suppose you invest 2500 and earn 4.2% annual interest, compounded quarterly. how long will it tak for your investment to double?
Solve this equation for the number of quarters, n.
[1 + (0.042/4)]^n = 2
1.0105^n = 2
n = 66.36
That many quarters is 16.58 years
You will have to wait for the first interest-paying date after 16.5 years.
To calculate the time it will take for an investment to double, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value (double the initial investment)
P = the principal amount (initial investment)
r = annual interest rate
n = number of times interest is compounded per year
t = time in years
In this case, let's plug in the given values:
A = 2P (as we want the investment to double)
P = $2,500
r = 4.2% or 0.042 (convert to decimal)
n = 4 (compounded quarterly)
t = time we want to find
Now, we can rewrite the formula as:
2P = P(1 + r/n)^(nt)
Simplifying further, we get:
2 = (1 + 0.042/4)^(4t)
Next, we can isolate the exponential term:
(1 + 0.042/4)^(4t) = 2
Now, we can solve for t by taking the natural logarithm (ln) of both sides:
ln((1 + 0.042/4)^(4t)) = ln(2)
Using the logarithmic property, we can bring down the exponent:
4t * ln(1 + 0.042/4) = ln(2)
Finally, solving for t:
t = ln(2) / (4 * ln(1 + 0.042/4))
Using a calculator, we can find the value of t. In this case, t is approximately 16.6 years.
So, it will take approximately 16.6 years for your investment to double with a 4.2% annual interest, compounded quarterly.