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:
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.