The APR for monthly compounding is 13.13%. (Rounded to two decimal places.)
The formula for the future value (FV) of a sum of money (P) invested at an annual interest rate of r for t years, with n compounding periods per year, is:FV = P(1 + r/n)^(nt)Thus, we can apply the formula and use it to calculate the APR or Annual Percentage Rate. a. What APR did you receive, if the interest was compounded semiannually?
Semiannually means twice a year, thus, there are two compounding periods per year. Therefore, we have:109(1 + r/2)^(2*5) = 135.26Dividing both sides by 109 and then taking the fifth root of both sides we have:(1 + r/2)^2 ≈ 1.23767 (rounded to five decimal places)1 + r/2 ≈ √1.23767 = 1.11208 (rounded to five decimal places)r/2 ≈ 0.11208 (subtracting 1 from both sides) r ≈ 0.22416 or 22.416% (multiplying both sides by 2)Therefore, the APR for semiannual compounding is 22.416%. (Rounded to four decimal places.)b. What APR did you receive if the interest was compounded monthly?
Monthly means there are 12 compounding periods per year. Thus, we can apply the same formula to solve for the APR in this case:109(1 + r/12)^(12*5) = 135.26Dividing both sides by 109 and then taking the one-twelfth power of both sides we have:(1 + r/12)^60 ≈ 1.23767 (rounded to five decimal places)1 + r/12 ≈ ∛1.23767 = 1.010942 (rounded to six decimal places)r/12 ≈ 0.010942 (subtracting 1 from both sides) r ≈ 0.1313 or 13.13% (multiplying both sides by 12)Therefore, the APR for monthly compounding is 13.13%. (Rounded to two decimal places.)
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