Assume that a savings account earns interest at the rate of

Assume that a savings account earns interest at the rate of 2% compounded continuously. If this account contains $1000 now, how many months will it take for this amount to double if no withdrawals are made?

Solution

The continuous compounding interest formula is A = Pe^rt , where P is the principal amount( initial investment),r is the annual interest rate in decimals, t is the time period and A is the maturity amount i.e. the amount after time t. Here, P = $ 1000, r = 0.02 and A = 2*1000 = $ 2000. Then, 2000 = 1000 e^0.2t or, e^0.2t = 2. On taking natural logarithms of both the sides, we have 0.2t ln e = ln 2 or, 0.2t = ln 2 = 0.69314718 ( as ln e = 1). Then t = 0.69314718 / 0.2 = 34.65735903 years = 415.888 months =415. 89 months approx. or, 416 months on rounding off to the nearest whole number.