Difference between revisions of "2018 AMC 10A Problems/Problem 6"
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== Problem == | == Problem == | ||
− | Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of 0, and the score increases by 1 for each like vote and decreases by 1 for each dislike vote. At one point Sangho saw that his video had a score of 90, and that <math>65\%</math> of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? | + | Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of <math>0</math>, and the score increases by <math>1</math> for each like vote and decreases by <math>1</math> for each dislike vote. At one point Sangho saw that his video had a score of <math>90</math>, and that <math>65\%</math> of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point? |
<math>\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 </math> | <math>\textbf{(A) } 200 \qquad \textbf{(B) } 300 \qquad \textbf{(C) } 400 \qquad \textbf{(D) } 500 \qquad \textbf{(E) } 600 </math> | ||
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==Solution 3== | ==Solution 3== | ||
− | Let <math>x</math> be the amount of votes cast, <math>35\%</math> of <math>x</math> would be dislikes and <math>65\%</math> of <math>x</math> would be likes. Since a like earns the video 1 point and a dislike takes 1 point away from the video, the amount of points Sangho's video will have in terms of <math>x</math> | + | Let <math>x</math> be the amount of votes cast, <math>35\%</math> of <math>x</math> would be dislikes and <math>65\%</math> of <math>x</math> would be likes. Since a like earns the video 1 point and a dislike takes 1 point away from the video, the amount of points Sangho's video will have in terms of <math>x</math> is <math>65\%-35\%=30\%</math> of <math>x</math>, or <math>\frac{3}{10}</math> of <math>x</math>. Because Songho's video had a score of 90 points, <math>\frac{3x}{10} = 90</math>. After we solve for <math>x</math>, we get <math>x = \boxed{\textbf{(B) } 300}</math>. |
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+ | -Mathandski | ||
==Solution 4== | ==Solution 4== | ||
− | + | Let x be the amount of like votes and y be the amount of dislike votes. Then <math>x-y=90</math>. Since <math>65\%</math> of the total votes are like votes, | |
<math> 0.65(x+y)=x</math>. then <math>.65y = .35x</math> or <math>13/7</math>y = <math>x</math>. Plugging that in to the original equation, <math> 6/7 y</math> = <math>90</math> and <math> y =105</math> | <math> 0.65(x+y)=x</math>. then <math>.65y = .35x</math> or <math>13/7</math>y = <math>x</math>. Plugging that in to the original equation, <math> 6/7 y</math> = <math>90</math> and <math> y =105</math> | ||
− | Then <math>x=195</math> and total votes is <math> 105+195 = 300</math>. The answer is B. | + | Then <math>x=195</math> and the total number of votes is <math> 105+195 = 300</math>. The answer is <math>\boxed{\textbf{(B) } 300}</math>. |
-goldenn | -goldenn | ||
+ | |||
+ | ==Solution 5== | ||
+ | Obviously, <math>90/(65\%-35\%) = 90/30\% = 300</math>. The answer is <math>\boxed{\textbf{(B) } 300}</math>. | ||
+ | |||
+ | (Note from <B+: please explain why this solution starts with the word "obviously." May the author or some other editor explain the "obviously" preceding the fraction which is the answer.) | ||
+ | |||
+ | ==Video Solution (HOW TO THINK CREATIVELY!)== | ||
+ | https://youtu.be/JiTC-7LbxpE | ||
+ | |||
+ | Education, the Study of Everything | ||
+ | |||
+ | |||
+ | |||
+ | ==Video Solutions== | ||
+ | https://youtu.be/ZiZVIMmo260 | ||
+ | |||
+ | https://youtu.be/ry6Bg3qTq8k | ||
+ | |||
+ | ~savannahsolver | ||
== See Also == | == See Also == |
Latest revision as of 10:31, 18 August 2024
Contents
Problem
Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of , and the score increases by
for each like vote and decreases by
for each dislike vote. At one point Sangho saw that his video had a score of
, and that
of the votes cast on his video were like votes. How many votes had been cast on Sangho's video at that point?
Solution 1
If of the votes were likes, then
of the votes were dislikes.
, so
votes is
of the total number of votes. Doing quick arithmetic shows that the answer is
.
Solution 2
Let's consider that Sangho has received 100 votes. This means he has received 65 upvotes and 35 downvotes. Part of these upvotes and downvotes cancel out, so Sangho is now left with a total of 30 upvotes, or a score increase of 30. In order for his score to be 90, Sangho must receive three sets of 100 votes. Therefore, the answer is .
-tryanotherangle
Solution 3
Let be the amount of votes cast,
of
would be dislikes and
of
would be likes. Since a like earns the video 1 point and a dislike takes 1 point away from the video, the amount of points Sangho's video will have in terms of
is
of
, or
of
. Because Songho's video had a score of 90 points,
. After we solve for
, we get
.
-Mathandski
Solution 4
Let x be the amount of like votes and y be the amount of dislike votes. Then . Since
of the total votes are like votes,
. then
or
y =
. Plugging that in to the original equation,
=
and
Then
and the total number of votes is
. The answer is
.
-goldenn
Solution 5
Obviously, . The answer is
.
(Note from <B+: please explain why this solution starts with the word "obviously." May the author or some other editor explain the "obviously" preceding the fraction which is the answer.)
Video Solution (HOW TO THINK CREATIVELY!)
Education, the Study of Everything
Video Solutions
~savannahsolver
See Also
2018 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.