Goldfish

Problem

 

The Problem:

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).

Analysis:

Use a spreadsheet to represent the amount of goldfish owned by Hansel and Gretel. Let
one column represent the number of months, let the second column represent the
number of Hansel's goldfish, and the third column represent the number of Gretel's
goldfish. Hansel's goldfish are multiplied by 4 each month, and Gretel's goldfish are
increased (addition) by 20 each month.

Summary of Analysis:

When you extend the table, you will see that Hansel and Gretel will have the same
number of goldfish in less than three months.

You can get a better approximation of 2.75 months if you create smaller intervals
between months, such as 0.05 (1/20) month. In this case, you will need to change your
formulas because Hansel's goldfish would only increase by 7.1773% every .05 of a
month (solve (1+ x)20 = 4), and Gretel's goldfish would only increase by 1 every one
0.05 of a month (20*.05=1).

Alternate Analysis: (we'll be studying this type of model in two weeks)

You can use functions to model the growth of Hansel and Gretel's goldfish, and then
examine their graphs to locate the time when they have the same amount of goldfish.

Hansel's fish increase exponentially, using the function H = 4(4)t, where H is the number
of goldfish owned by Hansel after t months.

Gretel's fish increase linearly, using the function G = 128 + 20t, where G is the number of
goldfish owned by Gretel after t months.

The intersection of the two functions at t = 2.7585 months illustrates that they both have
approximately 183 goldfish at this time period. Gretel has more fish before that time
period, and Hansel will have more fish after that time period, since both functions are
increasing for all values in their domain.