Suppose 1200 dollars is borrowed for five years at an interest rate of 6.5 percent per year, compounded continuously. Find the amount owed at the end, rounded to the nearest cent.

Step-by-step solution

1. 1

   Choose the continuous-compounding formula. Interest compounded continuously is not the $\left(1+\tfrac{r}{n}\right)^{nt}$ formula with a large $n$ — it is the limit of that expression:

   $A=Pe^{rt}.$

2. 2

   Identify the three inputs, converting the rate to a decimal.

   $P=1200,\qquad r=6.5\%=0.065,\qquad t=5\ \text{years}.$

   Leaving $r$ as $6.5$ instead of $0.065$ is the most common wrecking mistake here.

3. 3

   Compute the exponent.

   $rt=0.065\times 5=0.325.$

4. 4

   Evaluate the exponential, keeping full precision.

   $e^{0.325}=1.38403064\ldots$

   Rounding this to, say, $1.384$ would shift the final answer by about half a cent, which is exactly why the problem says not to round intermediate values.

5. 5

   Multiply and round to the nearest cent.

   $A=1200\times 1.38403064=1660.83678\ \Longrightarrow\ \$1660.84.$

6. 6

   Sanity-check against simple interest. Simple interest would give 1200(1+0.325)=\1590$, and continuous compounding must exceed that because interest earns interest. The gap of about$$71$over five years is a plausible size for a$6.5%$ rate.