Question

(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.)

Answer 1

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).

Answer 2

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.

Answer 3

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.