Final answer:
To find the original amount borrowed with an annual interest rate of 38% compounded semiannually that totals $11,350 after 407 weeks, we use the formula for compound interest with semiannual compounding.
By substituting the given values into the formula, we can solve for the initial principal amount.
Step-by-step explanation:
To calculate the original amount borrowed, we need to use the formula for compound interest. The formula for the future value of an investment compounded semiannually is P = A / (1 + r/n)^(nt), where:
- P is the principal amount (the initial amount of money)
- A is the amount of money accumulated after n years, including interest.
- r is the annual interest rate (decimal)
- n is the number of times that interest is compounded per year
- t is the time the money is invested or borrowed for, in years.
In this case, A is $11,350, r is 38% or 0.38, n is 2 (since the interest is compounded semiannually), and t is 407 weeks, which we convert to years by dividing by 52 weeks/year, giving us approximately 7.827 years.
To find the original amount borrowed (P), we rearrange the formula and solve:
P = 11350 / (1 + 0.38/2)^(2*7.827)
By calculating this expression, we can find the original amount that was borrowed.