*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.

(Source: Adapted from Mathematics Teaching in the Middle
School, Nov-Dec 1995).

*Problem setup *

Hansel’s goldfish increase four
times in a month, therefore in a geometric sequence, whereas Gretel’s goldfish
increase in an arithmetic sequence by adding 20 per month. Even though Hansel begins with only four
goldfish to Gretel’s 128 goldfish, the number of Hansel’s goldfish quickly numbers in
the thousands, whereas the numbers of Gretel’s goldfish increases only by 20
each month. The question is in how many months Hansel and
Gretel will have the same number of goldfish.

*Plans to Solve/Investigate
the Problem*

We first investigated the problem using a
spreadsheet. Using the values determined
with the spreadsheet, we created a graph of the two equations, one Hansel’s and
one Gretel’s.

*Investigation/Exploration of
the Problem*

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 |

11 |
16777216 |
348 |

12 |
67108864 |
368 |

13 |
268435456 |
388 |

14 |
1073741824 |
408 |

15 |
4294967296 |
428 |

16 |
17179869184 |
448 |

17 |
68719476736 |
468 |

18 |
274877906944 |
488 |

We then observed the point of intersection of the two line graphs by entering the two expressions in Excel and creating a graph using Chart Wizard:

Next, by using the graphing calculator program of our computer, we set two equations to represent our problem:

y = 4^{x + 1}

y = 128 + 20x.

*To
find the number of Gretel’s fish*, beginning with 128 fish and growing by 20
fish each month, we set up a chart:

__Months____ # of Gretel’s fish__

0 128

1 128 + 20

2 128 + 20 + 20

3 128 + 20 + 20 + 20

4 128 + 20 + 20 + 20 + 20 …

therefore y = 128 + 20x since x represents the number of fish that Gretel would have at any month x.

*Hansel
started with only four fish, and these four quadrupled every one month*. Therefore the expression y = 4 x+1 represents
the number of Hansel’s fish.

When we graphed these two expressions using the graphing calculator, and by zooming in to see the exact point of intersection, we found that the Hansel and Gretel would have 183.17 fish (y) at the 2.75 month (x):

**(purple)**

**(red) **

Click Here to see a Graphing Calculator file.

The problem is similar to a money financial problem comparing earning compounded interest versus adding money to a non-interest bearing savings account. For instance, Raggedy Ann and Raggedy Andy each wanted to set up separate savings accounts. Raggedy Ann is a frugal young lady and begins her account with all her birthday money of $128.00. She plans on adding $20.00 each month to her non-interest bearing account. Raggedy Andy, on the other hand, invests in Enron and believes that if he is steady in his deposits, his small initial investment will quadruple every month. In how many months will Raggedy Ann and Raggedy Andy have the same amount of savings?

*Author & Contact*

Jill Jackson and Shirley Crawford

Rockdale InterMath Algebra

jjackson@rockdale.k12.ga.us.
Or scrawford@rockdale.k12.ga.us.

*Link(s) to resources, references, lesson plans, and/or other materials*

www.math2.org/math/general/interest.htm