Brian buses tables at a local cafe To bus a table he must cl

Brian buses tables at a local cafe. To bus a table, he must clear the dirty dishes and reset the table for the next set of customers. One night he noticed that for every three-fifths of a table that he bused, another table of customers would get up and leave. He also noticed that right after he finished busing a table, a new table of customers would come into the restaurant. However, once every table was empty (no diners were left in the restaurant), nobody else came into the restaurant. Suppose there were six tables with customers and one unused table. How many new tables of customers would come in before the restaurant was empty? After the last table of customers had left, how many tables were unused?

Solution

dT=Dirty tables (or tables the need to be bussed).
Tc=Tables with customers

Now, when a customer leaves, you need to add +1 to dT and -1 Tc.
When you finish busing a table, you need to +1 to Tc.

Since you only clear 3/5 of the table at a time before another dT is added, we will keep this in fractions. We start out with 5/5 dT.

To start this is what we have:

5/5 dT : 6Tc

Now you buss 3/5 of that table so you have:

2/5dT. One customer will now leave. You subtract 1 from Tc, but you HAVE TO REMEMBER to add 1 to dT.

Now we have:

7/5dT : 5Tc

Now heres the next step. We\'re going to buss another 3/5 of the tables. We are now down to 4/5dT. However, we just finished bussing a whole table. So someone else is going to come in (+1Tc), but someone is going to leave as well since we have bussed 3/5 of a table (-1Tc). Essentially, Tc stays the same at 5Tc. So since we have bussed 3/5 of the table, we have 4/5 dT...but we can\'t forget to add one since a customer left. So, 9/5dT.

We now have:

9/5 dT : 5Tc

now that you understand it, ill do line by line.

9/5 dT : 5Tc
(minus 3/5 dT and minus 1Tc)
6/5dT : 4Tc
(But you have +1 dT for the -1Tc) so
11/5dt: 4Tc
8/5dt : 4Tc
-1 Tc +1 Tc + 1dT
15/5dT : 4Tc

keep going with that and you end up with 6 and 1/5 tables that are unbussed when all customers leave.