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Difference between revisions of "2021 AMC 12B Problems/Problem 12"

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{{duplicate|[[2021 AMC 10B Problems#Problem 19|2021 AMC 10B #19]] and [[2021 AMC 12B Problems#Problem 12|2021 AMC 12B #12]]}}
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==Problem==
 
==Problem==
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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 great 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>
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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>?
  
 
<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>
 
<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>
  
 
==Solution 1==
 
==Solution 1==
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Let the lowest value be <math>L</math> and the highest <math>G</math>, and let the sum be <math>Z</math> and the amount of numbers <math>n</math>. We have <math>\frac{Z-G}{n-1}=32</math>, <math>\frac{Z-L-G}{n-2}=35</math>, <math>\frac{Z-L}{n-1}=40</math>, and <math>G=L+72</math>. Clearing denominators gives <math>Z-G=32n-32</math>, <math>Z-L-G=35n-70</math>, and <math>Z-L=40n-40</math>. We use <math>G=L+72</math> to turn the first equation into <math>Z-L=32n+40</math>. Since <math>Z-L=40(n-1)</math> we substitute it into the equation which gives <math>n=10</math>. Turning the second into <math>Z-2L=35n+2</math> using <math>G=L+72</math> we see <math>L=8</math> and <math>Z=368</math> so the average is <math>\frac{Z}{n}=\boxed{\textbf{(D) }36.8}</math> ~aop2014
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==Solution 2==
 
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.
 
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.
  
 
Then, <math>S=x+y+z</math>
 
Then, <math>S=x+y+z</math>
  
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Firstly, when the greatest integer is removed, <math>\frac{S-x}{k-1}=32</math>
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First, when the greatest integer is removed, <math>\frac{S-x}{k-1}=32</math>
  
 
When the smallest integer is also removed, <math>\frac{S-x-y}{k-2}=35</math>
 
When the smallest integer is also removed, <math>\frac{S-x-y}{k-2}=35</math>
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This can be easily solved to yield <math>k=10</math>, <math>y=8</math>, <math>S=368</math>.
 
This can be easily solved to yield <math>k=10</math>, <math>y=8</math>, <math>S=368</math>.
  
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<math>\therefore</math> average value of all integers in the set <math>=S/k = 368/10 = 36.8</math>, D)
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<math>\therefore</math> average value of all integers in the set <math>=S/k = 368/10 = \boxed{\textbf{(D) }36.8}</math>
  
 
~ SoySoy4444
 
~ SoySoy4444
  
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==Solution 2==
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==Solution 3==
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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
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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{\textbf{(D) }36.8}</math>. You can work backwards because it is multiple choice and you don't have to do critical thinking. ~Lopkiloinm
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==Solution 4==
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Let <math>S = \{a_1, a_2, a_3, \hdots, a_n\}</math> with <math>a_1 < a_2 < a_3 < \hdots < a_n.</math> We are given the following: <cmath>{\begin{cases} \sum_{i=1}^{n-1} a_i = 32(n-1) = 32n-32, \\ \sum_{i=2}^n a_i = 40(n-1) = 40n-40, \\ \sum_{i=2}^{n-1} a_i = 35(n-2) = 35n-70, \\ a_n-a_1 = 72 \implies a_1 + 72 = a_n. \end{cases}}</cmath> Subtracting the third equation from the sum of the first two, we find that <cmath>\sum_{i=1}^n a_i = \left(32n-32\right) + \left(40n-40\right) - \left(35n-70\right) = 37n - 2.</cmath> Furthermore, from the fourth equation, we have <cmath>\sum_{i=2}^{n} a_i - \sum_{i=1}^{n-1} a_i = \left[\left(a_1 + 72\right) + \sum_{i=2}^{n-1} a_i\right] - \left[\left(a_1\right) + \sum_{i=2}^{n-1} a_i\right] = \left(40n-40\right)-\left(32n-32\right).</cmath> Combining like terms and simplifying, we have <cmath>72 = 8n-8 \implies 8n = 80 \implies n=10.</cmath> Thus, the sum of the elements in <math>S</math> is <math>37 \cdot 10 - 2 = 368,</math> and since there are 10 elements in <math>S,</math> the average of the elements in <math>S</math> is <math>\tfrac{368}{10}=\boxed{\textbf{(D) }36.8}</math>
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~peace09
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==Solution 5==
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Let <math>n</math> be the number of elements in <math>S, m_l = 32, \Sigma_l = m_l \cdot (n-1), m_g = 40,  \Sigma_g = m_g \cdot (n-1), m_{lg} = 35,  \Sigma_lg = m_{lg} \cdot (n-2).</math>
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<cmath>\Sigma_g - \Sigma_l = 72 = (n-1) \cdot (m_g - m_l) \implies n = 1 + \frac {72}{m_g - m_l} = 1 + \frac {72}{40-32}=10.</cmath>
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<cmath>m = \frac {\Sigma_g + \Sigma_l - \Sigma_{lg}}{n} = \frac {40 \cdot 9 + 32 \cdot 9 - 35 \cdot 8}{10} = 36.8.</cmath>
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'''vladimir.shelomovskii@gmail.com, vvsss'''
  
 
== Video Solution by OmegaLearn (System of equations) ==
 
== Video Solution by OmegaLearn (System of equations) ==
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~IceMatrix
 
~IceMatrix
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==Video Solution by Interstigation==
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https://youtu.be/TbHluJQoy8s
  
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~Interstigation
  
 
==See Also==
 
==See Also==

Latest revision as of 08:56, 20 May 2023

The following problem is from both the 2021 AMC 10B #19 and 2021 AMC 12B #12, so both problems redirect to this page.

