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Difference between revisions of "2012 AMC 10A Problems/Problem 13"

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{{duplicate|[[2012 AMC 12A Problems|2012 AMC 12A #8]] and [[2012 AMC 10A Problems|2012 AMC 10A #13]]}}
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== Problem ==
 
== Problem ==
  
 
An ''iterative average'' of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
 
An ''iterative average'' of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
  
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<math> \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{4}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} </math>
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<math> \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} </math>
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== Solution 1 ==
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The iterative average of any 5 integers <math>a,b,c,d,e</math> is defined as:
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<cmath>\frac{\frac{\frac{\frac{a+b} 2+c} 2+d} 2+e} 2=\frac{a+b+2c+4d+8e}{16}</cmath>
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Plugging in <math>1,2,3,4,5</math> for <math>a,b,c,d,e</math>, we see that in order to maximize the fraction,
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<math>a=1,b=2,c=3,d=4,e=5</math>,
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and in order to minimize the fraction,
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<math>a=5,b=4,c=3,d=2,e=1</math>.
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After plugging in these values and finding the positive difference of the two fractions, we arrive with <math>\frac{34}{16} \Rightarrow \frac{17}{8}</math>, which is our answer of <math>\boxed{\textbf{(C)}}</math>
  
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== Solution ==
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== Solution 2 ==
  
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The minimum and maximum can be achieved with the orders <math>5, 4, 3, 2, 1</math> and <math>1, 2, 3, 4, 5</math>.
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The minimum and maximum can be achieved with the orders <math>5, 4, 3, 2, 1</math> and <math>1, 2, 3, 4, 5</math> respectively. We can see this because the iterative average is like a weighted average that gives more weight to later numbers.  
  
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<math>5,4,3,2,1 \Rightarrow \frac92,3,2,1 \Rightarrow \frac{15}{4},2,1 \Rightarrow \frac{23}{8},1 \Rightarrow \frac{31}{8}</math>
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<math>5,4,3,2,1 \Rightarrow \frac92,3,2,1 \Rightarrow \frac{15}{4},2,1 \Rightarrow \frac{23}{8},1 \Rightarrow \frac{31}{16}</math>
  
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<math>1,2,3,4,5 \Rightarrow \frac32,3,4,5 \Rightarrow \frac94,4,5 \Rightarrow \frac{25}{8},5 \Rightarrow \frac{65}{8}</math>
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<math>1,2,3,4,5 \Rightarrow \frac32,3,4,5 \Rightarrow \frac94,4,5 \Rightarrow \frac{25}{8},5 \Rightarrow \frac{65}{16}</math>
  
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The difference between the two is <math>\frac{65}{8}-\frac{31}{8}=\frac{34}{8}=\boxed{\textbf{(C)}\ \frac{17}{4}}</math>.
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The difference between the two is <math>\frac{65}{16}-\frac{31}{16}=\frac{34}{16}=\boxed{\textbf{(C)}\ \frac{17}{8}}</math>.
  
 
== See Also ==
 
== See Also ==
  
 
{{AMC10 box|year=2012|ab=A|num-b=12|num-a=14}}
 
{{AMC10 box|year=2012|ab=A|num-b=12|num-a=14}}
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{{AMC12 box|year=2012|ab=A|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 00:58, 16 February 2021

The following problem is from both the 2012 AMC 12A #8 and 2012 AMC 10A #13, so both problems redirect to this page.

Problem

An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?

$\textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16}$

Solution 1

The iterative average of any 5 integers $a,b,c,d,e$ is defined as:

\[\frac{\frac{\frac{\frac{a+b} 2+c} 2+d} 2+e} 2=\frac{a+b+2c+4d+8e}{16}\]


Plugging in $1,2,3,4,5$ for $a,b,c,d,e$, we see that in order to maximize the fraction,

$a=1,b=2,c=3,d=4,e=5$,

and in order to minimize the fraction,

$a=5,b=4,c=3,d=2,e=1$.

After plugging in these values and finding the positive difference of the two fractions, we arrive with $\frac{34}{16} \Rightarrow \frac{17}{8}$, which is our answer of $\boxed{\textbf{(C)}}$

Solution 2

The minimum and maximum can be achieved with the orders $5, 4, 3, 2, 1$ and $1, 2, 3, 4, 5$ respectively. We can see this because the iterative average is like a weighted average that gives more weight to later numbers.

$5,4,3,2,1 \Rightarrow \frac92,3,2,1 \Rightarrow \frac{15}{4},2,1 \Rightarrow \frac{23}{8},1 \Rightarrow \frac{31}{16}$

$1,2,3,4,5 \Rightarrow \frac32,3,4,5 \Rightarrow \frac94,4,5 \Rightarrow \frac{25}{8},5 \Rightarrow \frac{65}{16}$

The difference between the two is $\frac{65}{16}-\frac{31}{16}=\frac{34}{16}=\boxed{\textbf{(C)}\ \frac{17}{8}}$.

See Also

2012 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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
2012 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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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