Two persons Alif and Laila start at the same time from Allahabad and Lucknow and proceeds towards each other at 45km/h and 54 km/h respectively when they meet it is found that one of them has travelled 72 km more than the other.The distance between the place (in km) is:

Options

A. 729
B. 792
C. 540
D. none of these

Answer

To solve the problem, we need to find the distance between Allahabad and Lucknow based on the information given about Alif and Laila's speeds and the distance one of them traveled more than the other.

### Step-by-Step Solution:

1. **Define Variables:**
Let the distance traveled by Alif be \( d_A \) and the distance traveled by Laila be \( d_L \). According to the problem, we know:
\[ d_A = d_L + 72 \]

2. **Use the Speeds:**
Alif travels at a speed of 45 km/h and Laila at 54 km/h. The time taken by both to meet is the same, so we can express the time taken by each in terms of distance and speed:
\[ \text{Time taken by Alif} = \frac{d_A}{45} \]
\[ \text{Time taken by Laila} = \frac{d_L}{54} \]

3. **Set the Times Equal:**
Since both start at the same time and meet at the same time, we can set the two time expressions equal to each other:
\[ \frac{d_A}{45} = \frac{d_L}{54} \]

4. **Substitute \( d_A \):**
Substitute \( d_A \) from the first equation into the time equation:
\[ \frac{d_L + 72}{45} = \frac{d_L}{54} \]

5. **Cross Multiply to Solve for \( d_L \):**
Cross-multiplying gives us:
\[ 54(d_L + 72) = 45d_L \]
Expanding this:
\[ 54d_L + 3888 = 45d_L \]

6. **Rearranging the Equation:**
Rearranging the equation to isolate \( d_L \):
\[ 54d_L - 45d_L = -3888 \]
\[ 9d_L = 3888 \]
\[ d_L = \frac{3888}{9} = 432 \text{ km} \]

7. **Calculate \( d_A \):**
Now, using the value of \( d_L \) to find \( d_A \):
\[ d_A = d_L + 72 = 432 + 72 = 504 \text{ km} \]

8. **Total Distance:**
The total distance between Allahabad and Lucknow is the sum of the distances traveled by Alif and Laila:
\[ \text{Total Distance} = d_A + d_L = 504 + 432 = 936 \text{ km} \]

### Final Answer:
The distance between Allahabad and Lucknow is **936 km**.