It will take approximately 5 years for the $138,450 investment to grow to $148,540 with a 3.98% annual interest rate compounded quarterly. To calculate the time it will take for an investment to grow to a specific amount, we can use the formula for compound interest: A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the initial investment amount
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case, the initial investment (P) is $138,450 and the future value (A) is $148,540. The interest rate (r) is 3.98% or 0.0398, and interest is compounded quarterly, so n = 4.
Now, let's solve for t, the number of years:
148,540 = 138,450(1 + 0.0398/4)^(4t)
To simplify the equation, let's divide both sides by 138,450:
148,540/138,450 = (1 + 0.0398/4)^(4t)
1.071937 = (1 + 0.00995)^(4t)
Now, let's take the natural logarithm (ln) of both sides to isolate t:
ln(1.071937) = ln((1 + 0.00995)^(4t))
Using the property of logarithms, we can bring the exponent down:
ln(1.071937) = 4t * ln(1 + 0.00995)
Now, divide both sides by 4 * ln(1 + 0.00995):
t = ln(1.071937) / (4 * ln(1 + 0.00995))
Using a calculator, we find that t ≈ 5.00002 years.
So the time it will take is 5 years.