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Difference between revisions of "2002 AMC 12A Problems/Problem 15"

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If the largest one is <math>16</math>, then the smallest one is <math>8</math>, and thus the mean is strictly larger than <math>8</math>, which is a contradiction.
 
If the largest one is <math>16</math>, then the smallest one is <math>8</math>, and thus the mean is strictly larger than <math>8</math>, which is a contradiction.
  
−
If the largest one is <math>15</math>, then the smallest one is <math>7</math>. This means that we already know four of the values: <math>8</math>, <math>8</math>, <math>7</math>, <math>15</math>. Since the mean of all the numbers is <math>8</math>, their sum must be <math>64</math>. Thus the sum of the missing four numbers is <math>64-8-8-7-15=26</math>. But if <math>7</math> is the smallest number, then the sum of the missing numbers must be at least <math>4\cdot 7=28</math>, which is again a contradiction.
+
If we have 2 8's we can add find the numbers 4, 6, 7, 8, 8, 9, 10, 12.
 +
This is a possible solution but has not reached the maximum.
  
−
If the largest number is <math>14</math>, we can easily find the solution <math>(6,6,6,8,8,8,8,14)</math>. Hence, our answer is <math>\boxed{\text{(D)}\ 14 }</math>.
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If we have 4 8's we can find the numbers 6, 6, 6, 8, 8, 8, 8, 14.
  
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===Note===
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We can also see that they satisfy the need for the mode, median, and range to be 8. This means that the answer will be
 +
<math>\boxed{\text{(D)}\ 14 }</math>. ~By QWERTYUIOPASDFGHJKLZXCVBNM
  
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The solution for <math>14</math> is, in fact, unique. As the median must be <math>8</math>, this means that both the <math>4^\text{th}</math> and the <math>5^\text{th}</math> number, when ordered by size, must be <math>8</math>s. This gives the partial solution <math>(6,a,b,8,8,c,d,14)</math>. For the mean to be <math>8</math> each missing variable must be replaced by the smallest allowed value.
+
== Solution 2 ==
−
The solution that works is <math>(6,6,6,8,8,8,8,14)</math>
 
  
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==Solution 2 ==
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We could express this collection as integers <math>a\textsubscript{1}</math> through <math>a\textsubscript{8}</math>, with <math>a\textsubscript{1}</math> being the smallest and <math>a\textsubscript{8}</math> being the largest.
−
Let the 8 numbers be <math>a, b, c, d, e, f, g, h</math>, arranged in increasing order. Since the range of the eight numbers is 8, <math>h=a+8</math>.
 
  
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I claim that <math>d</math>, <math>e</math> must both be <math>8</math>. Since the median is 8, the mean of <math>d</math> and <math>e</math> must be 8. Let's assume that <math>d</math> and <math>e</math> aren't <math>8</math>. The mode of the collection is <math>8</math>, and if <math>d</math> and <math>e</math> aren't, then <math>8</math> must be between <math>d</math> and <math>e</math> (i.e. not in the collection). This is a contradiction, so <math>d</math> and <math>e</math> have to be 8.
+
Since the mode is <math>8</math>, we know that <math>a\textsubscript{4}</math> and <math>a\textsubscript{5}</math> must also be <math>8</math>. If they were not, the other numbers, which are lesser and greater than <math>a\textsubscript{4}</math> and <math>a\textsubscript{5}</math> respectively, would not be able to satisfy the condition that <math>8</math> is the mode.
  
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Now, we have the eight numbers are <math>a, b, c, 8, 8, f, g, a+8</math>.
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There are <math>8</math> terms and the mean is <math>8</math>. This tells us that the sum of all the numbers is <math>64</math>.
  
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Since the mean is <math>8</math>, we have <math>a+b+c+8+8+f+g+a+8=8 \times 8 = 64</math>, giving us <math>2a+b+c+f+g=40</math>.
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We want to maximize the value of <math>a\textsubscript{8}</math>, so we set  <math>a\textsubscript{6}</math> and <math>a\textsubscript{7}</math> to <math>8</math> as well.
  
