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Difference between revisions of "2020 AMC 8 Problems/Problem 20"

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==Problem 20==
 
A scientist walking through a forest recorded as integers the heights of <math>5</math> trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
 
A scientist walking through a forest recorded as integers the heights of <math>5</math> trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
  
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\end{tabular}
 
\end{tabular}
 
\endgroup</cmath>
 
\endgroup</cmath>
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<math>\textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2</math>
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==Solution 1==
 
==Solution 1==
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It is not too hard to construct <math>22, 11, 22, 44, 22</math> as the heights of the trees from left to right. The average is <math>\frac{121}{5}=24.2\rightarrow\boxed{\textbf{(B)}}</math>. ~icematrix Note: this is the result of realizing that the 1st one must be greater then the 2nd, <math>\frac{121}{5}=24.2 or {\textbf{(B)}}</math>. ~oceanxia
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We will show that <math>22</math>, <math>11</math>, <math>22</math>, <math>44</math>, and <math>22</math> meters are the heights of the trees from left to right. We are given that all tree heights are integers, so since Tree 2 has height <math>11</math> meters, we can deduce that Trees 1 and 3 both have a height of <math>22</math> meters. There are now three possible cases for the heights of Trees 4 and 5 (in order for them to be integers), namely heights of <math>11</math> and <math>22</math>, <math>44</math> and <math>88</math>, or <math>44</math> and <math>22</math>. Checking each of these, in the first case, the average is <math>17.6</math> meters, which doesn't end in <math>.2</math> as the problem requires. Therefore, we consider the other cases. With <math>44</math> and <math>88</math>, the average is <math>37.4</math> meters, which again does not end in <math>.2</math>, but with <math>44</math> and <math>22</math>, the average is <math>24.2</math> meters, which does. Consequently, the answer is <math>\boxed{\textbf{(B) }24.2}</math>.
  
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==Solution 2==
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Notice the average height of the trees ends with <math>0.2</math>; therefore, the sum of all five heights of the trees must end with <math>1</math> or <math>6</math>. (<math>0.2</math> <math>\cdot</math> <math>5</math> = <math>1</math>)
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We already know Tree <math>2</math> is <math>11</math> meters tall. Both Tree <math>1</math> and Tree <math>3</math> must <math>22</math> meters tall - since neither can be <math>5.5</math>.
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Once again, apply our observation for solving for the Tree <math>4</math>'s height. Tree <math>4</math> can't be <math>11</math> meters for the sum of the five tree heights to still end with <math>1</math>. Therefore, the Tree <math>4</math> is <math>44</math> meters tall.
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Now, Tree <math>5</math> can either be <math>22</math> or <math>88</math>. Find the average height for both cases of Tree <math>5</math>. Doing this, we realize the Tree <math>5</math> must be <math>22</math> for the average height to end with <math>0.2</math> and that the average height is <math>\boxed{\textbf{(B)}\ 24.2}</math>.
  
