**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:

# 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.