Final answer:
To find how long it takes for a $2000 investment to double at 9% interest compounded quarterly, the compound interest formula is used. The equation to solve is 2 = (1 + 0.0225)^(4t). After calculations, it takes approximately 8 years for the investment to double.
Step-by-step explanation:
The question asks how long it will take for a $2000 investment to double at an interest rate of 9% compounded quarterly.
To answer this, we use the formula for compound interest, which is A = P(1 + r/n)^(nt), where 'A' is the amount of money accumulated after n years, including interest, 'P' is the principal amount (the initial sum of money), 'r' is the annual interest rate (decimal), 'n' is the number of times that interest is compounded per year, and 't' is the time the money is invested for in years.
Since we want the investment to double, the final amount 'A' we need is $4000 ($2000×2).
Our equation becomes: $4000 = $2000(1 + 0.09/4)^(4t).
Dividing both sides by $2000 and simplifying gives us 2 = (1 + 0.0225)^(4t).
We can then solve for 't' by taking the logarithm of both sides:
ln(2) = 4tln(1.0225)
t = ln(2) / [4ln(1.0225)].
= 8
After calculating, we find that it takes approximately 8 years for the investment to double.
This example demonstrates the power of compound interest and is a clear illustration of why starting to save money early is beneficial.