# Problem of the Day

Jennifer invests $2,000 at an annual interest rate of r% compounded quarterly for 9 years. At the end of 9 years her account is worth$16,000. What is the value of ?

A.
B.
C.
D.
E.

[latexpage]

D. $25\cdot2^{\frac{49}{12}}-400$

See the Solution

### Solution

[latexpage]

The formula for compound interest is $A=P(1+\frac{r\%}{n})^{nt}$. From the given information we can set up the equation $2000(1+\frac{r\%}{4})^{36}=16000$

$2000(1+\frac{r\%}{4})^{36}=16000$

$(1+\frac{r\%}{4})^{36}=8$

$1+\frac{r\%}{4}=8^{\frac{1}{36}}$

$\frac{r\%}{4}=8^{\frac{1}{36}}-1$

$r\%=4\cdot8^{\frac{1}{36}}-4$

$r=400\cdot8^{\frac{1}{36}}-400$

Now the challenge is to convert it to one of the answer choices.

$r=25\cdot16\cdot8^{\frac{1}{36}}-400$

$r=25\cdot2^4\cdot2^{\frac{1}{12}}-400$

$r=25\cdot2^{\frac{49}{12}}-400$