Person A opens an IRA at age 25, contributes $2000/yr for 10 years, but makes no additional contributions thereafter. Person B waits until age 35 to open an IRA and contributes $2000/yr for 30 years. There is no initial investment in either case, and the contributions are made continuously. Determine the return rate r for which the two IRA's have equal value at age 65.

For Person 1: IRA = (P ((1 + r) ^n1 - 1) / r) (1 + r) ^n2; where P = 2000, n1 = 10, n2 = 30

For Person 2: IRA = P ((1 + r) ^n - 1) / r; where P = 2000, n = 30

(2000 ((1 + r) ^10 - 1) / r) (1 + r) ^30 = 2000 ((1 + r) ^30 - 1) / r

log (2000 ((1 + r) ^10 - 1) / r) (1 + r) ^30 = log (2000 ((1 + r) ^30 - 1) / r)

log 2000 + log ((1 + r) ^10 - 1) - log r + log (1 + r) ^30 = log 2000 + log ((1 + r) ^30 - 1) - log r

log ((1 + r) ^30 - 1) - log ((1 + r) ^10 - 1) = log (1 + r) ^30

log (((1 + r) ^30 - 1) / ((1 + r) ^10 - 1)) = log (1 + r) ^30

log ((1 + r) ^20 + (1 + r) ^10 + 1) = log (1 + r) ^30

(1 + r) ^20 + (1 + r) ^10 + 1) = (1 + r) ^30

(1 + r) ^30 - (1 + r) ^20 - (1 + r) ^10 - 1 = 0

let (1 + r) ^10 be x, then

x^3 - x^2 - x - 1 = 0

x = 1.839286755

(1 + r) ^10 = 1.839286755

1 + r = 10th root of 1.839286755 = 1.063

r = 1.063 - 1 = 0.063

Therefore, the required rate is 6.3%.