Question

At the flower show, one trolley arrives every 4 minutes to take people to the rose pavilion. Another trolley arrives every 10 minutes to take people to the exotic flower greenhouse. Both trollies just arrived at the main building. How many minutes from now will be the next time the trollies arrive at the main building together?

minutes

Answer 1

To determine when the two trolleys will arrive at the main building together again, we need to find the least common multiple (LCM) of their arrival times. The first trolley arrives every 4 minutes, and the second trolley arrives every 10 minutes.

1. First, we can list the multiples of each time:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
- Multiples of 10: 10, 20, 30, 40, 50, ...

2. Next, we can see that both lists include the number 20, which is the first common number, indicating that the LCM of 4 and 10 is 20.

Alternatively, we can calculate the LCM using prime factorization:
- The prime factorization of 4 is \(2^2\).
- The prime factorization of 10 is \(2^1 \times 5^1\).
- To find the LCM, we take the highest power of each prime:
- For \(2\), the highest power is \(2^2\).
- For \(5\), the highest power is \(5^1\).

Thus, the LCM is:
\[
LCM = 2^2 \times 5^1 = 4 \times 5 = 20
\]

This means the next time both trolleys will arrive together at the main building is in **20 minutes**.

Final Answer:
20 minutes