Problem:
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?
TreeTree 1Tree 2Tree 3Tree 4Tree 5Average height​Height (meters)_meters11meters_meters_meters_meters_.​2meters​​
Answer Choices:
A. 22.2
B. 24.2
C. 33.2
D. 35.2
E. 37.2
Solution:
We know that tree 2 is 11 meters tall, and since the trees on either side of any given tree were said to be either double the height or half the height, we can conclude that tree 1 and tree 3 must both be 22 meters tall. If they were half that of tree 2, they would not be integers. This gives us only four possibilities for the remaining two trees, of which we simply test each case for an average that ends in "0.2":
- Tree 4 is 44 meters; Tree 5 is 88 meters:
555+44+88​=37.4
- Tree 4 is 44 meters; Tree 5 is 22 meters:
555+44+22​=24.2
- Tree 4 is 11 meters; Tree 5 is 22 meters:
555+11+22​=17.6
- Tree 4 is 11 meters; Tree 5 is 5.5 meters: This means that Tree 5 is not an integer, so this case is invalid and discarded.
The only one of these cases that has an average that ends in "0.2" is the 11,22 case. As such, the average height is 24.2 meters.
Thus, the correct answer is B.
Answer: B​.
The problems on this page are the property of the MAA's American Mathematics Competitions