Quincy and Celine have to move 16 small boxes and 10 large boxes. The chart indicates the time that each person takes to move each type of box. They start moving the boxes at 9:00 a.m. The earliest time at which they can be finished moving all of the boxes is
(A) 9:41 a.m.
(B) 9:42 a.m.
(C) 9:43 a.m.
(D) 9:44 a.m.
(E) $9: 45$ a.m.

|  | Celine | Quincy |
| :--- | :---: | :---: |
| small  box | 2 min. | 3 min. |
| large  box | 6 min. | 5 min. |
First, we do some experimentation.

Since Celine moves small boxes faster and Quincy moves large boxes faster, then suppose Celine moves all 16 small boxes (taking 32 minutes) and Quincy moves all 10 large boxes (taking 50 minutes). Thus, they would finish in 50 minutes.

We could transfer two large boxes from Quincy to Celine, who now moves 16 small and 2 large boxes, taking 44 minutes. Quincy would then move 8 large boxes, taking 40 minutes. So they would finish in 44 minutes. (So (E) is not the answer.)

If we transfer one small box from Celine to Quincy, then Quincy moves 8 large boxes and 1 small box, taking 43 minutes, and Celine moves 15 small and 2 large boxes, taking 42 minutes. So they would finish in 43 minutes. (So (D) is not the answer.)

Why is 43 minutes the smallest possible total time?

Suppose that it took them at most 42 minutes to finish the job. Then the total amount of working time would be at most 84 minutes.

Suppose that Celine moves $x$ small boxes and $y$ large boxes, which would take $2 x+6 y$ minutes. Then Quincy moves $16-x$ small boxes and $10-y$ large boxes, which would take $3(16-x)+$ $5(10-y)=98-3 x-5 y$ minutes.

Since the total working time is at most 84 minutes, then $(2 x+6 y)+(98-3 x-5 y) \leq 84$ or $14 \leq x-y$.

Since $0 \leq x \leq 16$ and $0 \leq y \leq 10$, then the possible pairs of $x$ and $y$ are $(16,0),(16,1),(16,2)$, $(15,0),(15,1),(14,0)$, which produce working times as follows:

|  |  | Celine | Celine  Small | Celine  Large | Quincy  Small | Quincy  Large | Quincy  Time |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 16 | 0 | 16 | 0 | 32 | 0 | 10 | 50 |
| 16 | 1 | 16 | 1 | 38 | 0 | 9 | 45 |
| 16 | 2 | 16 | 2 | 44 | 0 | 8 | 40 |
| 15 | 0 | 15 | 0 | 30 | 1 | 10 | 53 |
| 15 | 1 | 15 | 1 | 36 | 1 | 9 | 48 |
| 14 | 0 | 14 | 0 | 28 | 2 | 10 | 56 |

In each of these cases, while the total working time is no more than 84 minutes, it takes longer than 43 minutes to finish.

Therefore, it is impossible for them to finish in 42 minutes or less, so the earliest possible finishing time is 9:43 a.m.

ANSWER: (C)

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