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Since <math>f, g \ge 8</math>, <math>f+g \ge 16</math>. Plugging that in, we have <math>2a+b+c \le 24</math>. Note that we can't do the same for <math>b</math> and <math>c</math>, but we can do <math>b+c \ge 2a</math>, giving us <math>4a \le 24</math>, which means <math>a\le 6</math>.
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Knowing that we want to minimize numbers and that the range is <math>8</math> , we set <math>a\textsubscript{1}</math>, <math>a\textsubscript{2}</math>, and <math>a\textsubscript{3}</math> equal to <math>a\textsubscript{8} - 8</math>.
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We want to find <math>h=a+8\le 6+8=14</math>
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This is our answer, so <math>\boxed{(D)14}</math>
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{<math>a\textsubscript{1}</math>, <math>a\textsubscript{2}</math>, <math>a\textsubscript{3}</math>, <math>a\textsubscript{4}</math>, <math>a\textsubscript{5}</math>, <math>a\textsubscript{6}</math>, <math>a\textsubscript{7}</math>, <math>a\textsubscript{8}</math>} <math>=</math> {<math>a\textsubscript{8} - 8</math>, <math>a\textsubscript{8} - 8</math>, <math>a\textsubscript{8} - 8</math>, <math>8</math>, <math>8</math>, <math>8</math>, <math>8</math>, <math>a\textsubscript{8}</math>}
 +
 
 +
Letting the sum of all the numbers be <math>64</math>, we find that <math>32 + 4a_8 - 24 = 64</math>, which simplifies to <math>4a_8 = 56</math>. Solving, we get <math>\boxed{\text{(D)}\ 14 }</math>. ~By SK80, mod_x and huniiya for minor edits
 +
 
 +
== Solution 3 ==
 +
 
 +
Let's say that we choose <math>k</math> numbers to be 8, and the largest number to be <math>L</math>. There will therefore be <math>7-k</math> numbers with the value of <math>L - 8</math>.
 +
 
 +
This gives the following expression <math>\frac{(7-k)(L-8)+ 8k + L}{8} = 8</math>. Solving for <math>k</math> gives <math>\frac{120-8L}{16-L} = k</math>. The bank of choices we have for <math>L</math> is represented by answer choices (A), (B), (C), (D), and (E).
 +
 
 +
We find that the value for <math>14</math> gives <math>k = 4</math> creating the set <math>666-8888-14</math>, which satisfies all conditions.
 +
 
 +
This gives <math>\boxed{\text{(D)}\ 14 }</math>.
 +
 
 +
== Solution 4 ==
 +
 
 +
Since the mean of the numbers is <math>8</math> and there are <math>8</math> numbers, the sum of all <math>8</math> numbers is <math>64</math>. Since the range is also <math>8</math>, let the smallest value in the set <math>= x</math>. Therefore the largest number in the set is <math>x+8</math>. The mode is the number which appears most often, so we can let the <math>4</math>th to <math>7</math>th largest integers <math>= 8</math>. To maximize the largest value, we will let the <math>2</math>nd and <math>3</math>rd smallest integers also equal <math>x</math>.
 +
 
 +
In order this will be: <math>x, x, x, 8, 8, 8, 8, x+8</math>
 +
 
 +
Now we can set the expression equal to <math>64</math> (because of what we determined in the <math>1</math>st step): <math>x+x+x+8+8+8+8+x+8=64</math>
 +
 
 +
Simplyifying,
 +
 
 +
<math>4x+40=64</math>
 +
 
 +
<math>4x=24</math>
 +
 
 +
<math>x=6</math>
 +
 
 +
Therefore, since <math>x</math> is the smallest number in the set, the largest number in the set will be <math>x+8</math>, which is <math>\boxed{\text{(D)}\ 14 }</math>.
 +
 
 +
~goofytaipan
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/xqo0PgH-h8Y?t=848
 +
 
 +
~ pi_is_3.14
 +
 
 +
== Video Solution ==
 +
https://youtu.be/BNa1Ffu517I
 +
 
 +
https://www.youtube.com/watch?v=Sic_pM_cmrA  ~David
  
 
== See Also ==
 
== See Also ==

Latest revision as of 10:35, 3 October 2024

The following problem is from both the 2002 AMC 12A #15 and 2002 AMC 10A #21, so both problems redirect to this page.

Problem

The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is

$\text{(A) }11 \qquad \text{(B) }12 \qquad \text{(C) }13 \qquad \text{(D) }14 \qquad \text{(E) }15$

Solution 1

As the unique mode is $8$, there are at least two $8$s.

