The distance between two stations 'X and 'Y is 450 km. A train L starts at 6:00 pm from X and moves towards Y at an average speed of 60 km/h. Another train M starts from Y at 5 : 20 pm and moves towards Xat an average speed of 80 km/h. How far fromxwill the two trains meet and at what time?

Options

A. 170 km, 8 : 50 pm
B. 150 km, 7 : 50 pm
C. 170 km, 6 : 50 pm
D. 150 km, 9 : 50 pm

Answer

To solve the problem step by step, we will determine how far from station X the two trains meet and at what time they meet.

### Step 1: Determine the time difference between the two trains.
Train M starts from Y at 5:20 PM, while Train L starts from X at 6:00 PM. Therefore, Train M has a 40-minute head start.

**Hint:** Calculate the time difference between the two starting times.

### Step 2: Convert the time difference into hours.
40 minutes can be converted into hours:
\[ \text{Time difference in hours} = \frac{40}{60} = \frac{2}{3} \text{ hours} \]

**Hint:** Remember that 1 hour = 60 minutes.

### Step 3: Calculate the distance covered by Train M before Train L starts.
Train M travels at a speed of 80 km/h. The distance covered by Train M in 40 minutes is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 80 \times \frac{2}{3} = \frac{160}{3} \text{ km} \approx 53.33 \text{ km} \]

**Hint:** Use the formula for distance: Distance = Speed × Time.

### Step 4: Determine the remaining distance between the two trains when Train L starts.
The total distance between X and Y is 450 km. The distance remaining when Train L starts is:
\[ \text{Remaining distance} = 450 - \frac{160}{3} = \frac{1350 - 160}{3} = \frac{1190}{3} \text{ km} \approx 396.67 \text{ km} \]

**Hint:** Subtract the distance covered by Train M from the total distance.

### Step 5: Set up the equation for the meeting point.
Let \( d \) be the distance from X where the two trains meet. The distance covered by Train L will be \( d \), and the distance covered by Train M will be \( 450 - d \).

The time taken for both trains to meet can be expressed as:
\[ \text{Time taken by Train L} = \frac{d}{60} \]
\[ \text{Time taken by Train M} = \frac{450 - d}{80} \]

Since Train M started 40 minutes (or \(\frac{2}{3}\) hours) earlier, we can set up the equation:
\[ \frac{450 - d}{80} = \frac{d}{60} + \frac{2}{3} \]

**Hint:** Remember to account for the head start of Train M in the time equation.

### Step 6: Solve the equation.
Cross-multiplying to eliminate the fractions:
\[ 60(450 - d) = 80d + 80 \times \frac{2}{3} \times 60 \]
\[ 27000 - 60d = 80d + 3200 \]
\[ 27000 - 3200 = 80d + 60d \]
\[ 23800 = 140d \]
\[ d = \frac{23800}{140} = 170 \text{ km} \]

**Hint:** Simplify the equation carefully to isolate \( d \).

### Step 7: Calculate the time at which they meet.
Now, we can find the time taken by Train L to cover 170 km:
\[ \text{Time} = \frac{d}{\text{Speed}} = \frac{170}{60} \text{ hours} = \frac{17}{6} \text{ hours} \approx 2.83 \text{ hours} \]

Train L starts at 6:00 PM, so:
\[ 6:00 PM + 2.83 \text{ hours} \approx 8:50 PM \]

**Hint:** Add the time taken to the starting time to find the meeting time.

### Final Answer:
The two trains will meet 170 km from station X at approximately 8:50 PM.