Two places P and Q are 162 km apart. A train leaves P for Q and simultaneously another train leaves Q to P. They meet at the end of 6 hours. If the former train travels 8 km/hr faster than the other, then speed of train from Q is

Options

A. `12 (5)/(6)` km/hr
B. `10 (5)/(6)` km/hr
C. `9 (1)/(2)` km/hr
D. `8 (1)/(2)` km/hr

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the speed of the train leaving from P to Q be \( x \) km/hr. Therefore, the speed of the train leaving from Q to P will be \( x - 8 \) km/hr, since it travels 8 km/hr slower than the first train.

**Hint:** Define the speeds of both trains in terms of a single variable.

### Step 2: Calculate the Total Distance Covered
The total distance between P and Q is 162 km. When both trains meet after 6 hours, they would have collectively covered this distance.

**Hint:** Remember that distance = speed × time.

### Step 3: Set Up the Equation
The distance covered by the train from P to Q in 6 hours is:
\[ \text{Distance from P to Q} = \text{Speed} \times \text{Time} = x \times 6 \]
The distance covered by the train from Q to P in 6 hours is:
\[ \text{Distance from Q to P} = (x - 8) \times 6 \]

Since the total distance is 162 km, we can set up the equation:
\[ 6x + 6(x - 8) = 162 \]

**Hint:** Combine the distances covered by both trains into one equation.

### Step 4: Simplify the Equation
Now simplify the equation:
\[ 6x + 6x - 48 = 162 \]
Combine like terms:
\[ 12x - 48 = 162 \]

**Hint:** Combine all terms involving \( x \) on one side.

### Step 5: Solve for \( x \)
Add 48 to both sides:
\[ 12x = 162 + 48 \]
\[ 12x = 210 \]
Now divide by 12:
\[ x = \frac{210}{12} \]
\[ x = 17.5 \text{ km/hr} \]

**Hint:** Make sure to perform the arithmetic carefully when isolating \( x \).

### Step 6: Find the Speed of the Train from Q
Now that we have \( x \), we can find the speed of the train from Q:
\[ \text{Speed of train from Q} = x - 8 = 17.5 - 8 = 9.5 \text{ km/hr} \]

**Hint:** Substitute back to find the speed of the second train.

### Final Answer
The speed of the train from Q is **9.5 km/hr**.