120} = \frac{31}{120} \]

### Step 5: Solve for \( x \)
Since the denominators are the same, we can equate the numerators:
\[ 31x = 31 \]
Dividing both sides by 31 gives:
\[ x = 1 \text{ km} \]

### Step 6: Calculate Total Distance
Since Vishal traveled equal distances at each speed and there are three segments:
\[ \text{Total Distance} = 3x = 3 \times 1 = 3 \text{ km} \]

### Final Answer
The total distance Vishal traveled is **3 km**.

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