https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Mixi2004&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T00:15:30ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_12B_Problems/Problem_12&diff=1477142021 AMC 12B Problems/Problem 122021-02-22T02:24:14Z<p>Mixi2004: /* Problem */</p>
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<div>==Problem==<br />
Suppose that <math>S</math> is a finite set of positive integers. If the greatest integer in <math>S</math> is removed from <math>S</math>, then the average value (arithmetic mean) of the integers remaining is <math>32</math>. If the least integer in <math>S</math> is also removed, then the average value of the integers remaining is <math>35</math>. If the greatest integer is then returned to the set, the average value of the integers rises to <math>40.</math> The greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>. What is the average value of all the integers in the set <math>S?</math><br />
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<math>\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37</math><br />
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==Solution 1==<br />
Let <math>x</math> be the greatest integer, <math>y</math> be the smallest, <math>z</math> be the sum of the numbers in S excluding <math>x</math> and <math>y</math>, and <math>k</math> be the number of elements in S.<br />
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Then, <math>S=x+y+z</math><br />
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Firstly, when the greatest integer is removed, <math>\frac{S-x}{k-1}=32</math><br />
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When the smallest integer is also removed, <math>\frac{S-x-y}{k-2}=35</math><br />
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When the greatest integer is added back, <math>\frac{S-y}{k-1}=40</math><br />
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We are given that <math>x=y+72</math><br />
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After you substitute <math>x=y+72</math>, you have 3 equations with 3 unknowns <math>S,</math>, <math>y</math> and <math>k</math>.<br />
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<math>S-y-72=32k-32</math><br />
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<math>S-2y-72=35k-70</math><br />
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<math>S-y=40k-40</math><br />
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This can be easily solved to yield <math>k=10</math>, <math>y=8</math>, <math>S=368</math>.<br />
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<math>\therefore</math> average value of all integers in the set <math>=S/k = 368/10 = 36.8</math>, D)<br />
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~ SoySoy4444<br />
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==Solution 2==<br />
We should plug in <math>36.2</math> and assume everything is true except the <math>35</math> part. We then calculate that part and end up with <math>35.75</math>. We also see with the formulas we used with the plug in that when you increase by <math>0.2</math> the <math>35.75</math> part decreases by <math>0.25</math>. The answer is then <math>\boxed{(D) 36.8}</math>. You can work backwards because it is multiple choice and you don't have to do critical thinking. ~Lopkiloinm<br />
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== Video Solution by OmegaLearn (System of equations) ==<br />
https://youtu.be/dRdT9gzm-Pg<br />
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~ pi_is_3.14<br />
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==Video Solution by Hawk Math==<br />
https://www.youtube.com/watch?v=p4iCAZRUESs<br />
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==Video Solution by TheBeautyofMath==<br />
https://youtu.be/FV9AnyERgJQ?t=676<br />
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~IceMatrix<br />
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==See Also==<br />
{{AMC12 box|year=2021|ab=B|num-b=11|num-a=13}}<br />
{{AMC10 box|year=2021|ab=B|num-b=18|num-a=20}}<br />
{{MAA Notice}}</div>Mixi2004