Question

Please solve the problem step by step, to check my final answers.

Answer 1

We are asked to determine the future value and the time for a quarterly compounded interest. To do that we will use the following formula:

$A=P(1+\frac{r}{400})^{4t}$

Where:

$\begin{gathered} A=\text{ future value} \\ P=\text{ initial value} \\ r=\text{ interest rate} \\ t=\text{ time} \end{gathered}$

Part A. We are asked to determine the time in 7 years. To do that we will substitute the value of "t = 7" and "r = 2", we get:

$A=4000(1+\frac{2}{400})^{(4)(7)}$

Solving the operations:

$A=4599.49$

Therefore, in 7 years there will be the amount of $4599.49

Part B. We are asked to determine the time to get the amount of $5000. To do that we will substitute the value of "A = 5000", and we get:

$5000=4000(1+\frac{2}{400})^{4t}$

Now, we solve for "t". First, we divide both sides by 4000:

$\frac{5000}{4000}=(1+\frac{2}{400})^{4t}$

Now, we take the natural logarithm to both sides:

$\ln(\frac{5000}{4000})=\ln(1+\frac{2}{400})^{4t}$

Now, we use the following property of logarithms:

$\ln x^y=y\ln x$

Applying the property we get:

$\operatorname{\ln}(\frac{5,000}{4,000})=4t\operatorname{\ln}(1+\frac{2}{400})$

Now, we divide both sides by the natural logarithm and by 4:

$\frac{1}{4}\frac{\ln(\frac{5000}{4000})}{\ln(1+\frac{2}{400})}=t$

Solving the operations:

$11.19=t$

Therefore, the amount of 5000 will be obtained after 11.19 years.