Interest rate. Upstate Bank is offering long-term certificates of deposit with a face value of
(future value). Bank customers can buy these CDs today for and will receive the in
20 years. What interest rate is the bank paying on these CDs?
What interest rate is the bank paying on these CDs?
(Round to two decimal plices.)

Ask by Craig Simpson. in the United States

Jan 22,2025

Question

Interest rate. Upstate Bank is offering long-term certificates of deposit with a face value of
(future value). Bank customers can buy these CDs today for and will receive the in
20 years. What interest rate is the bank paying on these CDs?
What interest rate is the bank paying on these CDs?
(Round to two decimal plices.)

Ask by Craig Simpson. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

To find the interest rate that the bank is paying on these CDs, we can use the formula for compound interest:

$$ A = P(1 + \frac{r}{100})^n $$

where:
- $$ A $$ is the future value of the investment (in this case, $100,000),
- $$ P $$ is the present value of the investment (in this case, $45,000),
- $$ r $$ is the interest rate (which we need to find),
- $$ n $$ is the number of years (in this case, 20 years).

Given that the present value of the investment is $45,000 and the future value is $100,000, we can rearrange the formula to solve for the interest rate $$ r $$:

$$ 100,000 = 45,000(1 + \frac{r}{100})^{20} $$

Now, we can solve for $$ r $$ to find the interest rate that the bank is paying on these CDs.
Solve the equation by following steps:
- step0: Solve for $$r$$:
  $$100000=45000\left(1+\frac{r}{100}\right)^{20}$$
- step1: Swap the sides:
  $$45000\left(1+\frac{r}{100}\right)^{20}=100000$$
- step2: Divide both sides:
  $$\frac{45000\left(1+\frac{r}{100}\right)^{20}}{45000}=\frac{100000}{45000}$$
- step3: Divide the numbers:
  $$\left(1+\frac{r}{100}\right)^{20}=\frac{20}{9}$$
- step4: Simplify the expression:
  $$1+\frac{r}{100}=\pm \sqrt[20]{\frac{20}{9}}$$
- step5: Simplify:
  $$1+\frac{r}{100}=\pm \frac{\sqrt[20]{20\times 19683^{2}}}{3}$$
- step6: Separate into possible cases:
  $$\begin{align}&1+\frac{r}{100}=\frac{\sqrt[20]{20\times 19683^{2}}}{3}\&1+\frac{r}{100}=-\frac{\sqrt[20]{20\times 19683^{2}}}{3}\end{align}$$
- step7: Calculate:
  $$\begin{align}&r=\frac{100\sqrt[20]{20\times 19683^{2}}-300}{3}\&r=-\frac{100\sqrt[20]{20\times 19683^{2}}+300}{3}\end{align}$$
- step8: Rewrite:
  $$r_{1}=-\frac{100\sqrt[20]{20\times 19683^{2}}+300}{3},r_{2}=\frac{100\sqrt[20]{20\times 19683^{2}}-300}{3}$$
The interest rate that the bank is paying on these CDs is approximately 4.07%.
The bank is paying an interest rate of approximately 4.07% on these CDs.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Extra Insights

preguntas relacionadas

Latest Economics Questions