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Difference between revisions of "2018 AMC 12A Problems/Problem 1"
 
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A large urn contains <math>100</math> balls, of which <math>36 \%</math> are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be <math>72 \%</math>? (No red balls are to be removed.)
 
A large urn contains <math>100</math> balls, of which <math>36 \%</math> are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be <math>72 \%</math>? (No red balls are to be removed.)
  
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<math> \textbf{(A)}\ 28 \qquad\textbf{(B)}\  32 \qquad\textbf{(C)}\  36 \qquad\textbf{(D)}\  
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<math> \textbf{(A)}\ 28 \qquad\textbf{(B)}\  32 \qquad\textbf{(C)}\  36 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 64 </math>
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50 \qquad\textbf{(E)}\ 64 </math>
 
  
 
== Solution 1 ==
 
== Solution 1 ==
  
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There are <math>36</math> red balls; for these red balls to comprise <math>72 \%</math> of the urn, there must be only <math>14</math> blue balls. Since there are currently <math>64</math> blue balls, this means we must remove <math>50 = \boxed{ \textbf{(D)}.}</math>
+
There are <math>36</math> red balls; for these red balls to comprise <math>72 \%</math> of the urn, there must be only <math>14</math> blue balls. Since there are currently <math>64</math> blue balls, this means we must remove <math>\boxed{\textbf{(D)}\ 50}</math>.
  
 
== Solution 2 ==
 
== Solution 2 ==
  
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There are <math>36</math> red balls and <math>64</math> blue balls. For the percentage of the red balls to double from <math>36 \%</math> to <math>72 \%</math> of the urn, half of the total number of balls must be removed. Therefore, the number of blue balls that need to be removed is <math>50 = \boxed{ \textbf{(D)}}</math>
+
There are <math>36</math> red balls and <math>64</math> blue balls. For the percentage of the red balls to double from <math>36 \%</math> to <math>72 \%</math> of the urn, half of the total number of balls must be removed. Therefore, the number of blue balls that need to be removed is <math>\boxed{\textbf{(D)}\ 50}</math>.
  
 
== Solution 3 ==
 
== Solution 3 ==
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We want to continuously remove blue balls until the percentage of red balls in the urn is 72%.
 
We want to continuously remove blue balls until the percentage of red balls in the urn is 72%.
 
Therefore, we want
 
Therefore, we want
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<math>\frac{36}{100-x}=\frac{72}{100}</math>
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<cmath>\frac{36}{100-x}=\frac{72}{100}.</cmath>
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Solving for <math>x</math> gives that we must remove <math>\boxed{50}</math> blue balls.
+
Solving for <math>x</math> gives that we must remove <math>\boxed{\textbf{(D)}\ 50}</math> blue balls.
 +
 
 +
== Video Solution 1 ==
 +
https://youtu.be/u26afAqgTKw
 +
 
 +
~Education, the Study of Everything
  
 
== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2018|ab=A|before = First Problem|num-a=2}}
 
{{AMC12 box|year=2018|ab=A|before = First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:57, 28 October 2022

Problem

A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.)

$\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 64$

Solution 1

There are $36$ red balls; for these red balls to comprise $72 \%$ of the urn, there must be only $14$ blue balls. Since there are currently $64$ blue balls, this means we must remove $\boxed{\textbf{(D)}\ 50}$.

Solution 2

There are $36$ red balls and $64$ blue balls. For the percentage of the red balls to double from $36 \%$ to $72 \%$ of the urn, half of the total number of balls must be removed. Therefore, the number of blue balls that need to be removed is $\boxed{\textbf{(D)}\ 50}$.

Solution 3

There are $36$ red balls out of the total $100$ balls. We want to continuously remove blue balls until the percentage of red balls in the urn is 72%. Therefore, we want \[\frac{36}{100-x}=\frac{72}{100}.\] Solving for $x$ gives that we must remove $\boxed{\textbf{(D)}\ 50}$ blue balls.

Video Solution 1

https://youtu.be/u26afAqgTKw

~Education, the Study of Everything

See Also

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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 12 Problems and Solutions

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