Question 1118220
When {{{13}}} sets of customers had left the cafe
( {{{6}}} sets who where seating at their tables initially,
and {{{13-6=7}}} sets who entered and left the cafe),
Brian has bused {{{13(3/5)=39/5=7&4/5}}} tables,
so {{{highlight(7)}}} new tables of customers had come in,
and Brian rushed to lock the door before an 8th set of customers could come in.
At that point the number of unbused tables was
{{{1+13-7&4/5=highlight(6&1/5)}}} .
 
STEPWISE:
As Brian has bused {{{3/5}}} of a table, {{{1}}} table-ful of customers leaves, so now Brian has {{{1}}} more table to bus,
to be added to the {{{2/5}}} left to do of the table he is busing.
For each set of customers that leaves the restaurant,
the number of tables unbused increases by {{{2/5}}}
(compared to the number unbused when the previous set of customers left).
As Brian has bused {{{5(3/5)=3}}} tables, {{{5*1=5}}} table-ful of customers have left,
but at the same time new customers have filled {{{3}}} tables.
The number of tables in use has decreased by {{{5-3=2}}} as {{{5}}} tables emptied.
As Brian has bused {{{10(3/5)=6}}} tables, {{{10*1}}} table-ful of customers have left,
but at the same time new customers have filled {{{6}}} tables,
leaving {{{6-10+6=2}}} tables occupied.
After that, {{{2+1=3}}} more table-fuls of diners will have to leave for the cafe to be empty,
because by the time the customers in those {{{2}}} occupied tables leave,
Brian will have bused {{{2(3/5)=6/5=1&1/5}}} table,
but as he finished busing one table, another table became occupied.
As the last {{{3}}} table-fuls of customers left,
a total of {{{10+3=13}}} table-fuls of customers had left,
Brian had bused {{{13(3/5)=39/5=7&4/5}}} tables,
while the number of tables to be bused had increased by {{{13}}} ,
so after Brian locked the door, the number of unbused tables was
{{{1+13-7&4/5=6&1/5}}} .