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.