Sammy's Chipmunk and Squirrel Observations

Task

For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide?

IM Commentary

This task provides a context for setting up a linear equation whose solution requires some algebraic manipulation. Because the numbers involved are not too large, students can also experiment with some small values and eventually find the solution this way; a first solution with a table is provided showing this method. On the other hand, the reasoning required without using an equation is complex enough that the simplicity and elegance of the algebraic approach can be highlighted.

For the algebraic solution, since a variable is not provided, students might choose different variables providing an opportunity for an interesting class discussion. For example, they could let $x$ be the number of holes the squirrel has made and this would lead to the equation $$ 4x = 3(x + 4). $$ This is slightly less natural because the question is about the number of acorns buried by the chipmunk, but this is the same as the number of acorns buried by the squirrel and so this approach also works.

This task was adapted from problem #7 on the 2012 American Mathematics Competition (AMC) 10B Test. For the 2012 AMC 10B, which was taken by 35,086 students, the multiple choice answers for the problem had the following distribution:

Choice	Answer	Percentage of Answers
(A)	30	1
(B)	36	8
(C)	42	2
(D)*	48	81
(E)	54	1
Omit	--	7

Of the 35,086 students, 17,169 (49%) were in 10th grade, 9,928 (28%) were in 9th grade, and the remainder were below than 9th grade.

Solutions

Solution: 1 Table

We start by making a table with the number of holes dug by the chipmunk and squirrel and the number of acorns they have buried. We are looking for a common number of acorns and then need to study the number of holes.

Holes for Chipmunk	Chipmunk's acorns	Holes for Squirrel	Squirrel's acorns
1	3	1	4
2	6	2	8
3	9	3	12
4	12	4	16

Notice that the first common number of acorns we find is 12. The chipmunk hides 12 acorns in 4 holes while the squirrel hides 12 acorns in 3 holes. This is a difference of only one hole so if we want a difference of 4 holes we can repeat this scenario four times to get:

Holes for Chipmunk	Chipmunk's acorns	Holes for Squirrel	Squirrel's acorns
16	48	12	48

So the chipmunk and squirrel each buried 48 acorns. This is the only answer that works because with more acorns the difference in the number of holes dug goes up and with fewer acorns this difference becomes smaller.

Solution: 2 Using equations

We can introduce a variable $h$ for the number of holes that the chipmunk has dug. Since the chipmunk hides 3 acorns in each hole this make a total of $3h$ acorns that the chipmunk hides. The squirrel, on the other hand, has dug 4 fewer holes than the chipmunk: this is represented by the expression $h - 4$. The squirrel hides 4 acorns in each hole so this means thas squirrel has hidden $4\times (h - 4)$ acorns in total. We are given that the chipmunk and squirrel have hidden the same number of acorns so $$ 3h = 4 \times (h - 4). $$ Using the distributive property on the right hand side gives $$ 3h = 4h - 16. $$ Subtracting 3 from both sides and adding 16 to both sides gives $$ 16 = h. $$ So the squirrel and chipmunk have each made 16 holes. Since the chipmunk hides 3 acorns in each hole this means that the chipmunk (and squirrel) have each hidden 48 acorns.