2010 AMC 12A Problems/Problem 2

Problem

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?

$\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694$

Solution 1

It is easy to see that the ferry boat takes $6$ trips total. The total number of people taken to the island is

$\begin{align*}&100+99+98+97+96+95\\ &=6(100)-(1+2+3+4+5)\\ &=600 - 15\\ &=\boxed{585\ \textbf{(A)}}\end{align*}$

Solution 2

You can notice that: $\begin{align*}&100 + 95 = 99 + 96 = 98 + 97 = 195\end{align*}$ This means that the sum is: $\begin{align*}&100+99+98+97+96+95\\ &=195 * 3\\ &=\boxed{585\ \textbf{(A)}}\end{align*}$

Video Solution

https://youtu.be/TLtqy-62TKo?si=vWBMLRKfpbOZ3C9P

~Charles3829

2010 AMC 12A (Problems • Answer Key • Resources)
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2010 AMC 12A Problems/Problem 2

Contents

Problem

Solution 1

Solution 2

Video Solution

See also