Difference between revisions of "2019 AMC 8 Problems/Problem 23"
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==Solution 1== | ==Solution 1== | ||
| − | Since <math>\frac{\text{total points}}{4}</math> and <math>\frac{2(\text{total points})}{7}</math> are integers, we have <math>28 | \text{total points}</math>. We see that the number of points scored by the other team members is less than or equal to <math>14</math> and greater than or equal to <math>0</math>. We let the total number of points be <math>t</math> and the total number of points scored by the other team members, which means that <math>\frac{t}{4} + \frac{2t}{7} + 15 + x = t \quad \implies \quad 0 \le \frac{13t}{28} - 15 = x \le 14</math>, which means <math>15 \le \frac{13t}{28} \le 29</math>. The only value of <math>t</math> that satisfies all conditions listed is <math>56</math>, so <math>x=\boxed{\textbf{(B)} 11}</math>. - juliankuang (lol im smart.) | + | Since <math>\frac{\text{total points}}{4}</math> and <math>\frac{2(\text{total points})}{7}</math> are integers, we have <math>28 | \text{total points}</math>. We see that the number of points scored by the other team members is less than or equal to <math>14</math> and greater than or equal to <math>0</math>. We let the total number of points be <math>t</math> and the total number of points scored by the other team members be <math>x</math>, which means that <math>\frac{t}{4} + \frac{2t}{7} + 15 + x = t \quad \implies \quad 0 \le \frac{13t}{28} - 15 = x \le 14</math>, which means <math>15 \le \frac{13t}{28} \le 29</math>. The only value of <math>t</math> that satisfies all conditions listed is <math>56</math>, so <math>x=\boxed{\textbf{(B)} 11}</math>. - juliankuang (lol im smart.) |
==Solution 2== | ==Solution 2== | ||
Revision as of 20:24, 12 April 2020
Problem 23
After Euclid High School's last basketball game, it was determined that 
of the team's points were scored by Alexa and 
were scored by Brittany. Chelsea scored 
points. None of the other 
team members scored more than 
points What was the total number of points scored by the other team members?
Solution 1
Since 
and 
are integers, we have 
. We see that the number of points scored by the other team members is less than or equal to 
and greater than or equal to 
. We let the total number of points be 
and the total number of points scored by the other team members be 
, which means that 
, which means 
. The only value of that satisfies all conditions listed is 
, so 
. - juliankuang (lol im smart.)
Solution 2
Starting from the above equation 
where is the total number of points scored and 
is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation 
, or 
. Since is necessarily divisible by 28, let 
where 
and divide by 28 to obtain 
. Then it is easy to see 
(
) is the only candidate, giving . -scrabbler94
Solution 3
Fakesolve: We first start by setting the total number of points as 
, since 
. However, we see that this does not work since we surpass the number of points just with the information given (
). Next, we assume that the total number of points scored is . With this, we have that Alexa, Brittany, and Chelsea score: 
, and thus, the other seven players would have scored a total of 
(We see that this works since we could have 
of them score points, and the other 
of them score 
point) -aops5234
See Also
| 2019 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 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. 
