Joe was in charge of lights for a dance. The red light blinks every two seconds, the yellow light every three seconds, and the blue light every five seconds. If we include the very beginning and very end of the dance, how many times during a seven minute dance will all the lights come on at the same time? (Assume that all three lights blink simultaneously at the very beginning of the dance.)

To solve the problem, we need to determine how often all three lights (red, yellow, and blue) will blink simultaneously during the seven-minute dance.

### Step 1: Determine the Least Common Multiple (LCM)
The red light blinks every 2 seconds, the yellow light every 3 seconds, and the blue light every 5 seconds. To find out when all three lights blink together, we need to calculate the least common multiple (LCM) of the three time intervals: 2, 3, and 5 seconds.

- The prime factorization of \(2\) is \(2\).
- The prime factorization of \(3\) is \(3\).
- The prime factorization of \(5\) is \(5\).

Since these are all distinct prime numbers, the LCM is simply the product of these numbers:

\[
\text{LCM}(2, 3, 5) = 2 \times 3 \times 5 = 30 \text{ seconds}
\]

So, all three lights will blink together every 30 seconds.

### Step 2: Calculate the Total Number of Simultaneous Blinks
Now, we need to determine how many times this simultaneous blink will occur during a 7-minute dance.

- Convert 7 minutes to seconds:

\[
7 \times 60 = 420 \text{ seconds}
\]

- The first simultaneous blink occurs at the start of the dance (at 0 seconds), and then every 30 seconds thereafter. To find the number of times they blink together within 420 seconds, including the blink at the very end:

\[
\text{Number of simultaneous blinks} = \frac{420}{30} + 1 = 14 + 1 = 15
\]

### Final Answer
The three lights will blink simultaneously \( \boxed{15} \) times during the 7-minute dance, including the start and the end.

Would you like more details or have any questions?

Here are 5 related questions:

1. If the blue light blinked every 4 seconds instead of 5, how many simultaneous blinks would there be?
2. How would the number of simultaneous blinks change if the dance lasted 10 minutes instead of 7?
3. What is the LCM of 4, 6, and 9 seconds?
4. How can you calculate the LCM of more than two numbers?
5. If a fourth light blinked every 6 seconds, how many simultaneous blinks would occur during the dance?

**Tip:** When calculating the number of events in a time interval, remember to include both the start and the end times if events occur exactly at those moments.

Discover how to calculate the number of times red, yellow, and blue lights blink simultaneously during a dance lasting 7 minutes. Learn about least common multiple (LCM) and its application in synchronizing periodic events. Suitable for students in Grades 6-8.

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