Difference between revisions of "2019 AMC 8 Problems/Problem 23"
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Given the information above, we start with the 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 remaining, giving <math>x=\boxed{\textbf{(B)} 11}</math>. | Given the information above, we start with the 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 remaining, giving <math>x=\boxed{\textbf{(B)} 11}</math>. | ||
| − | -scrabbler94 | + | -scrabbler94 |
==Solution 2== | ==Solution 2== | ||
Revision as of 16:23, 4 January 2024
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
Given the information above, we start with the 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 remaining, 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 3 — 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
Solution 4: Answer choices
We can rewrite the question as an algebraic equation: 
, where represents the total amount of points and the amount of points the other players scored. From there, we add the two fractions to get 
. Subtracting 
from both sides, we get 
. We multiply each side by to get rid of the denominator, in which we get 
. Now let’s think of this logically. This equation is telling us that if you add 
and times the amount of points scored by the extra players, you get 
times the amount of points total. And since we have to have a whole number of points total, this means that 
must be divisible by . Plugging in all the answer choices for , we find that the only answer that makes divisible by is .
~ilee0820
Videos Explaining Solution
Video Solution by Math-X (Learn to do this under a minute!!!)
https://youtu.be/IgpayYB48C4?si=JjQHbrlBpeox9TFq&t=7063
~Math-X
https://www.youtube.com/watch?v=fKjmw_zzCUU
- Happytwin
https://www.youtube.com/watch?v=jE-7Se7ay1c
Associated video
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
https://youtu.be/HISL2-N5NVg?t=4115
~ pi_is_3.14
~savannahsolver
Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)
~Hayabusa1
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. 
