A valuable quarter

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.

Depending on the exact condition, one of these rare quarters can be worth far more than \$60,000 or substantially less. The all time record for sales price of any quarter was set in 1990 when a 1901 San Francisco quarter reportedly sold for an astonishing \$550,000. The teacher could give this data point for an additional calculation to see if the annual rate of return for this coin exceeds 15%. Since this task was written, a new record price of \$1,527,000 was set in November, 2013 by an extraordinary 1796 quarter dollar, the first date produced by the United States of America.

Students will engage in MP8 ''Look For and Express Regularity in Repeated Reasoning'' in solving this problem, not only in applying a similar exponential expression with different interest rates but also these formulas themselves come from compounding the annual interest 112 times in each case. Students also have an opportunity to ''Look For and Make Use of Structure'' (MP7) when they produce the exponential expressions used to solve part (a): the exponential growth of the bank account balance comes from the fact that it grows by an equal factor each year. Finally, if the teacher wishes to discuss MP6 ''Attend to Precision'' there is an ideal opportunity here because banks, when they credit interest, round off to the nearest cent: this is not taken into account in the exponential formulas in the solution. Therefore the actual bank balance after 112 years will likely differ, and could differ substantially, from the answers given. For the 5% interest rate, for example, for the first several years, the account will only be credited with 1 cent (as opposed to 1 cent plus a fraction of a cent). But for the period where the interest is between one and a half cents and two cents, the amount of credited interest will be two cents. It would be interesting to study how much these rounding errors tend to cancel each other out.

Solution

1. From January 1, 1901 to January 1, 2013 is 2013 - 1901 = 112 years. To develop an expression for the bank account balance after 112 years, we can start by looking at the balance after 1, 2, and 3 years. We begin with the 5% interest rate. So on January 1, 1902 we will have our initial \$0.25 and an additional 5%: $$ \$0.25 \times (1.05) = \$0.2625. $$ After 2 years we will have the \$0.2625 from the first year plus an additional 5% interest on this amount: $$ \$0.25 \times (1.05)^2 = \$0.2625 \times (1.05) = \$0.275625. $$ After 3 years we will have the \$0.275625 from the first year plus an additional 5% interest on this amount: $$ \$0.25 \times (1.05)^3 = \$0.275625\times (1.05) = \$0.28940625. $$ Note that it is difficult to see a pattern in the actual account balances after 1, 2, and 3 years. The expressions on the left, however, follow a simple pattern. Each year, the account grows by an additional multiplicative factor of 1.05: this makes sense as the 1 in 1.05 carries the beginning balance through to the next year and the 0.05 adds the 5% annual interest. So at 5% annual interest, the account balance at the end of $t$ years would be $$ \$0.25 \times (1.05)^t. $$ From January 1, 1901 to January 1, 2013 is 2013 - 1901 = 112 years so the account balance on January 1, 2013 would be $$ \$ 0.25 \times (1.05)^{112} \approx \$59.04. $$ Note that this is only an approximation: as can be seen from the cases calculated above, each additional year of interest adds two more decimals to the value of the expression $\$0.25 \times (1.05)^t$.

   The same method applies to the 10% and 15% interest rate: the only part of the expressions that changes is the 1.05 term: with a 10% interest rate this becomes 1.10 and with a 15% interest rate it is 1.15.

   \begin{align} \$ 0.25 \times (1.1)^{112} &\approx \$10,812.37 \\ \$ 0.25 \times (1.15)^{112} &\approx \$1,570,717.67 \end{align}

    

2. A higher appreciation rate brings more money to the account while a lower appreciation rate brings less money to the account. The value of the \$0.25 as a rare coin is \$60,000 which is greater than \$10,812.37 but less than \$1,570,717.67 so this means that the annual appreciation rate of the quarter as a rare coin is greater than 10% but less than 15%.

3. Consider the equation $$ \$ 0.25 \times x^{112} = \$ 60,000 $$ If we solve for $x$ we will know what number \$0.25 must be multiplied by 112 times, in order to reach \$ 60,000. To solve the equation we can divide both sides by \$0.25 giving $$ x^{112} = \frac{\$ 60,000}{\$ 0.25} = 240,000. $$ We can solve this by extracting a root: $$ x = \sqrt[112]{240,000}. $$ Alternatively, we could think of it as raising both sides of the equation to the $\frac{1}{112}$ power: $$ x=240,000^\frac{1}{112}. $$ Using a calculator we find $$ x \approx 1.117. $$ From the calculations above, we see that the decimal part of $x$ is the annual appreciation rate so the coin has appreciated at a rate of about 11.7% per year, in line with our conclusion from part (b).