Problem

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. What is the average value of all the integers in the set $S$?

$\textbf{(A) }36.2 \qquad \textbf{(B) }36.4 \qquad \textbf{(C) }36.6\qquad \textbf{(D) }36.8 \qquad \textbf{(E) }37$

Solution 1

Let the lowest value be $L$ and the highest $G$, and let the sum be $Z$ and the amount of numbers $n$. We have $\frac{Z-G}{n-1}=32$, $\frac{Z-L-G}{n-2}=35$, $\frac{Z-L}{n-1}=40$, and $G=L+72$. Clearing denominators gives $Z-G=32n-32$, $Z-L-G=35n-70$, and $Z-L=40n-40$. We use $G=L+72$ to turn the first equation into $Z-L=32n+40$. Since $Z-L=40(n-1)$ we substitute it into the equation which gives $n=10$. Turning the second into $Z-2L=35n+2$ using $G=L+72$ we see $L=8$ and $Z=368$ so the average is $\frac{Z}{n}=\boxed{\textbf{(D) }36.8}$ ~aop2014

Solution 2

Let $x$ be the greatest integer, $y$ be the smallest, $z$ be the sum of the numbers in S excluding $x$ and $y$, and $k$ be the number of elements in S.

Then, $S=x+y+z$

First, when the greatest integer is removed, $\frac{S-x}{k-1}=32$

When the smallest integer is also removed, $\frac{S-x-y}{k-2}=35$

When the greatest integer is added back, $\frac{S-y}{k-1}=40$

We are given that $x=y+72$


After you substitute $x=y+72$, you have 3 equations with 3 unknowns $S,$, $y$ and $k$.

$S-y-72=32k-32$

$S-2y-72=35k-70$

$S-y=40k-40$

This can be easily solved to yield $k=10$, $y=8$, $S=368$.

$\therefore$ average value of all integers in the set $=S/k = 368/10 = \boxed{\textbf{(D) }36.8}$

~ SoySoy4444

Solution 3

We should plug in $36.2$ and assume everything is true except the $35$ part. We then calculate that part and end up with $35.75$. We also see with the formulas we used with the plug in that when you increase by $0.2$ the $35.75$ part decreases by $0.25$. The answer is then $\boxed{\textbf{(D) }36.8}$. You can work backwards because it is multiple choice and you don't have to do critical thinking. ~Lopkiloinm

Solution 4

Let $S = \{a_1, a_2, a_3, \hdots, a_n\}$ with $a_1 < a_2 < a_3 < \hdots < a_n.$ We are given the following: \[{\begin{cases} \sum_{i=1}^{n-1} a_i = 32(n-1) = 32n-32, \\ \sum_{i=2}^n a_i = 40(n-1) = 40n-40, \\ \sum_{i=2}^{n-1} a_i = 35(n-2) = 35n-70, \\ a_n-a_1 = 72 \implies a_1 + 72 = a_n. \end{cases}}\] Subtracting the third equation from the sum of the first two, we find that \[\sum_{i=1}^n a_i = \left(32n-32\right) + \left(40n-40\right) - \left(35n-70\right) = 37n - 2.\] Furthermore, from the fourth equation, we have \[\sum_{i=2}^{n} a_i - \sum_{i=1}^{n-1} a_i = \left[\left(a_1 + 72\right) + \sum_{i=2}^{n-1} a_i\right] - \left[\left(a_1\right) + \sum_{i=2}^{n-1} a_i\right] = \left(40n-40\right)-\left(32n-32\right).\] Combining like terms and simplifying, we have \[72 = 8n-8 \implies 8n = 80 \implies n=10.\] Thus, the sum of the elements in $S$ is $37 \cdot 10 - 2 = 368,$ and since there are 10 elements in $S,$ the average of the elements in $S$ is $\tfrac{368}{10}=\boxed{\textbf{(D) }36.8}$

~peace09

Solution 5

Let $n$ be the number of elements in $S, m_l = 32, \Sigma_l = m_l \cdot (n-1), m_g = 40, \Sigma_g = m_g \cdot (n-1), m_{lg} = 35, \Sigma_lg = m_{lg} \cdot (n-2).$ \[\Sigma_g - \Sigma_l = 72 = (n-1) \cdot (m_g - m_l) \implies n = 1 + \frac {72}{m_g - m_l} = 1 + \frac {72}{40-32}=10.\] \[m = \frac {\Sigma_g + \Sigma_l - \Sigma_{lg}}{n} = \frac {40 \cdot 9 + 32 \cdot 9 - 35 \cdot 8}{10} = 36.8.\] vladimir.shelomovskii@gmail.com, vvsss

Video Solution by OmegaLearn (System of equations)

https://youtu.be/dRdT9gzm-Pg

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=p4iCAZRUESs

Video Solution by TheBeautyofMath

https://youtu.be/FV9AnyERgJQ?t=676

~IceMatrix

Video Solution by Interstigation

https://youtu.be/TbHluJQoy8s

~Interstigation

See Also

2021 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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
2021 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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. AMC logo.png

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