Final answer:
It takes approximately 8.04 years for a $1,400 investment to double when it's invested at 9% interest compounded annually, derived using the formula for compound interest and taking the logarithm of both sides to solve for time.
Step-by-step explanation:
To calculate how long it takes for an investment to double when compounded annually, we use the formula A = P(1 + r)^t, where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- t is the number of years the money is invested.
Since you want to double your investment, A should be equal to 2P. We are trying to find the value of t when P is $1,400 and r is 9% (or 0.09). The equation then becomes $2,800 = $1,400(1 + 0.09)^t.
By dividing both sides by $1,400, we get 2 = (1 + 0.09)^t. Taking the logarithm of both sides gives us the equation log(2) = t log(1.09).
Dividing both sides by log(1.09) to isolate t, we get:
t = log(2) / log(1.09)
Using a calculator:
t ≈ 8.0432 years
Therefore, it will take approximately 8.04 years for your $1,400 investment to double at an annual compound interest rate of 9%.