Abdi arrived at 5:02 a.m., and so Abdi paid $5.02.
Caleigh arrived at 5:10 a.m., and so Caleigh paid $5.10.
In total, Abdi and Caleigh paid $5.02+$5.10=$10.12.
If both Daniel and Emily had arrived at the same time, then they each would have paid the same amount, or $12.34÷2=$6.17.
In this case, they would have both arrived at 6:17 a.m.
If Daniel arrived 5 minutes earlier, at 6:12 a.m., and Emily arrived 5 minutes later, at 6:22 a.m., then they would have arrived 10 minutes apart and in total they would have still paid $12.34.
(We may check that these arrival times are 10 minutes apart, and that Daniel and Emily’s total price is $6.12+$6.22=$12.34, as required.)
To minimize the amount that Karla could have paid, we maximize the amount that Isaac and Jacob pay.
Isaac and Jacob arrived together and Karla arrived after.
Since Karla arrived at a later time than Isaac and Jacob, then Karla paid more than Isaac and Jacob.
If Isaac and Jacob both arrived together at 6:18 a.m., then they would each have paid $6.18, and Karla would have paid $18.55−$6.18−$6.18=$6.19.
This is the minimum amount that Karla could have paid. Why?
If Isaac and Jacob arrived at 6:19 a.m. or later, then Karla would have arrived at a time earlier than 6:19 a.m. (since $18.55−$6.19−$6.19=$6.17).
Since Karla arrived after Isaac and Jacob, this is not possible.
If Isaac and Jacob arrived earlier than 6:18 a.m., then they would have each paid less than $6.18, and so Karla would have paid more than $6.19.
Therefore, the minimum amount that Karla could have paid is $6.19.
If Larry arrived earlier than 5:39 a.m., then he would have paid less than $5.39 and so Mio would have paid more than $11.98−$5.39=$6.59.
Since Mio arrived during the time of the special pricing, it is not possible for Mio to have paid more than $6.59, and so Larry must have arrived at 5:39 a.m. or later.
If Larry arrived between 5:39 a.m. and 5:59 a.m. (inclusive), then Larry would have paid the amount between $5.39 and $5.59 corresponding to his arrival time.
Therefore, Mio would have paid an amount between $11.98−$5.59=$6.39 and $11.98−$5.39=$6.59 (inclusive).
Each of the amounts between $6.39 and $6.59 corresponds to an arrival time for Mio between 6:39 a.m. and 6:59 a.m., each of which is a possible time that Mio could have arrived during the special pricing period.
That is, each arrival time for Larry from 5:39 a.m. to 5:59 a.m. corresponds to an arrival time for Mio from 6:39 a.m. to 6:59 a.m.
Each of these times is during the period of the special pricing and each corresponding pair of times gives a total price of $11.98.
To see this, consider that if Larry arrived x minutes after 5:39 a.m. (where x is an integer and 0≤x≤20), then Mio arrived x minutes before 6:59 a.m., and in total they paid $5.39+x + $6.59−x =$11.98.
Since Larry’s arrival time and Mio’s arrival time may be switched to give the same total, $11.98, then Larry could also have arrived between 6:39 a.m. and 6:59 a.m.
The only times left to consider are those from 6:00 a.m. to 6:38 a.m.
If Larry arrived at one of these times, his price would have been between $6.00 and $6.38, and so Mio’s price would have been between $11.98−$6.38=$5.60 and $11.98−$6.00=$5.98.
Since there are no arrival times which correspond to Mio having to pay an amount between $5.60 and $5.98, then it is not possible that Larry arrived at any time from 6:00 a.m. to 6:38 a.m.
Therefore, the ranges of times during which Larry could have arrived are 5:39 a.m. to 5:59 a.m or 6:39 a.m. to 6:59 a.m.