, 2017

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 $\$12.34\div2=\$6.17$.

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 $0\leq x\leq20$), 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.