2010 AMC 12A Problem 2

Question

A ferry boat shuttles tourists to an island every hour starting at

10

am until its last trip, which starts at

3

pm. One day the boat captain notes that on the

10

am trip there were

100

tourists on the ferry boat, and that on each successive trip, the number of tourists was

1

fewer than on the previous trip. How many tourists did the ferry take to the island that day?

585

594

672

679

694

Count how many trips run from

10

am through

3

pm

The trip counts are

100,99,\ldots,95;

add them

The ferry makes

6

trips: at

10,

11,

12,

1,

2,

and

3.

The numbers of tourists are

100,99,98,97,96,95,

so the total is

\begin{gathered} 6\cdot100-(1+2+3+4+5) \\ =600-15 \\ =585. \end{gathered}

Thus, A is the correct answer.

Problem 2 in other years