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==Solution 3==
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As in Solution 1, we shall show that the heights of the trees are <math>22</math>, <math>11</math>, <math>22</math>, <math>44</math>, and <math>22</math> meters. Let <math>S</math> be the sum of the heights, so that the average height will be <math>\frac{S}{5}</math> meters. We note that <math>0.2 = \frac{1}{5}</math>, so in order for <math>\frac{S}{5}</math> to end in <math>.2</math>, <math>S</math> must be one more than a multiple of <math>5</math>. Moreover, as all the heights are integers, the heights of Tree 1 and Tree 3 are both <math>22</math> meters. At this point, our table looks as follows:
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<cmath>
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\begingroup
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\setlength{\tabcolsep}{10pt}
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\renewcommand{\arraystretch}{1.5}
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\begin{tabular}{|c|c|}
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\hline Tree 1 & 22 meters \\
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Tree 2 & 11 meters \\
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Tree 3 & 22 meters \\
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Tree 4 & \rule{0.5cm}{0.15mm} meters \\
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Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline
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Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\
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\hline
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\end{tabular}
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\endgroup</cmath>
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If Tree 4 now has a height of <math>11</math>, then Tree 5 would need to have height <math>22</math>, but in that case <math>S</math> would equal <math>88</math>, which is not <math>1</math> more than a multiple of <math>5</math>. So we instead take Tree 4 to have height <math>44</math>. Then the sum of the heights of the first 4 trees is <math>22+11+22+44 = 99</math>, so using a height of <math>22</math> for Tree 5 gives <math>S=121</math>, which is <math>1</math> more than a multiple of <math>5</math> (whereas <math>88</math> gives <math>S = 187</math>, which is not). Thus the average height of the trees is <math>\frac{121}{5} = \boxed{\textbf{(B) }24.2}</math> meters.
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==Solution 4==
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Since we know that the tree heights have to be integers, then it is immediate that Tree 1 and 3 have a height of <math>22</math>. Now using the information given by the last column (that the average of the heights of the trees ends in <math>.2</math>), we can tell that the sum of all the heights of the trees ends in either <math>1</math> or <math>6</math>, because those are the only numbers from <math>0</math> to <math>9</math> that are congruent to <math>1</math> after taking modulo <math>5</math>. The two multiples of eleven (eleven because all of the tree heights have to be a multiple of eleven if they are integers) that come to mind are <math>66</math> and <math>121</math>. Since the sum of the heights of Tree 1, 2, and 3 is already <math>55</math>, we know that <math>66</math> is impossible to obtain. Then, we can decide with relative confidence that the answer should be <math>\frac{121}{5} = \boxed{\textbf{(B) }24.2}</math>.
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 +
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P.S. : We can check our solution by trying to obtain <math>121</math>. Surely enough, we have Tree 1 = <math>22</math>, Tree 2 = <math>11</math>, Tree 3 = <math>22</math>, Tree 4 = <math>44</math>, and Tree 5 = <math>22</math>. These all add up to get <math>121</math>.
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==Solution 5 (basic modular arithmetic)==
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Since her average ends with 0.2, the sum of her tree heights must be <math>\equiv 1 \pmod {5}</math>.
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If Tree 1's height is <math>a</math>, Tree 3's is <math>b</math>, Tree 4's is <math>c</math>, and Tree 5's is <math>d</math>, then we <math>a</math> and <math>b</math> are both 22, since all of the tree heights are integers.
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Now we have <math>22+11+22+c+d \equiv 1 \pmod {5}</math>. Simplifying, we get <math>c+d \equiv 1 \pmod {5}</math>.
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The only possible combination of <math>c</math> and <math>d</math> that abides the condition are 44 and 22, respectively.
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Adding these up, we get 121, and <math>\frac{121}{5}</math> is <math>\boxed{\textbf{(B) }24.2}</math>.
  
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==Solution 2==
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-kn07
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For the heights of the trees to be integers, we must have the height of Tree 1, Tree 2, Tree 3 as <math>22, 11, 22</math>. Now, there are two possible cases.
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 +
==Video Solution by NiuniuMaths (Easy to understand!)==
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https://www.youtube.com/watch?v=bHNrBwwUCMI
  
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~NiuniuMaths
  
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==Video Solution by Math-X (First understand the problem!!!)==
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https://youtu.be/UnVo6jZ3Wnk?si=1OAhRlAjzYP_EzBa&t=3750
  
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Case 1: The length of tree 4 is <math>11</math>-
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~Math-X
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So, we have the first 4 tree lengths as <math>22, 11, 22, 11</math>. For the length of Tree 5 to be an integer, we must have the length of Tree <math>5</math> as <math>22</math>. But, when we take the average of these <math>5</math> integers, it results in <math>17.6</math>, which doesn't satisfy our conditions.
 
  
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==Video Solution (🚀 Very fast 🚀)==
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https://youtu.be/swfsLlZVGbM
  
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~Education, the Study of Everything
  
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Case 2: The length of tree 4 is <math>44</math>-
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==Video Solution==
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So, we have the first <math>4</math> tree lengths as <math>22, 11, 22, 44</math>. Now, using quick modular arithmetic, we see that when the length of Tree 5 is <math>88</math>, the average of the heights of the 5 trees is <math>\boxed{24.2}\rightarrow\boxed{\textbf{(B)}}</math>. This is where our condition is satisfied.
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https://youtu.be/PCb5vOIioJA
  
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~ATGY
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Please like and subscribe!
  
