Two men A and B walk round a circle 1300 metres in circumference. A walks at the rate of 150 metres per minute and B at the rate 80 metres per minute. If they both start at the same time from the same point and walk in the same direction, after how much time will they meet each other?

The correct answer is 18 $\frac{4}{7}$ min.

Step 1: Understand the Problem
We are given two men, A and B, walking around a circular track in the same direction.
- Total circumference of the circular path = 1300 metres
- Speed of A ($v_A$) = 150 metres per minute
- Speed of B ($v_B$) = 80 metres per minute

Step 2: Calculate the Relative Speed
Since both A and B start at the same point and move in the same direction, the effective rate at which A gains distance over B is their relative speed:

$Relative\; Speed=v_A-v_B=150-80=70\text{metres per minute}$

Step 3: Calculate the Time Taken to Meet
For them to meet each other for the first time after starting from the same point, the faster person (A) must complete one extra full round (1300 metres) relative to the slower person (B).

$\text{Time taken}=\frac{\text{Circumference}}{\text{Relative Speed}}=\frac{1300}{70}=\frac{130}{7}\text{minutes}$

Step 4: Convert to a Mixed Fraction
Dividing 130 by 7:
130 ÷ 7 = 18 with a remainder of 4.

$\frac{130}{7}=18\frac{4}{7}\text{minutes}$

Thus, they will meet each other after 18 $\frac{4}{7}$ minutes.