Final answer:
The principal is calculated by rearranging the compound interest formula to solve for 'P', with the future value given as $5700, annual interest rate of 5%, and a compounding period of eight years and two months, compounded quarterly.
Step-by-step explanation:
Finding the Principal Amount from Future Value with Compound Interest
To solve for the principal that will grow to $5700 in eight years and two months at an interest rate of 5% compounded quarterly, we use the compound interest formula:
Future Value (FV) = Principal (P) × (1 + interest rate per period)^number of periods
The first step is to adjust the interest rate and time according to the compounding frequency. The annual rate of 5% (or 0.05 as a decimal) is compounded quarterly, so we divide it by the number of quarters per year:
Quarterly interest rate = Annual rate / 4
Next, we need to convert the time into quarters. Eight years and two months is the same as 8 years and 1/6 of a year (since 2 months is approximately 1/6 of a year), in total 8.1667 years, which is:
Number of quarters = 8.1667 years × 4 quarters/year
Now, we plug these values into the formula and solve for the principal:
FV = P × (1 + 0.05/4)^(8.1667×4)
After calculating the number of periods and the interest rate per period, we isolate P (Principal) and solve for it using algebra:
P = FV / (1 + 0.05/4)^(8.1667×4)
By substituting $5700 for FV and the previously calculated values, we can find the principal amount.