Present and Future Value Example 3

Hi everyone! Welcome back to integralcalc.com. We're gonna be doing another future value
problem today. This one is a little bit more interesting because we're actually asked to
solve for the interest rate, r as opposed to future value.
The reason that we know that we're using this formula is first-off because they asked us
to assume that the interest is compounded continuously. And whenever you see continuously
compounded interests, you use the formula with e as opposed to 1 plus r over n to
the nt. So, we're using the formula with e in it.
And the information they give us, we're trying to find the interest rate that has to be paid
so that an initial deposit doubles in 7 years, assuming that the interest is compounded continuously.
So, we're not given explicit numbers for future and present value. All we know is that our
initial deposit doubles in 7 years. First of all, we have our initial deposit, so that 
clearly is our present value and  it's doubling in 7 years so our future value is gonna be
double what it was in 7 years. So we can fairly easily go ahead and plug things in.
We're gonna call our initial deposit D instead of PV so that we can simplify this a little
bit. So our initial deposit, our present value is gonna be D. We leave the e. We're trying
to solve for the interest rate, the rate that has to be paid in order for this thing to
happen. The time they've given us for the deposit to double is 7 years, so we go ahead
and plug in 7 years for time, for t. And then they said that our initial deposit has
to double in 7 years. So, if our present value, our initial deposit is D, and we're looking
for double that, then our future value will be 2D. So, 2D equals D e to the r times
7. And now we can solve this formula for r or
the rate that must be paid in order to double our deposit in 7 years. So the first thing
that we go ahead and do is divide both sides by D and we will  be left with D over
here equals e and  then we can make this 7 a coefficient on the r just to simplify.
So we have D equals (er, sorry. Just kidding.) This is not D squared. This is 2D so we
have 2 over here. If we divide both sides by D, the D's will cancel; we'll be left
with 2. So we have 2 equals e to the 7r. And to get rid of this e so that we can
get the r out of the exponent, we're gonna take the natural log or ln of both sides
because when we do that, we'll cancel out the natural log and the e. e and natural
log always cancel each other out, so when you're trying to get something out of the
exponent on e, that's always what you go ahead and do.
So now we have natural log of 2 over here. These will cancel and we'll be left with 7r on
this side. We can go ahead and divide both sides by 7 and we'll be left with r equals
natural log of 2 over 7. So you could leave the rate like this or if
you want to find what it is in terms of a decimal approximation, you can do natural
log of 2/7 into your calculator and you get r equals point 0990 approximately. So if you're trying
to find the percentage and you wanna get it in terms of you know, 6% or 4% or whatever,
we move the decimal point over two places and we can see then that our annual percentage
rate must be 9.9%, roughly for an initial deposit with interest compounded continuously
to double in 7 years. So that's how you solve that problem. Hope
that helps. I'll see you guys next time! Bye!