Difference between revisions of "2010 AMC 12A Problems/Problem 2"
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| − | == Problem | + | == 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? | 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? | ||
<math>\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694</math> | <math>\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694</math> | ||
| − | [[ | + | == Solution 1 == |
| + | It is easy to see that the ferry boat takes <math>6</math> trips total. The total number of people taken to the island is | ||
| + | |||
| + | <cmath>\begin{align*}&100+99+98+97+96+95\\ | ||
| + | &=6(100)-(1+2+3+4+5)\\ | ||
| + | &=600 - 15\\ | ||
| + | &=\boxed{585\ \textbf{(A)}}\end{align*}</cmath> | ||
| + | ==Solution 2== | ||
| + | You can notice that: | ||
| + | <cmath>\begin{align*}&100 + 95 = 99 + 96 = 98 + 97 = 195\end{align*}</cmath> | ||
| + | This means that the sum is: | ||
| + | <cmath>\begin{align*}&100+99+98+97+96+95\\ | ||
| + | &=195 * 3\\ | ||
| + | &=\boxed{585\ \textbf{(A)}}\end{align*}</cmath> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/TLtqy-62TKo?si=vWBMLRKfpbOZ3C9P | ||
| + | |||
| + | ~Charles3829 | ||
| + | |||
| + | == See also == | ||
| + | {{AMC12 box|year=2010|num-b=1|num-a=3|ab=A}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 10:54, 18 September 2024
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?
Solution 1
It is easy to see that the ferry boat takes 
trips total. The total number of people taken to the island is
Solution 2
You can notice that:
This means that the sum is:
Video Solution
https://youtu.be/TLtqy-62TKo?si=vWBMLRKfpbOZ3C9P
~Charles3829
See also
| 2010 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 1 |
Followed by Problem 3 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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