There are 8 equidistant points $A, B, C, D, E, F, G$ and $H$ in the clockwise direction on the periphery of a circle. In a time interval $t$, a person reaches from $A$ to $C$ with uniform motion while another person reaches the point $E$ from the point $B$ during the same time interval with uniform motion. Both the persons move in the same direction along the circumference of the circle and start at the same instant. How much time after the start, will the two persons meet each other?

### Concept & Relative Distance on Circular Tracks When two objects move on a circular track in the same direction starting from different positions, the time to meet depends on the initial gap and their relative speed. The faster object must close the "rear" gap to lap the slower one. $$Time = \frac{Distance\ to\ Gain}{Relative\ Speed}$$ ### Step-by-Step Solution 1. **Define units:** Let the distance between any two adjacent points be 1 unit. Total circumference = 8 units. 2. **Determine speeds:** Person 1 (P1) travels from A to C (2 units) in time $t$. Speed of P1 = $2/t$ units/time. Person 2 (P2) travels from B to E (3 units) in time $t$. Speed of P2 = $3/t$ units/time. 3. **Analyze initial positions:** P1 starts at A. P2 starts at B. Since points are clockwise (A, B, C...), P2 is ahead of P1 by 1 unit in the direction of motion. 4. **Calculate relative speed:** Since both move clockwise (same direction), relative speed = Speed of P2 - Speed of P1 = $(3/t) - (2/t) = 1/t$ units/time. 5. **Determine distance to gain:** P2 is faster and is ahead. For P2 to meet P1, P2 must lap P1. This means P2 has to cover the entire circle minus the initial 1-unit lead to catch up from behind. Distance to gain = Total Circumference - Initial Lead = $8 - 1 = 7$ units. 6. **Calculate meeting time:** Time = (Distance to gain) / Relative Speed = $7 / (1/t) = 7t$. ### Exam Strategy & Shortcut Think in terms of "gaps" instead of speed formulas. In time $t$, P1 covers 2 gaps, P2 covers 3 gaps. P2 gains 1 gap every $t$ time. P2 is initially 1 gap ahead, so P2 needs to gain 7 gaps to catch P1 from behind. $7 \text{ gaps} \times (t \text{ per gap}) = 7t$. ### Common Pitfall Assuming the faster person just needs to close the 1-unit gap. Since P2 is faster AND starts ahead, P2 is pulling away. P2 must go all the way around the circle to catch P1 from behind, which requires gaining $8 - 1 = 7$ units. ### Final Answer Therefore, the correct answer is **$7t$**.