Difference between revisions of "2020 AMC 8 Problems/Problem 6"
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==Video Solution== | ==Video Solution== | ||
+ | https://youtu.be/SPNobOd4t1c (Includes all the problems and has a free class update) | ||
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https://youtu.be/61c1MR9tne8 | https://youtu.be/61c1MR9tne8 | ||
Revision as of 16:47, 14 December 2020
Problem
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
Solution 1
Write the order of the cars as , where the left end of the row represents the back of the train and the right end represents the front. Call the people
,
,
,
, and
respectively. The first condition gives
, so we try
,
, and
. In the first case, as
sat in front of
, we must have
or
, both of which do not comply with the last condition. In the second case, we obtain
, which works, while the third case is obviously impossible, since it results in there being no way for
to sit in front of
. It follows that, with the only possible arrangement being
, the person sitting in the middle car is
.
Solution 2
Follow the first few steps of Solution 1. We must have , and also have
. There are only
spaces available for
, so the only possible arrangement of them is
, so the arrangement is
, so the person in the middle car is
.
Video Solution
https://youtu.be/SPNobOd4t1c (Includes all the problems and has a free class update)
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 AJHSME/AMC 8 Problems and Solutions |
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