Eric deposits 100 into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Mike deposits 200 into a different savings account at time 0, which pays simple interest at an annual rate of i. Eric and Mike earn the same amount of interest during the last six months of the 8th year. Calculate i.

9.46%

Explanation:

Eric gets compounded interest = principal x (1 + interest rate)ⁿ

Mike gets simple interest = principal x [1 + (interest rate x n)]

during the first 7.5 years, Eric will get: 100 x (1 + 0.5i)¹⁵, in order to simplify the calculations we can call this Principal₇.₅

during the last 6 months Eric will earn: [Principal₇.₅ x (1 + 0.5i)] - Principal₇.₅ *WE ONLY WANT TO CALCULATE THE INTEREST, NOT THE PRINCIPAL

Principal₇.₅ + Principal₇.₅ (0.5i) - Principal₇.₅ = Principal₇.₅ (0.5i)

now we replace Principal₇.₅ (0.5i):

100 x (1 + 0.5i)¹⁵ x 0.5i = 50i x (1 + 0.5i)¹⁵

since Mike earns simple interest, during the last 6 months he will earn:

200 x 0.5i = 100i

now we equal both equations:

50i x (1 + 0.5i)¹⁵ = 100i

(1 + 0.5i)¹⁵ = 100i / 50i = 2

(1 + 0.5i)¹⁵ = 2

¹⁵√(1 + 0.5i)¹⁵ = ¹⁵√2

1 + 0.5i = 1.04729

0.5i = 1.04729 - 1 = 0.04729

i = 0.04729 / 0.5 = 0.09458 = 9.46%

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