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==Solution 3==
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==Video Solution by WhyMath==
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Let the sum of the heights of the trees be denoted by <math>S</math>. The average height will then be <math>\frac{S}{5}</math>. Since the average height has decimal part <math>.2</math>, it follows that <math>S</math> must have a remainder of <math>1</math> when divided by <math>5</math>. Thus, <math>S</math> must be <math>1</math> more than a multiple of 5. Since <math>S</math> is an integer, it follows that the heights of Tree 1 and Tree 3 are likely both 22. At this point, our table looks as follows:
+
https://youtu.be/uCV9-WpaIjw
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<cmath>\begin{tabular}{|c|c|}
 
−
\hline Tree 1 & 22 meters \\
 
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Tree 2 & 11 meters \\
 
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Tree 3 & 22 meters \\
 
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Tree 4 & \rule{0.2cm}{0.15mm} meters \\
 
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Tree 5 & \rule{0.2cm}{0.15mm} meters \\ \hline
 
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Average height & \rule{0.2cm}{0.15mm}.2 meters \\
 
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\hline
 
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\end{tabular}</cmath>
 
  
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If we guess that Tree 4 has a height of 11, we would need to make Tree 5 have a height of 22. In this case <math>S</math> would equal <math>88</math> which is not one more than a multiple of <math>5</math>. So we instead guess that Tree 4 has a height of <math>44</math>. Then we only need to try out heights of <math>22</math> and <math>88</math> for Tree 5. Using a height of <math>22</math> for Tree 5 gives us <math>S=121</math> which is <math>1</math> more than a multiple of <math>5</math>. Thus, the average height of the trees is <math>\frac{121}{5}=24.2</math> meters <math>\implies\boxed{\textbf{(B) }24.2}</math>.<br>
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~savannahsolver
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~[http://artofproblemsolving.com/community/user/jmansuri junaidmansuri]
 
  
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==Solution 4==
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==Video Solution==
 +
https://youtu.be/VnOecUiP-SA
  
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The problem states that all tree heights are integers. Therefore, we can deduce that the first and third trees have a height of <math>22</math> meters. Trees four and five must have heights of either <math>11,22</math> or <math>44,88</math> or <math>44,22</math>. Checking which ones match the answer choices, we find that the trees four and five have heights of <math>44</math> and <math>22</math> meters, respectively. Thus, the answer is <math>\frac{22+11+22+44+22}5=\textbf{(B) }24.2</math>.
+
==Video Solution by Interstigation==
 +
https://youtu.be/YnwkBZTv5Fw?t=1045
  
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-franzliszt
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~Interstigation
  
 
==See also==  
 
==See also==  
 
{{AMC8 box|year=2020|num-b=19|num-a=21}}
 
{{AMC8 box|year=2020|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:36, 26 January 2024

Problem 20

A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

\[\begingroup \setlength{\tabcolsep}{10pt} \renewcommand{\arraystretch}{1.5} \begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.4cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.5cm}{0.15mm} meters \\ Tree 4 & \rule{0.5cm}{0.15mm} meters \\ Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\ \hline \end{tabular} \endgroup\] $\textbf{(A) }22.2 \qquad \textbf{(B) }24.2 \qquad \textbf{(C) }33.2 \qquad \textbf{(D) }35.2 \qquad \textbf{(E) }37.2$

Solution 1

We will show that $22$, $11$, $22$, $44$, and $22$ meters are the heights of the trees from left to right. We are given that all tree heights are integers, so since Tree 2 has height $11$ meters, we can deduce that Trees 1 and 3 both have a height of $22$ meters. There are now three possible cases for the heights of Trees 4 and 5 (in order for them to be integers), namely heights of $11$ and $22$, $44$ and $88$, or $44$ and $22$. Checking each of these, in the first case, the average is $17.6$ meters, which doesn't end in $.2$ as the problem requires. Therefore, we consider the other cases. With $44$ and $88$, the average is $37.4$ meters, which again does not end in $.2$, but with $44$ and $22$, the average is $24.2$ meters, which does. Consequently, the answer is $\boxed{\textbf{(B) }24.2}$.

Solution 2

Notice the average height of the trees ends with $0.2$; therefore, the sum of all five heights of the trees must end with $1$ or $6$. ($0.2$ $\cdot$ $5$ = $1$) We already know Tree $2$ is $11$ meters tall. Both Tree $1$ and Tree $3$ must $22$ meters tall - since neither can be $5.5$. Once again, apply our observation for solving for the Tree $4$'s height. Tree $4$ can't be $11$ meters for the sum of the five tree heights to still end with $1$. Therefore, the Tree $4$ is $44$ meters tall. Now, Tree $5$ can either be $22$ or $88$. Find the average height for both cases of Tree $5$. Doing this, we realize the Tree $5$ must be $22$ for the average height to end with $0.2$ and that the average height is $\boxed{\textbf{(B)}\ 24.2}$.

