Question

Find how much money needs to be deposited now into an account to obtain $5,100 (Future Value) in 8 years if the interest rate is 7% per year compounded quarterly (4 times per year). The final amount is $ Round your answer to 2 decimal places

Answer 1

solution:

Let the money needs to be deposited now =
T

time = 8 years = 32 quarters

rate of interest
Ꭱ)
= 7% per year = $\frac{7}{4}\%$ per quarter

amount after 8 years (32 quarters) = $5100

Now by using

$A=P\left ( 1+\frac{R}{100} \right )^T$

$\Rightarrow 5100=x\left ( 1+\frac{\frac{7}{4}}{100} \right )^{32}$

by taking log on both sides, we get

$log (5100)=log\left [x\left ( 1+\frac{\frac{7}{4}}{100} \right )^{32} \right ]$

by using $log(mn)=log(m)+log(n)$ , we have

$\Rightarrow log (5100)=log(x)+log\left ( 1+\frac{\frac{7}{4}}{100} \right )^{32}$

by using $log(m^n)=nlog(m)$ , we have

$log (5100)=log(x)+32log\left ( 1+\frac{7}{400} \right )$

$\Rightarrow log (5100)=log(x)+32log\left ( \frac{407}{400} \right )$

$\Rightarrow log (5100)=log(x)+32log\left ( 1.0175 \right )$

$\Rightarrow 3.7076=log(x)+0.2411$

$\Rightarrow log(x)=3.7076-0.2411$

$\Rightarrow log(x)=3.4665$

taking antilog on both sides, we get

$x=2927.52$

Thus, the money needs to be deposited now = $2927.52

this complete the solution.