Title†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††
Gretelís 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††††††††††

 

Hanselís goldfish increase four times in a month, therefore in a geometric sequence, whereas Gretelís goldfish increase in an arithmetic sequence by adding 20 per month.Even though Hansel begins with only four goldfish to Gretelís 128 goldfish, the number ofHanselís goldfish quickly numbers in the thousands, whereas the numbers of Gretelís 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 Hanselís and one Gretelís.

 

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 = 4x + 1

y = 128 + 20x.

 

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

 

Months†††††††††††††††††††††††† # of Gretelís fish

0                                                                                        128

1                                                                                        128 + 20

2                                                                                        128 + 20 + 20

3                                                                                        128 + 20 + 20 + 20

4                                                                                        128 + 20 + 20 + 20 + 20 Ö

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 Hanselís 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.

 

Extensions of the Problem††† Click To Download

 

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

Edwards Middle School

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