Question

Four friends, Ashish, Brian, Chaitra, and Dorothy, decide to jog for 30 minutes inside a stadium with a circular running track that is 200 metres long. The friends run at different speeds. Ashish completes a lap exactly every 60 seconds. Likewise, Brian, Chaitra and Dorothy complete a lap exactly every 1 minute 30 seconds, 40 seconds and 1 minute 20 seconds respectively. The friends begin together at the start line exactly at 4 p.m. What is the total of the numbers of laps the friends would have completed when they next cross the start line together ?

• A. 43
• B. 36
• C. They will never be at the start line together again before 4:30 p.m.
• D. 47
• E. 28

Answer 1

The Correct Option is D

Step 1: Determine Lap Times 
- Ashish: \(60 \, \text{sec}\) - Brian: \(90 \, \text{sec}\) - Chaitra: \(40 \, \text{sec}\) - Dorothy: \(80 \, \text{sec}\)

Step 2: Find Common Meeting Time
They meet at the start line when time is a common multiple of all lap times. So we calculate: \[ \text{LCM}(60, \; 90, \; 40, \; 80) \] - Prime factorizations: \[ 60 = 2^2 \cdot 3 \cdot 5,\quad 90 = 2 \cdot 3^2 \cdot 5,\quad 40 = 2^3 \cdot 5,\quad 80 = 2^4 \cdot 5 \] - Take maximum powers of each prime: \[ 2^4 \cdot 3^2 \cdot 5 = 16 \cdot 9 \cdot 5 = 720 \] Thus, they meet at the start line again after \(720 \, \text{seconds} = 12 \, \text{minutes}\).

Step 3: Laps Completed by Each Friend in 720 Seconds
- Ashish: \( \frac{720}{60} = 12 \) laps - Brian: \( \frac{720}{90} = 8 \) laps - Chaitra: \( \frac{720}{40} = 18 \) laps - Dorothy: \( \frac{720}{80} = 9 \) laps

Step 4: Total Laps
\[ 12 + 8 + 18 + 9 = 47 \]

Step 5: Conclusion
The total number of laps when they next cross the start line together is 47.

Final Answer:
\[ \boxed{\text{47}} \]