Ram travels from P to Q at 10 km/hr and returns at 15 km/hr. Shyam travels from P to Q and returns at 12.5 km/hr. If he takes 12 minutes less than Ram, then what is the distance between P and Q?

Options

A. 60 km
B. 45 km
C. 36 km
D. 30 km

Answer

To solve the problem step by step, we will first establish the variables and equations based on the information provided.

### Step 1: Define the Variables
Let the distance between points P and Q be \( x \) km.

### Step 2: Calculate Ram's Time
Ram travels from P to Q at a speed of 10 km/hr and returns at a speed of 15 km/hr.
- Time taken to travel from P to Q:
\[ \text{Time}_{PQ} = \frac{x}{10} \text{ hours} \]
- Time taken to return from Q to P:
\[ \text{Time}_{QP} = \frac{x}{15} \text{ hours} \]
- Total time taken by Ram:
\[ \text{Time}_{\text{Ram}} = \frac{x}{10} + \frac{x}{15} \]

### Step 3: Find a Common Denominator for Ram's Time
To add the two fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30.
- Convert the fractions:
\[ \text{Time}_{\text{Ram}} = \frac{3x}{30} + \frac{2x}{30} = \frac{5x}{30} = \frac{x}{6} \text{ hours} \]

### Step 4: Calculate Shyam's Time
Shyam travels the same distance at a speed of 12.5 km/hr for both directions.
- Time taken by Shyam:
\[ \text{Time}_{\text{Shyam}} = \frac{x}{12.5} + \frac{x}{12.5} = \frac{2x}{12.5} = \frac{2x}{12.5} = \frac{16x}{100} = \frac{4x}{25} \text{ hours} \]

### Step 5: Set Up the Equation Based on the Time Difference
According to the problem, Shyam takes 12 minutes less than Ram. We need to convert 12 minutes into hours:
\[ 12 \text{ minutes} = \frac{12}{60} = \frac{1}{5} \text{ hours} \]
Thus, we can set up the equation:
\[ \text{Time}_{\text{Ram}} - \text{Time}_{\text{Shyam}} = \frac{1}{5} \]
Substituting the expressions we found:
\[ \frac{x}{6} - \frac{4x}{25} = \frac{1}{5} \]

### Step 6: Solve the Equation
To solve the equation, we need a common denominator for the fractions on the left side. The LCM of 6 and 25 is 150.
- Convert the fractions:
\[ \frac{25x}{150} - \frac{24x}{150} = \frac{1}{5} \]
This simplifies to:
\[ \frac{25x - 24x}{150} = \frac{1}{5} \]
\[ \frac{x}{150} = \frac{1}{5} \]

### Step 7: Cross Multiply to Solve for x
Cross multiplying gives us:
\[ x = 150 \times \frac{1}{5} = 30 \]

### Conclusion
The distance between P and Q is \( x = 30 \) km.