As the range is $8$ and one of the numbers is $8$, the largest one can be at most $16$.

If the largest one is $16$, then the smallest one is $8$, and thus the mean is strictly larger than $8$, which is a contradiction.

If we have 2 8's we can add find the numbers 4, 6, 7, 8, 8, 9, 10, 12. This is a possible solution but has not reached the maximum.

If we have 4 8's we can find the numbers 6, 6, 6, 8, 8, 8, 8, 14.

We can also see that they satisfy the need for the mode, median, and range to be 8. This means that the answer will be $\boxed{\text{(D)}\ 14 }$. ~By QWERTYUIOPASDFGHJKLZXCVBNM

Solution 2

We could express this collection as integers $a\textsubscript{1}$ through $a\textsubscript{8}$, with $a\textsubscript{1}$ being the smallest and $a\textsubscript{8}$ being the largest.

Since the mode is $8$, we know that $a\textsubscript{4}$ and $a\textsubscript{5}$ must also be $8$. If they were not, the other numbers, which are lesser and greater than $a\textsubscript{4}$ and $a\textsubscript{5}$ respectively, would not be able to satisfy the condition that $8$ is the mode.

There are $8$ terms and the mean is $8$. This tells us that the sum of all the numbers is $64$.

We want to maximize the value of $a\textsubscript{8}$, so we set $a\textsubscript{6}$ and $a\textsubscript{7}$ to $8$ as well.

Knowing that we want to minimize numbers and that the range is $8$ , we set $a\textsubscript{1}$, $a\textsubscript{2}$, and $a\textsubscript{3}$ equal to $a\textsubscript{8} - 8$.

{$a\textsubscript{1}$, $a\textsubscript{2}$, $a\textsubscript{3}$, $a\textsubscript{4}$, $a\textsubscript{5}$, $a\textsubscript{6}$, $a\textsubscript{7}$, $a\textsubscript{8}$} $=$ {$a\textsubscript{8} - 8$, $a\textsubscript{8} - 8$, $a\textsubscript{8} - 8$, $8$, $8$, $8$, $8$, $a\textsubscript{8}$}

Letting the sum of all the numbers be $64$, we find that $32 + 4a_8 - 24 = 64$, which simplifies to $4a_8 = 56$. Solving, we get $\boxed{\text{(D)}\ 14 }$. ~By SK80, mod_x and huniiya for minor edits

Solution 3

Let's say that we choose $k$ numbers to be 8, and the largest number to be $L$. There will therefore be $7-k$ numbers with the value of $L - 8$.

This gives the following expression $\frac{(7-k)(L-8)+ 8k + L}{8} = 8$. Solving for $k$ gives $\frac{120-8L}{16-L} = k$. The bank of choices we have for $L$ is represented by answer choices (A), (B), (C), (D), and (E).

We find that the value for $14$ gives $k = 4$ creating the set $666-8888-14$, which satisfies all conditions.

This gives $\boxed{\text{(D)}\ 14 }$.

Solution 4

Since the mean of the numbers is $8$ and there are $8$ numbers, the sum of all $8$ numbers is $64$. Since the range is also $8$, let the smallest value in the set $= x$. Therefore the largest number in the set is $x+8$. The mode is the number which appears most often, so we can let the $4$th to $7$th largest integers $= 8$. To maximize the largest value, we will let the $2$nd and $3$rd smallest integers also equal $x$.

In order this will be: $x, x, x, 8, 8, 8, 8, x+8$

Now we can set the expression equal to $64$ (because of what we determined in the $1$st step): $x+x+x+8+8+8+8+x+8=64$

Simplyifying,

$4x+40=64$

$4x=24$

$x=6$

Therefore, since $x$ is the smallest number in the set, the largest number in the set will be $x+8$, which is $\boxed{\text{(D)}\ 14 }$.

~goofytaipan

Video Solution by OmegaLearn

https://youtu.be/xqo0PgH-h8Y?t=848

~ pi_is_3.14

Video Solution

https://youtu.be/BNa1Ffu517I

https://www.youtube.com/watch?v=Sic_pM_cmrA ~David

See Also

2002 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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
2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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

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