Intermath | Workshop Support
Write-up 2


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