# A valuable quarter

Alignments to Content Standards: F-LE.A.2

## Task

In 1901, the San Francisco mint produced only 72,664 quarters. By comparison, during other years around the turn of the century they made between 1 million and 2 million quarters. As a result these 1901 San Francisco quarters are extremely rare coins and today, in brand new condition, each one is worth about \$60,000. 1. Suppose you put \$0.25 in the bank, on the first of January 1901, at 5% interest compounded annually. How much money would you have on January 1, 2013? What if the annual interest rate were 10% or 15%?
2. What can you deduce about the annual appreciation rate of the quarter as a rare coin? Explain.
3. Find the annual appreciation rate of the quarter as a rare coin. Does this agree with your answer to part (b)?

## IM Commentary

Successful work on this task involves modeling a bank account balance with an exponential function and then solving an exponential equation arising from the given information. This can be done by extracting a root, which will require a calculator in order to evaluate the expressions. Students will also need to be familiar with the context of annual interest and of compounding interest. The teacher may wish to indicate, for part (a) of this problem, that students may assume that no other deposits or withdrawals are made from the account. For part (b) of the problem, the teacher may wish to provide extra guidance, indicating that the goal is to use the calculations from part (a) to situate the annual interest rate earned by the rare coin. Also worth discussing is the fact that banks do not credit interest to accounts on an annual basis but rather at the end of each month: the monthly interest rate will, however, correspond to some annual interest rate so this does not change the nature of the calculations.

It is good for everyone, teachers and students, to spend a moment seeing the power of compounding interest which is revealed in this problem. Annual interest rates of 5% or 10% sound relatively innocent but yield fantastic results over a long period of time: of course the results are magnified even further if money is being deposited or invested on a regular basis, as opposed to a single deposit in this situation. This will eventually be important in most people's lives when they consider loans for cars, homes, or education.