Question

How long does it take your money to double at 8% interest?

Answer 1

A close approximation is the "rule of 72" which is that the product of the interest rate and the time needed to double is 72
so 72 = 8 x and x = 9.

Else, (1+0.08)^x = 2 take log of both sides

x ln(1.08) = ln 2

x = ln(2)/ln(1.08) = 9.006

Answer 2

they want me to use the formula... \[A=Pe^{rt}\]

Answer 3

Doubling means that A'/A = 2

A'/A = exp(rt')/exp(rt) = 2

ln 2 = rt' - rt = r (t'-t)

set t = 0 to start

ln 2 = r t' = 0.08 t'

t' = ln 2 / 0.08 = 8.66

The difference (9 vs. 8.66) is because the exp(rt) equation assumes continuous compounding and the (1+r)^t assumes compounding only at unit intervals of t, like years.

Answer 4

Glad to have been of help.

Answer 5

okay why does A= exponent rt?

Answer 6

how would we plug into the formula... meaning what does each variable represent in this case in the formula I gave you.

Answer 7

The amount = principal times growth factor

let N be the number of dollars in the account

dN/dt = N r growth equation for rate r

dN/N = r dt then integrate from t=0 to t

ln N - ln No = r t express as exponential

N = No exp(rt)

A = P exp(rt) in financial notation

Answer 8

Okay , I don't think I 'll ever fully understand how to set this up, but thanks for the help.

Answer 9

A is the amount you get after some time t ...

A = Pe^rt; well P is doubled when A = 2P

2P = Pe^rt ... divide and log out