Difference between revisions of "2015 AMC 8 Problems/Problem 24"
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− | + | == Problem == | |
− | <math> | + | 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 <math>76</math> game schedule. How many games does a team play within its own division? |
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− | </math> | ||
− | + | <math>\textbf{(A) } 36 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 54 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 72 </math> | |
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− | + | ==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>. | ||
− | + | Since <math>M>4</math> we start by trying M=5. This doesn't work because <math>56</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> does work by giving <math>N=16</math> ,<math>~M=7</math> and thus <math>3\times 16=\boxed{\textbf{(B)}~48}</math> games in their division. |
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+ | <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. | ||
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+ | ==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>. |
− | + | Testing all of the cases we get tha <math>M=7</math>. | |
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>. |
Latest revision as of 10:01, 23 July 2024
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
game schedule. How many games does a team play within its own division?
Solution 1
On one team they play games in their division and
games in the other. This gives
.
Since we start by trying M=5. This doesn't work because
is not divisible by
.
Next, does not work because
is not divisible by
.
We try does 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
.
Testing all of the cases we get tha
.
This gives , as desired. The answer is
.
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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