Difference between revisions of "2019 AMC 8 Problems/Problem 23"
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Starting from the above equation <math>\frac{t}{4}+\frac{2t}{7} + 15 + x = t</math> where <math>t</math> is the total number of points scored and <math>x\le 14</math> is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation <math>x+15 = \frac{13}{28}t</math>, or <math>28x+28\cdot 15=13t</math>. Since <math>t</math> is necessarily divisible by 28, let <math>t=28u</math> where <math>u \ge 0</math> and divide by 28 to obtain <math>x + 15 = 13u</math>. Then it is easy to see <math>u=2</math> (<math>t=56</math>) is the only candidate, giving <math>x=\boxed{\textbf{(B)} 11}</math>. -scrabbler94 | Starting from the above equation <math>\frac{t}{4}+\frac{2t}{7} + 15 + x = t</math> where <math>t</math> is the total number of points scored and <math>x\le 14</math> is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation <math>x+15 = \frac{13}{28}t</math>, or <math>28x+28\cdot 15=13t</math>. Since <math>t</math> is necessarily divisible by 28, let <math>t=28u</math> where <math>u \ge 0</math> and divide by 28 to obtain <math>x + 15 = 13u</math>. Then it is easy to see <math>u=2</math> (<math>t=56</math>) is the only candidate, giving <math>x=\boxed{\textbf{(B)} 11}</math>. -scrabbler94 | ||
− | ==Solution | + | ==Solution 2== |
We first start by setting the total number of points as <math>28</math>, since <math>\text{LCM}(4,7) = 28</math>. However, we see that this does not work since we surpass the number of points just with the information given (<math>28\cdot\frac{1}{4}+28\cdot\frac{2}{7} + 15 = 30</math> <math>(> 28)</math> ). Next, we can see that the total number of points scored is <math>56</math> as, if it is more than or equal to <math>84</math>, at least one of the others will score more than 2 points. With this, we have that Alexa, Brittany, and Chelsea score: <math>56\cdot\frac{1}{4}+56\cdot\frac{2}{7} + 15 = 45</math>, and thus, the other seven players would have scored a total of <math>56-45 = \boxed{\textbf{(B)} 11}</math> (We see that this works since we could have <math>4</math> of them score <math>2</math> points, and the other <math>3</math> of them score <math>1</math> point) -aops5234 -Edited by [[User: Penguin_Spellcaster|Penguin_Spellcaster]] | We first start by setting the total number of points as <math>28</math>, since <math>\text{LCM}(4,7) = 28</math>. However, we see that this does not work since we surpass the number of points just with the information given (<math>28\cdot\frac{1}{4}+28\cdot\frac{2}{7} + 15 = 30</math> <math>(> 28)</math> ). Next, we can see that the total number of points scored is <math>56</math> as, if it is more than or equal to <math>84</math>, at least one of the others will score more than 2 points. With this, we have that Alexa, Brittany, and Chelsea score: <math>56\cdot\frac{1}{4}+56\cdot\frac{2}{7} + 15 = 45</math>, and thus, the other seven players would have scored a total of <math>56-45 = \boxed{\textbf{(B)} 11}</math> (We see that this works since we could have <math>4</math> of them score <math>2</math> points, and the other <math>3</math> of them score <math>1</math> point) -aops5234 -Edited by [[User: Penguin_Spellcaster|Penguin_Spellcaster]] | ||
Revision as of 11:12, 30 April 2021
Contents
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
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 2
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 can see that the total number of points scored is
as, if it is more than or equal to
, at least one of the others will score more than 2 points. 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 -Edited by Penguin_Spellcaster
Solution 4 — Modular Arithmetic
Adding together Alexa's and Brittany's fractions, we get as the fraction of the total number of points they scored together. However, this is just a ratio, so we can introduce a variable:
where
is the common ratio. Let
and
and
be the number of people who scored 1, 2, and 0 points, respectively. Writing an equation, we have
We want all of our variables to be integers. Thus, we want
Simplifying,
The only possible value, as this integer sum has to be less than
must be 11. Therefore
and the answer is
- ab2024
Video explaining solution
https://www.youtube.com/watch?v=fKjmw_zzCUU - Happytwin
Associated video - https://www.youtube.com/watch?v=jE-7Se7ay1c
https://www.youtube.com/watch?v=3Mae_6qFxoU&t=204s ~ hi_im_bob
https://www.youtube.com/watch?v=o2mcnLOVFBA&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=5 ~ MathEx
Video Solution
https://youtu.be/HISL2-N5NVg?t=4115
~ pi_is_3.14
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.