Difference between revisions of "2020 AMC 8 Problems/Problem 6"
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− | ==Problem | + | ==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? | 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? | ||
<math>\textbf{(A) }\text{Aaron} \qquad \textbf{(B) }\text{Darren} \qquad \textbf{(C) }\text{Karen} \qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}</math> | <math>\textbf{(A) }\text{Aaron} \qquad \textbf{(B) }\text{Darren} \qquad \textbf{(C) }\text{Karen} \qquad \textbf{(D) }\text{Maren}\qquad \textbf{(E) }\text{Sharon}</math> | ||
− | ==See also== {{AMC8 box|year=2020| | + | ==Solution 1== |
+ | Write the order of the cars as <math>\square\square\square\square\square</math>, where the left end of the row represents the back of the train and the right end represents the front. Call the people <math>A</math>, <math>D</math>, <math>K</math>, <math>M</math>, and <math>S</math> respectively. The first condition gives <math>M\square\square\square\square</math>, so we try <math>MAS\square\square</math>, <math>M\square AS\square</math>, and <math>M\square\square AS</math>. In the first case, as <math>D</math> sat in front of <math>A</math>, we must have <math>MASDK</math> or <math>MASKD</math>, which do not comply with the last condition. In the second case, we obtain <math>MKASD</math>, which works, while the third case is obviously impossible since it results in there being no way for <math>D</math> to sit in front of <math>A</math>. It follows that, with the only possible arrangement being <math>MKASD</math>, the person sitting in the middle car is <math>\boxed{\textbf{(A) }\text{Aaron}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | Follow the first few steps of Solution 1. We must have <math>M\square\square\square\square</math>, and also have <math>AS\text{(anything)}D \text{ and } K\dots D</math>. There are only <math>4</math> spaces available for <math>A, S, D, K</math>, so the only possible arrangement of them is <math>KASD</math>, so the arrangement is <math>MKASD</math>, so the person in the middle car is <math>\boxed{\textbf{(A) }\text{Aaron}}</math>. | ||
+ | |||
+ | ==Video Solution by NiuniuMaths (Easy to understand!)== | ||
+ | https://www.youtube.com/watch?v=8hgK6rESdek&t=9s | ||
+ | |||
+ | ~NiuniuMaths | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING!!!)== | ||
+ | https://youtu.be/wnTfhfhS1VA | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution by Math-X (First understand the problem!!!)== | ||
+ | https://youtu.be/UnVo6jZ3Wnk?si=lE54gozu1zUD7xkh&t=567 | ||
+ | |||
+ | ~Math-X | ||
+ | |||
+ | ==Video Solution by North America Math Contest Go Go Go== | ||
+ | |||
+ | https://www.youtube.com/watch?v=RR6svhjdPEA | ||
+ | |||
+ | ==Video Solution by WhyMath== | ||
+ | https://youtu.be/G04TnUc5iAA | ||
+ | |||
+ | ~savannahsolver | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/61c1MR9tne8 ~ The Learning Royal | ||
+ | |||
+ | ==Video Solution by Interstigation== | ||
+ | https://youtu.be/YnwkBZTv5Fw?t=186 | ||
+ | |||
+ | ~Interstigation | ||
+ | |||
+ | ==Video Solution by STEMbreezy== | ||
+ | https://youtu.be/U27z1hwMXKY?list=PLFcinOE4FNL0TkI-_yKVEYyA_QCS9mBNS | ||
+ | |||
+ | ~STEMbreezy | ||
+ | |||
+ | ==See also== | ||
+ | {{AMC8 box|year=2020|num-b=5|num-a=7}} | ||
+ | {{MAA Notice}} |
Latest revision as of 07:15, 24 January 2024
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Video Solution by NiuniuMaths (Easy to understand!)
- 5 Video Solution (CREATIVE THINKING!!!)
- 6 Video Solution by Math-X (First understand the problem!!!)
- 7 Video Solution by North America Math Contest Go Go Go
- 8 Video Solution by WhyMath
- 9 Video Solution
- 10 Video Solution by Interstigation
- 11 Video Solution by STEMbreezy
- 12 See also
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
, 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 by NiuniuMaths (Easy to understand!)
https://www.youtube.com/watch?v=8hgK6rESdek&t=9s
~NiuniuMaths
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/UnVo6jZ3Wnk?si=lE54gozu1zUD7xkh&t=567
~Math-X
Video Solution by North America Math Contest Go Go Go
https://www.youtube.com/watch?v=RR6svhjdPEA
Video Solution by WhyMath
~savannahsolver
Video Solution
https://youtu.be/61c1MR9tne8 ~ The Learning Royal
Video Solution by Interstigation
https://youtu.be/YnwkBZTv5Fw?t=186
~Interstigation
Video Solution by STEMbreezy
https://youtu.be/U27z1hwMXKY?list=PLFcinOE4FNL0TkI-_yKVEYyA_QCS9mBNS
~STEMbreezy
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.