Problem A and Solution

Animals on Parade

The staff at a provincial park set up a trail camera. Over \(24\) hours they captured the number of foxes, chipmunks, mice, raccoons, and rabbits that crossed the trail. The bar chart below was drawn on lined paper and shows the results of their tallies, where the letters A, B, C, D, and E along the horizontal axis each represent one type of animal; however, the numbers on the vertical axis are missing.

Use the following clues to match the letters in the bar chart to the correct animal.

There were half as many foxes as there were chipmunks

The number of chipmunks is the median of the data. (Note: The

*median*is defined below.)There were twice as many mice as there were raccoons.

There are more rabbits than raccoons.

Note: The *Median* of a data
set refers to the middle number after the numbers have been arranged in
order. If a data set has an even number of values, then there are two
“middle numbers”. In this case we calculate the sum of the two numbers
and divide by \(2\) to determine the
median.

If the bars were ordered from shortest to tallest, then the bar
labelled **B** would be in the middle. Thus, the bar
labelled **B** represents the middle number of the data.
Since the number of chipmunks is the median (middle) of the data, then
bar **B** represents the number of chipmunks.

Since there were half as many foxes as there were chipmunks, then the
bar representing the number of foxes must be half as tall as bar
**B**. Since bar **C** is half the height of
bar **B**, bar **C** represents the number of
foxes.

Since there are more rabbits than raccoons, and there are twice as
many mice as raccoons, then the shortest bar of the remaining bars
**A**, **D**, and **E** must
represent the number of raccoons. Therefore, bar **E**
represents the number of raccoons. Since bar **D** is twice
the height of bar **E** and there were twice as many mice
as there were raccoons, bar **D** represents the number of
mice.

Bar **A** is the only one not yet connected to an
animal. This means that bar **A** must represent the number
of rabbits. The height of bar **A** indicates that there are more rabbits than
raccoons, which matches the last clue. Note that we did not need the
last clue in any of the earlier logic, but it is a good check for us and
it is important that the last clue is actually true for consistency.