Solution 3

As in Solution 1, we shall show that the heights of the trees are $22$, $11$, $22$, $44$, and $22$ meters. Let $S$ be the sum of the heights, so that the average height will be $\frac{S}{5}$ meters. We note that $0.2 = \frac{1}{5}$, so in order for $\frac{S}{5}$ to end in $.2$, $S$ must be one more than a multiple of $5$. Moreover, as all the heights are integers, the heights of Tree 1 and Tree 3 are both $22$ meters. At this point, our table looks as follows: \[\begingroup \setlength{\tabcolsep}{10pt} \renewcommand{\arraystretch}{1.5} \begin{tabular}{|c|c|} \hline Tree 1 & 22 meters \\ Tree 2 & 11 meters \\ Tree 3 & 22 meters \\ Tree 4 & \rule{0.5cm}{0.15mm} meters \\ Tree 5 & \rule{0.5cm}{0.15mm} meters \\ \hline Average height & \rule{0.5cm}{0.15mm}\text{ .}2 meters \\ \hline \end{tabular} \endgroup\]

If Tree 4 now has a height of $11$, then Tree 5 would need to have height $22$, but in that case $S$ would equal $88$, which is not $1$ more than a multiple of $5$. So we instead take Tree 4 to have height $44$. Then the sum of the heights of the first 4 trees is $22+11+22+44 = 99$, so using a height of $22$ for Tree 5 gives $S=121$, which is $1$ more than a multiple of $5$ (whereas $88$ gives $S = 187$, which is not). Thus the average height of the trees is $\frac{121}{5} = \boxed{\textbf{(B) }24.2}$ meters.

Solution 4

Since we know that the tree heights have to be integers, then it is immediate that Tree 1 and 3 have a height of $22$. Now using the information given by the last column (that the average of the heights of the trees ends in $.2$), we can tell that the sum of all the heights of the trees ends in either $1$ or $6$, because those are the only numbers from $0$ to $9$ that are congruent to $1$ after taking modulo $5$. The two multiples of eleven (eleven because all of the tree heights have to be a multiple of eleven if they are integers) that come to mind are $66$ and $121$. Since the sum of the heights of Tree 1, 2, and 3 is already $55$, we know that $66$ is impossible to obtain. Then, we can decide with relative confidence that the answer should be $\frac{121}{5} = \boxed{\textbf{(B) }24.2}$.


P.S. : We can check our solution by trying to obtain $121$. Surely enough, we have Tree 1 = $22$, Tree 2 = $11$, Tree 3 = $22$, Tree 4 = $44$, and Tree 5 = $22$. These all add up to get $121$.

Solution 5 (basic modular arithmetic)

Since her average ends with 0.2, the sum of her tree heights must be $\equiv 1 \pmod {5}$. If Tree 1's height is $a$, Tree 3's is $b$, Tree 4's is $c$, and Tree 5's is $d$, then we $a$ and $b$ are both 22, since all of the tree heights are integers. Now we have $22+11+22+c+d \equiv 1 \pmod {5}$. Simplifying, we get $c+d \equiv 1 \pmod {5}$. The only possible combination of $c$ and $d$ that abides the condition are 44 and 22, respectively. Adding these up, we get 121, and $\frac{121}{5}$ is $\boxed{\textbf{(B) }24.2}$.

-kn07

Video Solution by NiuniuMaths (Easy to understand!)

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

~NiuniuMaths

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/UnVo6jZ3Wnk?si=1OAhRlAjzYP_EzBa&t=3750

~Math-X

Video Solution (🚀 Very fast 🚀)

https://youtu.be/swfsLlZVGbM

~Education, the Study of Everything

Video Solution

https://youtu.be/PCb5vOIioJA

Please like and subscribe!

Video Solution by WhyMath

https://youtu.be/uCV9-WpaIjw

~savannahsolver

Video Solution

https://youtu.be/VnOecUiP-SA

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=1045

~Interstigation

See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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All AJHSME/AMC 8 Problems and Solutions

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