Joe and Peter have 360 day membership to a tennis club. Joe will use it every other day. Peter will use it every third day. They both use the club on day one. How many days total out of 360 will at least one of them use the club. Please explain.
To find out how many days at least one of them will use the club, we can use the principle of inclusion-exclusion.
First, let's find out how many days Joe will use the club. We know that Joe will use it every other day, so we need to determine the number of odd days between 1 and 360.
To find the number of odd days, we can use the formula (N+1)/2, where N is the total number of days (360 in this case). Applying the formula, we get (360+1)/2 = 181.
Therefore, Joe will use the club for 181 days.
Now, let's find out how many days Peter will use the club. We know that Peter will use it every third day, so we need to determine the number of days divisible by 3 between 1 and 360.
To find the number of days divisible by 3, we can use the formula N/3. Applying the formula, we get 360/3 = 120.
Therefore, Peter will use the club for 120 days.
At this point, we have counted the days for each of them individually, but we have counted the days when they both use the club twice. To account for this, we need to subtract the number of days when they both use the club once.
To find the number of days when they both use the club, we need to determine the number of days that are both odd and divisible by 3 between 1 and 360. To do this, we need to find the number of days that are multiples of 6, as every other odd day (Joe's days) will coincide with every third day (Peter's days) when it is a multiple of 6.
To find the number of days that are multiples of 6, we can use the formula N/6. Applying the formula, we get 360/6 = 60.
Therefore, both Joe and Peter will use the club on 60 days.
Finally, to find the number of days at least one of them uses the club, we can add the number of days Joe uses, the number of days Peter uses, and subtract the number of days they both use.
Total days = Joe's days + Peter's days - Days when both use the club
Total days = 181 + 120 - 60
Total days = 241
Therefore, at least one of them will use the club for 241 days out of the total 360 days.