High School

Jack, Art, Fran, and Megan work as volunteers at the local kennel.

- Jack gives the dogs baths every 4 days.
- Art cleans out cages every 6 days.
- Fran feeds the animals in Section B every 2 days.
- Megan helps the receptionist every 3 days.

How many times in 12 weeks will all 4 helpers be at the clinic on the same day?

(Note: Show all work and explain your process.)

Answer :

All 4 helpers will be at the clinic on the same day 7 times in 12 weeks.

What is Prime Factorization?

"Prime Factorization" is the process of determining which prime numbers multiply together to get the original number.

To find the number of times in 12 weeks that all 4 helpers will be at the clinic on the same day, we need to find the least common multiple (LCM) of the numbers 4, 6, 2, and 3.

First, we can find the prime factorization of each number:

4 = 2 x 2

6 = 2 x 3

2 = 2

3 = 3

Next, we can find the LCM by taking the highest power of each prime factor that appears in the factorizations:

LCM = 2 x 2 x 3 = 12

Therefore, all 4 helpers will be at the clinic on the same day every 12 days. Since 12 weeks is equivalent to 84 days, we can divide 84 by 12 to get the number of times:

84 ÷ 12 = 7

So, all 4 helpers will be at the clinic on the same day 7 times in 12 weeks.

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Final answer:

In 12 weeks, Jack, Art, Fran, and Megan will all be at the local kennel on the same day 7 times. This is determined by calculating the least common multiple (LCM) of their individual schedules and dividing the total days in 12 weeks by this LCM.

Explanation:

The question asks how many times in 12 weeks will all 4 volunteers at a local kennel be at the clinic on the same day based on their individual schedules. This is a least common multiple (LCM) problem. Let's calculate.

Jack works every 4 days

Art works every 6 days

Fran works every 2 days

Megan works every 3 days

To find when all four helpers will be at the clinic on the same day, we calculate the LCM of their schedules: 4, 6, 2, 3. The LCM of 4, 6, 2, 3 is 12. This means every 12 days, they all will work at the clinic on the same day. Now, 12 weeks equals 84 days (12 weeks × 7 days = 84 days). We divide 84 by the LCM, 12.

84 days ÷ 12 days/schedule = 7 times

Therefore, in 12 weeks, all four helpers will be at the clinic on the same day 7 times.