Difference between revisions of "2020 AMC 8 Problems/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?

$$\begin{tabular}{|c|c|} \hline Tree 1 & \rule{0.2cm}{0.15mm} meters \\ Tree 2 & 11 meters \\ Tree 3 & \rule{0.2cm}{0.15mm} meters \\ Tree 4 & \rule{0.2cm}{0.15mm} meters \\ Tree 5 & \rule{0.2cm}{0.15mm} meters \\ \hline Average height & \rule{0.2cm}{0.15mm}.2 meters \\ \hline \end{tabular}$$$\newline \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

It is not too hard to construct $22, 11, 22, 44, 22$ as the heights of the trees from left to right. The average is $\frac{121}{5}=24.2\rightarrow\boxed{\textbf{(B)}}$. ~icematrix Note: this is the result of realizing that the 1st one must be greater then the 2nd, $\frac{121}{5}=24.2 or {\textbf{(B)}}$. ~oceanxia

Solution 2

For the heights of the trees to be integers, we must have the height of Tree 1, Tree 2, Tree 3 as $22, 11, 22$. Now, there are two possible cases.

Case 1: The length of tree 4 is $11$- So, we have the first 4 tree lengths as $22, 11, 22, 11$. For the length of Tree 5 to be an integer, we must have the length of Tree $5$ as $22$. But, when we take the average of these $5$ integers, it results in $17.6$, which doesn't satisfy our conditions.

Case 2: The length of tree 4 is $44$- So, we have the first $4$ tree lengths as $22, 11, 22, 44$. Now, using quick modular arithmetic, we see that when the length of Tree 5 is $88$, the average of the heights of the 5 trees is $\boxed{24.2}\rightarrow\boxed{\textbf{(B)}}$. This is where our condition is satisfied.

~ATGY

Solution 3

Let the sum of the heights of the trees be denoted by $S$. The average height will then be $\frac{S}{5}$. Since the average height has decimal part $.2$, it follows that $S$ must have a remainder of $1$ when divided by $5$. Thus, $S$ must be $1$ more than a multiple of 5. Since $S$ 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: $$\begin{tabular}{|c|c|} \hline Tree 1 & 22 meters \\ Tree 2 & 11 meters \\ Tree 3 & 22 meters \\ Tree 4 & \rule{0.2cm}{0.15mm} meters \\ Tree 5 & \rule{0.2cm}{0.15mm} meters \\ \hline Average height & \rule{0.2cm}{0.15mm}.2 meters \\ \hline \end{tabular}$$

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 $S$ would equal $88$ which is not one more than a multiple of $5$. So we instead guess that Tree 4 has a height of $44$. Then we only need to try out heights of $22$ and $88$ for Tree 5. Using a height of $22$ for Tree 5 gives us $S=121$ which is $1$ more than a multiple of $5$. Thus, the average height of the trees is $\frac{121}{5}=24.2$ meters $\implies\boxed{\textbf{(B) }24.2}$.
~junaidmansuri

Solution 4

The problem states that all tree heights are integers. Therefore, we can deduce that the first and third trees have a height of $22$ meters. Trees four and five must have heights of either $11,22$ or $44,88$ or $44,22$. Checking which ones match the answer choices, we find that the trees four and five have heights of $44$ and $22$ meters, respectively. Thus, the answer is $\frac{22+11+22+44+22}5=\textbf{(B) }24.2$.

-franzliszt