Final answer:
To find the smallest amount Cynthia can invest to be worth over $8,000 at the end of 3 years with a 5% compound interest rate, we can use the formula for compound interest. By plugging in the values and solving for the principal amount, we find that Cynthia should invest approximately $6,923.08.
Step-by-step explanation:
To find out the smallest amount Cynthia can invest, we need to use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the total amount, P is the principal amount (initial investment), r is the annual interest rate (as a decimal), n is the number of times that interest is compounded per year, and t is the number of years.
In this case, Cynthia wants her investment to be worth over $8,000 after 3 years, so A = $8,000, P is what we are trying to find, r = 5% or 0.05, n = 1 (compounded annually), and t = 3.
Plugging in these values into the formula:
$8,000 = P(1 + 0.05/1)^(1*3)
Now, we can solve for P by isolating it:
- $8,000 = P(1.05)^3
- $8,000 = P(1.157625)
- P ≈ $8,000 / 1.157625
- P ≈ $6,923.08
Therefore, the smallest amount Cynthia can invest, to the nearest pound, is approximately $6,923.08.