Respuesta :
It takes 3 years to save an amount of $ 5000
Solution:
The formula for compound interest, including principal sum, is:
[tex]A = p(1+\frac{r}{n})^{nt}[/tex]
Where,
A = the future value of the investment/loan
P = the principal investment amount
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per unit t
t = the time the money is invested or borrowed for
From given,
p = 4000
[tex]r = 7.75 \% = \frac{7.75}{100} = 0.0775[/tex]
A = 5000
n = 2 ( compounded semiannually)
t = ?
Therefore,
[tex]5000 = 4000(1+\frac{0.0775}{2})^{2t}\\\\\frac{5}{4} = (\frac{2.0775}{2})^{2t}\\\\Take\ logarithms\ base\ e\\\\ln\frac{5}{4} = 2t\ ln(1.03875)\\\\ln\ 5 - ln\ 4 = 2t\ 0.03801\\\\1.6094 - 1.3862 = 0.07602t\\\\t = 2.93 \approx 3[/tex]
Thus it takes 3 years to save an amount of $ 5000
