Difference between revisions of "2015 AMC 8 Problems/Problem 24"
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+ | == Problem == | ||
+ | |||
A baseball league consists of two four-team divisions. Each team plays every other team in its division <math>N</math> games. Each team plays every team in the other division <math>M</math> games with <math>N>2M</math> and <math>M>4</math>. Each team plays a 76 game schedule. How many games does a team play within its own division? | A baseball league consists of two four-team divisions. Each team plays every other team in its division <math>N</math> games. Each team plays every team in the other division <math>M</math> games with <math>N>2M</math> and <math>M>4</math>. Each team plays a 76 game schedule. How many games does a team play within its own division? | ||
− | <math> | + | <math>\textbf{(A) } 36 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 54 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 72 </math> |
− | \textbf{(A) } 36 \qquad | + | |
− | \textbf{(B) } 48 \qquad | + | ==Solutions== |
− | \textbf{(C) } 54 \qquad | ||
− | \textbf{(D) } 60 \qquad | ||
− | \textbf{(E) } 72 | ||
− | </math> | ||
− | ==Solution 1== | + | ===Solution 1=== |
On one team they play <math>3N</math> games in their division and <math>4M</math> games in the other. This gives <math>3N+4M=76</math> | On one team they play <math>3N</math> games in their division and <math>4M</math> games in the other. This gives <math>3N+4M=76</math> | ||
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Next, <math>M=6</math> does not work because <math>52</math> is not divisible by <math>3</math> | Next, <math>M=6</math> does not work because <math>52</math> is not divisible by <math>3</math> | ||
− | We try <math>M=7</math> | + | We try <math>M=7</math> <math>does</math> work by giving <math>N=16,~M=7</math> and thus <math>3\times 16=\boxed{\textbf{(B)}~48}</math> games in their division. |
<math>M=10</math> seems to work, until we realize this gives <math>N=12</math>, but <math>N>2M</math> so this will not work. | <math>M=10</math> seems to work, until we realize this gives <math>N=12</math>, but <math>N>2M</math> so this will not work. | ||
− | ==Solution 2== | + | ===Solution 2=== |
<math>76=3N+4M > 10M</math>, giving <math>M \le 7</math>. | <math>76=3N+4M > 10M</math>, giving <math>M \le 7</math>. | ||
Since <math>M>4</math>, we have <math>M=5,6,7</math>. | Since <math>M>4</math>, we have <math>M=5,6,7</math>. | ||
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This gives <math>3N=48</math>, as desired. The answer is <math>\boxed{\textbf{(B)}~48}</math> | This gives <math>3N=48</math>, as desired. The answer is <math>\boxed{\textbf{(B)}~48}</math> | ||
− | =Video | + | ===Video Solutions=== |
https://youtu.be/LiAupwDF0EY - Happytwin | https://youtu.be/LiAupwDF0EY - Happytwin | ||
https://www.youtube.com/watch?v=bJSWtw91SLs - Oliver Jiang | https://www.youtube.com/watch?v=bJSWtw91SLs - Oliver Jiang | ||
+ | |||
+ | == Video Solution == | ||
+ | https://youtu.be/HISL2-N5NVg?t=4968 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
==See Also== | ==See Also== |
Latest revision as of 15:56, 17 July 2021
Contents
Problem
A baseball league consists of two four-team divisions. Each team plays every other team in its division games. Each team plays every team in the other division games with and . Each team plays a 76 game schedule. How many games does a team play within its own division?
Solutions
Solution 1
On one team they play games in their division and games in the other. This gives
Since we start by trying . This doesn't work because is not divisible by .
Next, does not work because is not divisible by
We try work by giving and thus games in their division.
seems to work, until we realize this gives , but so this will not work.
Solution 2
, giving . Since , we have . Since is , we must have equal to , so .
This gives , as desired. The answer is
Video Solutions
https://youtu.be/LiAupwDF0EY - Happytwin
https://www.youtube.com/watch?v=bJSWtw91SLs - Oliver Jiang
Video Solution
https://youtu.be/HISL2-N5NVg?t=4968
~ pi_is_3.14
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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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