Homework question compound interest

Question

If $ \$ 6000$ are invested at 7% compounded continuously, what amount after 2 years? I know how to set it up but at one point I get lost

$$A=Pe^rt$$ $$A=6000^{0.07}(2)$$

Somebody please help.

Answer 1

In order to calculate the interest over several years, the idea is to do the calculation for the first year and then continue the calculation for the next year with your new result. So let's do the calculation. At first, you'd have $6000\$$ and $6000\$ \cdot 0.07 = 420\$$ interest, so a total of $6420\$$. After the 2nd year, you'd have your $6420\$$ and an additional $6420\$ \cdot 0.07 = 449.40\$$ interest, so a total of $6869.40\$.

Note that we actually calculated the following: $$(6400 \$ + 6400 \$ \cdot 0.07) + (6400 \$ + 6400 \$ \cdot 0.07) \cdot 0.07 = \\ (6400 \$ \cdot 1.07) + (6400 \$ \cdot 1.07)\cdot 0.07 = \\ (6400 \$ \cdot 1.07) \cdot 1.07 = \\ 6400 \$ \cdot 1.07^2 $$

which hopefully sheds some light on the given formular, you've tried to apply.

Answer 2

If compounded continuously as you stated, at a nominal annual rate of 7%, then the amount accumulated is $$A = Pe^{rt},$$ where $P$ is the principal, $r$ is the nominal annual rate of interest, and $t$ is the time in years. Your error is that $t$ was not in the exponent along with $r$ when it should be. For $P = 6000$, $r = 0.07$, and $t = 2$, $A = 6901.64$. The other answers assume that 7% represents an effective annual rate, not a nominal one.

If the figure of 7% were assumed to be an effective rate of interest $i$, then specifying the way the interest is compounded, and hence the purpose of the question, becomes pointless.

Link to calculation in WolframAlpha