Title
Gretel's Goldfish
Problem Statement
Hansel has goldfish that quadruple, or become four times as many, every
month. Gretel has goldfish that increase by 20 every month. Right now,
Hansel has 4 goldfish and Gretel has 128 goldfish. In how many months
will they have the same number of goldfish? Show how you arrived at your
answer..
Problem setup
This problem uses two
different types of sequences (geometric and arithmetic). We are
trying to discover when, if at all, Hansel and Gretel will have the same
number of fish if their fish reproduce at different rates.
Hansel's fish use a geometric sequence and reproduce at a rate of 4
times the number of the previous month. Gretel's fish reproduce at
an arithmetic rate of 20 more each month. Hansel starts with 4
fish. Gretel starts with 128 fish.
This is similar to the
sequences I have looked at by Fibonacci.
Plans to
Solve/Investigate the Problem
Initially I set up a
chart using Excel to examine the growth in numbers of fish from month to
month. I then graphed the results to get a visual representation
of how the changes took place.
Investigation/Exploration of the Problem
Below you can find
the chart that shows how the numbers of the fish grew.
Months 
Hansel 
Gretel 
0 
4 
128 
1 
16 
148 
2 
64 
168 
3 
256 
188 
4 
1024 
208 
5 
4096 
228 
6 
16384 
248 
7 
65536 
268 
8 
262144 
288 
9 
1048576 
308 
10 
4194304 
328 
As you can see, Hansel's
fish numbers increase much faster then Gretel's. Although
Hansel starts off less fish his numbers quickly surpass Gretel's.
By looking at a chart of these numbers we can see that just before the
third month the graphs intersect showing when Hansel and Gretel have the
same number of fish and what that number is.
Author & Contact
Teresa Johnson
tjohnson1@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans,
and/or other materials
Link 1
Link 2
