(Solving for n with non-annual periods)
Approximately how many years would it take for an investment to grow sevenfold if it were invested at 12 percent compounded quarterly?
Assume that you invest $1 today.
If you invest $1 at 12 percent compounded quarterly, about how many years would it take for your investment to grow sevenfold to $7?
(Hint: Remember to convert your calculator solution to years.)
_ years (Round to one decimal place.)
We use the formula:
A=P(1+r/4)^4n
where
A=future value
P=present value
r=rate of interest
n=time period.
7=1*(1+0.12/4)^(4n)
7=(1.03)^4n
Taking log on both sides;
log 7=4n*log (1.03)
n=1/4[log 7/log(1.03)]
=16.5 years(Approx).
Using the formula for compound interest, we have:
A = P(1 + r/n)^(nt)
where: A = final amount = 7 P = principal = 1 r = annual interest rate = 12% n = number of times compounded per year = 4 (quarterly) t = number of years
Substituting the given values, we get:
7 = 1(1 + 0.12/4)^(4t) 7 = (1.03)^(4t) ln(7) = ln(1.03)^(4t) ln(7) = 4t ln(1.03) t = ln(7)/(4 ln(1.03)) t ≈ 14.2
Therefore, it would take approximately 14.2 years for the investment to grow sevenfold if it were invested at 12 percent compounded quarterly.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the future value, P is the present value, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.
We want to find t when A/P = 7 and r = 0.12, n = 4 (since there are 4 quarters in a year). Substituting the values and solving for t, we get:
7 = 1(1 + 0.12/4)^(4t) ln 7 = ln (1 + 0.03)^4t ln 7 = 4t ln 1.03 t = ln 7 / (4 ln 1.03) t ≈ 14.5 quarters
To convert quarters to years, we divide by 4:
t ≈ 3.625 years
Therefore, it would take approximately 3.6 years for the investment to grow sevenfold if it were invested at 12 percent compounded quarterly.
(Solving for n with non-annual periods) Approximately how many years would it take for an investment...
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