Answer:
The time (t) = 2.6 years
Step-by-step explanation:
To calculate the time for earning a compound interest, compounded on a certain amount of present value (PV), compounded periodically, the following formula is used:
[tex]FV=PV(1+\frac{i}{n} )^{n*t}[/tex]
where:
FV = future value = $6,100
PV = present value = $5,000
i = interest rate in decimal = 7.5% = 0.075
n = number of compounding periods per year = quarterly = 4 (4 quarters a year)
t = time of compounding in years = ???
Therefore the time is calculated thus:
[tex]6100=5000(1+\frac{0.075}{4} )^{4*t}[/tex]
[tex]6100=5000(1+0.01875 )^{4t}[/tex]
[tex]6100=5000(1.01875 )^{4t}[/tex]
Next, let us divide both sides of the equation by 5000
[tex]\frac{6100}{5000} = \frac{5000(1.01875)^{4t} }{5000}[/tex]
1.22 = [tex](1.01875)^{4t}[/tex]
Taking natural logarithm of both sides
㏑(1.22) = ㏑[tex](1.01875)^{4t}[/tex]
㏑(1.22) = 4t × ㏑(1.01875)
0.1989 = 4t × 0.01858
4t = [tex]\frac{0.1989}{0.01858} = 10.71[/tex]
∴ 4t = 10.71
t = 10.71 ÷ 4 = 2.6 years
