2010 AMC 12A Problem 2

Below is the professionally curated solution for Problem 2 of the 2010 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12A solutions, or check the answer key.

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Concepts:arithmetic sequence summation

Difficulty rating: 970

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?

$585$

$594$

$672$

$679$

$694$

Solution:

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 $6\cdot100-(1+2+3+4+5)=600-15=585.$

Thus, A is the correct answer.

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