Difference between revisions of "2019 AMC 8 Problems/Problem 23"
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==Solution 1== | ==Solution 1== | ||
− | + | 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 | ||
==Solution 2== | ==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]] | |
− | |||
− | ==Solution | + | ==Solution 3 — Modular Arithmetic == |
− | Adding together Alexa's and Brittany's fractions, we get <math>\frac{15}{28}</math> as the fraction of the total number of points they scored together. However, this is just a ratio, so we can introduce a variable: <math>\frac{15x}{28x}</math> where <math>x</math> is the common ratio. Let <math>y</math> and <math>z</math> and <math>w</math> be the number of people who scored 1, 2, and 0 points, respectively. Writing an equation, we have <math>\frac{13x}{28x} = 15 + y + 2z + 0w.</math> We want all of our variables to be integers. Thus, we want <math>15 + y + 2z = 0 \pmod {13}.</math> Simplifying, <math>y+2z = 11 \pmod {13}.</math> The only possible value, as this integer sum has to be less than <math>7 \cdot 2 + 1 = 15,</math> must be 11. Therefore <math>y+2z = 11,</math> and the answer is <math>\boxed{ \textbf{(B) 11}}</math> | + | Adding together Alexa's and Brittany's fractions, we get <math>\frac{15}{28}</math> as the fraction of the total number of points they scored together. However, this is just a ratio, so we can introduce a variable: <math>\frac{15x}{28x}</math> where <math>x</math> is the common ratio. Let <math>y</math> and <math>z</math> and <math>w</math> be the number of people who scored 1, 2, and 0 points, respectively. Writing an equation, we have <math>\frac{13x}{28x} = 15 + y + 2z + 0w.</math> We want all of our variables to be integers. Thus, we want <math>15 + y + 2z = 0 \pmod {13}.</math> Simplifying, <math>y+2z = 11 \pmod {13}.</math> The only possible value, as this integer sum has to be less than <math>7 \cdot 2 + 1 = 15,</math> must be 11. Therefore, <math>y+2z = 11,</math> and the answer is <math>\boxed{ \textbf{(B) 11}}</math>. |
− | + | - ab2024 | |
− | https://www.youtube.com/watch?v=3Mae_6qFxoU&t=204s ~ hi_im_bob | + | ==Solution 4: Answer choices== |
+ | We can rewrite the question as an algebraic equation: <math>\frac{1}{4} x + \frac{2}{7} x + 15 + y</math>, where <math>x</math> represents the total amount of points and <math>y</math> the amount of points the <math>7</math> other players scored. From there, we add the two fractions to get <math>\frac{15}{28} x + 15 = x</math>. Subtracting <math>\frac{15}{28} x</math> from both sides, we get <math>\frac{13}{28} x = y + 15</math>. We multiply each side by <math>28</math> to get rid of the denominator, in which we get <math>13x = 420 + 28y</math>. Now let’s think of this logically. This equation is telling us that if you add <math>420</math> and <math>28</math> times the amount of points scored by the extra <math>7</math> players, you get <math>13</math> times the amount of points total. And since we have to have a whole number of points total, this means that <math>420 + 28y</math> must be divisible by <math>13</math>. Plugging in all the answer choices for <math>y</math>, we find that the only answer that makes <math>420 + 28y</math> divisible by <math>13</math> is <math>\boxed{ \textbf{(B) 11}}</math>. | ||
+ | |||
+ | ~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://youtu.be/wsYCn2FqZJE | https://youtu.be/wsYCn2FqZJE | ||
− | https://www.youtube.com/watch?v= | + | https://www.youtube.com/watch?v=o2mcnLOVFBA&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=5 |
+ | |||
+ | ~ MathEx | ||
+ | |||
+ | https://youtu.be/HISL2-N5NVg?t=4115 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | https://youtu.be/dI8RzUHLqZc | ||
+ | |||
+ | ~savannahsolver | ||
− | https:// | + | ==Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)== |
+ | https://youtu.be/Xm4ZGND9WoY | ||
− | + | ~Hayabusa1 | |
==See Also== | ==See Also== |
Latest revision as of 12:36, 26 March 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.