Institution: Bangladesh Bank Exam: Officer (Cash) -Sonali Bank | BD Development | BD Krishi Bank - 12.10.2018 ||
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To solve this problem, we set up a few definitions. Let the distance be \( d \) km, and the time he should ideally take to cover that distance be \( t \) hours.

When he travels at 3 km/hr, the time taken is \( \frac{d}{3} \) hours. According to the problem, he is 15 minutes (or \( \frac{1}{4} \) hours) late, so we can express this as:
\[
\frac{d}{3} = t + \frac{1}{4}
\]

Conversely, when he travels at 4 km/hr, the time taken is \( \frac{d}{4} \) hours, and he arrives 15 minutes early:
\[
\frac{d}{4} = t - \frac{1}{4}
\]

Now we have two equations:
1. \( \frac{d}{3} = t + \frac{1}{4} \)
2. \( \frac{d}{4} = t - \frac{1}{4} \)

We can eliminate \( t \) by rearranging both equations. From the first equation, we can solve for \( t \):
\[
t = \frac{d}{3} - \frac{1}{4}
\]

Substituting this into the second equation gives:
\[
\frac{d}{4} = \left(\frac{d}{3} - \frac{1}{4}\right) - \frac{1}{4}
\]
\[
\frac{d}{4} = \frac{d}{3} - \frac{2}{4}
\]

Multiplying through by 12 (the least common multiple of 3 and 4) to eliminate the fractions:
\[
3d = 4d - 6
\]
\[
d = 6 \text{ km}
\]

Thus, the correct distance to travel is 6 km, confirming option B as the solution.