Question

In an episode from the TV show Futurama, Phillip J. Fry had $0.93 in his bank account in 2000. In the year 3000 it had grown to $4.3 billion! What was the percent interest rate (to three decimal places) if it had been continuously compounded? What was the percent interest rate (to three decimal places) if it had been compounded quarterly?

Answer 1

Formula for continuous interest:
A=Pe^(rt)
take log on both sides and solve for r:
r=ln(A/P)/t
where A=4300000000,P=0.93,t=1000

You should get around 2.2%, ASSUMING banks do not round interest deposits to the nearest cent.
If they deposit interest every two months and round the interest to the nearest cent, the money will not grow at all because interest for every 2 months is less than half a cent.

If it had been compounded quarterly, then the formula is
A=P(1+(r/4))^(4t)
Again, take log on both sides,
log10(A/P)=(4t)log10(1+r/4)
=>
r = (10^[log10(A/P)/(4t)]-1)*4
where log10=log to the base 10
A=4300000000
P=0.93
t=1000 (years)
You should get also about 2.2%, but a little less than continuous interest.

Again, it depends on how banks "round" down or round up the cent, and hope they don't have monthly charges! :)

Answer 2

continuous growth:

4300000= .93 e^1000r
4623655.91 = e^1000r
1000r = ln 4623655.91
r = .0153467
= 1.535%

compounded quarterly:
let the quarterly rate be i
.93(1+i)^4000 = 4300000
(1+i)^4000 = 4623655.91
i+i = 1.003844...
i = ..003844..

so the annual rate compounded quarterly is .. 0.015376 or 1.5376 %

check:
.93 e^(.01535(1000)) = 4, 314,230 using the rounded continuous rate

.93(1+.003844)^4000 = 4,299,999.76 not bad!

Answer 3

Andrea, which continent are you on?
In USA, 1 billion = 10^9, and I think in UK and France, 1 billion = 10^12. This makes a big difference.