Gretel's Goldfish

 

 

 

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).


If Hansel's goldfish increase 4 times as many each month, that would give the equation x = 4 * (4^y), where x is the total and y is the month number. Gretel's goldfish increase by only 20 each month, giving the equation x = 128 + 20y, where x is the total and y is the month number. Using Gretel's Goldfish spreadsheet, you can see the number of months it takes for both Hansel and Gretel to have the same number of goldfish:

 

Gretel's Goldfish Spreadsheet

# 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

As you can see by the spreadsheet, Hansel and Gretel never have the exact number of goldfish exactly on the same month, but it is somewhere in between months 2 and 3 that their numbers are equal.


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