Answer:
It will take approximately 14.2 years for the money to double at a 5% annual interest rate compounded quarterly.
Explanation:
To determine how many years it will take for the money to double, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal amount (initial investment), r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.
In this case, the final amount is twice the initial investment (2P), the annual interest rate is 5% (or 0.05 as a decimal), and interest is compounded quarterly (n = 4).
Let's substitute these values into the formula and solve for t:
2P = P(1 + 0.05/4)^(4t)
Cancelling out the P:
2 = (1 + 0.05/4)^(4t)
Now, we can solve for t by taking the logarithm of both sides:
log(2) = log((1 + 0.05/4)^(4t))
Using the logarithmic property:
log(2) = 4t * log(1 + 0.05/4)
Now, isolate t:
t = (log(2)) / (4 * log(1 + 0.05/4))
On solving the above equation, we get,
t ≈ 14.2 years
Therefore, it will take approximately 14.2 years for the money to double at a 5% annual interest rate compounded quarterly.