Find the missing values assuming continuously compound interest

Question

Initial investment=$650

annual rate=?

time to double=7 1/2 years

Amount after 10years=?

Answer 1

Let A = final amount, I = initial amount, r = annual interest rate, t = number of years.

The formula for continuous compounding is A = I * e^rt

The initial amount doubles in 7.5 years. Let's say the initial amount is $1. Then 2 = 1*e^7.5r. Take the natural log of both sides, so ln 2 = ln(e^7.5r). Because ln and e are inverses of each other, they "cancel" each other, so ln(e^7.5r) = 7.5r. This leaves ln 2 = 7.5r or r = ln 2 / 7.5 = .0924 or 9.24% annual interest rate.

To get the final amount in 10 years for an initial amount of $650, A = 650 * e^(.0924)*10 = $1,637.58

Answer 2

2 = e^7.5r; r = ln2/7.5 = 0.0924 = 9.24%;

FV = 650·e^0.924 = $163758