Difference between revisions of "2021 AMC 10B Problems/Problem 19"

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==Solution==
 
==Solution==
 
Let the lowest value be L and the highest G, and let the sum be Z and the amount of numbers n. 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>, which gives <math>n=10</math>. Turning the second into <math>Z-2L=35n+2</math> we see <math>L=8</math>  and <math>Z=368</math> so the average is <math>\frac{Z}{n}=\boxed{(D)36.8}</math> ~aop2014
 
Let the lowest value be L and the highest G, and let the sum be Z and the amount of numbers n. 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>, which gives <math>n=10</math>. Turning the second into <math>Z-2L=35n+2</math> we see <math>L=8</math>  and <math>Z=368</math> so the average is <math>\frac{Z}{n}=\boxed{(D)36.8}</math> ~aop2014
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==Solution 2==
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Let <math>S = {x_1, x_2, \dots, x_n}</math> in increasing order. There are <math>n</math> integers in this set.
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The greatest integer in <math>S</math> is <math>x_n</math>, so we need to remove this and get the average as <math>32</math>. There are <math>n-1</math> total terms. Thus, <cmath>\frac{x_1+x_2+\dots+x_{n-1}}{n-1} = 32</cmath><cmath>x_1+x_2+\dots+x_{n-1} = 32(n-1).</cmath>
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The least integer is <math>x_1.</math> Removing it, we have <cmath>x_2+\dots+x_{n-1} = 35(n-2).</cmath>
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Finally, adding back <math>x_n</math> we have <cmath>x_2+\dots+x_n = 40(n-1).</cmath>
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Since the greatest integer in the original set <math>S</math> is <math>72</math> greater than the least integer in <math>S</math>, we have <cmath>x_n = x_1 + 72.</cmath>
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Substituting this into the third equation, we have <math>x_2 + \dots + x_{n-1} + x_1 + 72 = 40(n-1).</math> Rearranging, this becomes <math>x_1 + \dots + x_{n-1} = 40(n-1) - 72.</math>
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But the LHS of this equation is the same as the LHS of the first equation! Equating them, we have <math>40(n-1)-72 = 32(n-1)</math>, which means <math>n = 10.</math>
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Let's plug <math>n = 10</math> into all of our previous equations. We have the following system:
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<cmath>x_1+x_2+\dots+x_9 = 288</cmath><cmath>x_2+\dots+x_9 = 280</cmath><cmath>x_2+\dots+x_{10} = 360</cmath>
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We're asked to find the average of <math>S</math>, which is <math>\frac{x_1+x_2+\dots+x_{10}}{10}.</math> Let's focus on the numerator. We can find <math>x_1</math> by subtracting the second equation from the first, to get <math>x_1 = 8.</math> We can also find <math>x_{10}</math> by subtracting the second equation from the third, to get <math>x_{10} = 80.</math>
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We also know <math>x_2+\dots+x_9</math> to equal <math>280</math>. So when we sum up our three values, we get <cmath>x_1 + (x_2 + \dots + x_9) + x_{10} = 8 + 280 + 80 = 368.</cmath>
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Therefore, the sum of the values is <math>368</math>, and finding the average in <math>10</math> terms we have <cmath>\frac{368}{10} = \boxed{(D)\text{ }36.8}.</cmath>
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-PureSwag

Revision as of 16:48, 11 February 2021

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 is $S$ is [i]also[/i] 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 of $40$. 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

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 $Z-L=32n+40$, which gives $n=10$. Turning the second into $Z-2L=35n+2$ we see $L=8$ and $Z=368$ so the average is $\frac{Z}{n}=\boxed{(D)36.8}$ ~aop2014

Solution 2

Let $S = {x_1, x_2, \dots, x_n}$ in increasing order. There are $n$ integers in this set.

The greatest integer in $S$ is $x_n$, so we need to remove this and get the average as $32$. There are $n-1$ total terms. Thus, \[\frac{x_1+x_2+\dots+x_{n-1}}{n-1} = 32\]\[x_1+x_2+\dots+x_{n-1} = 32(n-1).\]

The least integer is $x_1.$ Removing it, we have \[x_2+\dots+x_{n-1} = 35(n-2).\]

Finally, adding back $x_n$ we have \[x_2+\dots+x_n = 40(n-1).\]

Since the greatest integer in the original set $S$ is $72$ greater than the least integer in $S$, we have \[x_n = x_1 + 72.\]

Substituting this into the third equation, we have $x_2 + \dots + x_{n-1} + x_1 + 72 = 40(n-1).$ Rearranging, this becomes $x_1 + \dots + x_{n-1} = 40(n-1) - 72.$

But the LHS of this equation is the same as the LHS of the first equation! Equating them, we have $40(n-1)-72 = 32(n-1)$, which means $n = 10.$

Let's plug $n = 10$ into all of our previous equations. We have the following system: \[x_1+x_2+\dots+x_9 = 288\]\[x_2+\dots+x_9 = 280\]\[x_2+\dots+x_{10} = 360\]

We're asked to find the average of $S$, which is $\frac{x_1+x_2+\dots+x_{10}}{10}.$ Let's focus on the numerator. We can find $x_1$ by subtracting the second equation from the first, to get $x_1 = 8.$ We can also find $x_{10}$ by subtracting the second equation from the third, to get $x_{10} = 80.$

We also know $x_2+\dots+x_9$ to equal $280$. So when we sum up our three values, we get \[x_1 + (x_2 + \dots + x_9) + x_{10} = 8 + 280 + 80 = 368.\]

Therefore, the sum of the values is $368$, and finding the average in $10$ terms we have \[\frac{368}{10} = \boxed{(D)\text{ }36.8}.\]

-PureSwag

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