This video is provided as supplementary
material for courses taught at Howard Community
College and in this video I’m going to show how to calculate the
growth rate of an investment. The problem says An investment $4000
grows to $6800 in 10 years. What annual nominal rate would produce the same growth if the
interest was compound annually, or compounded monthly or compounded continuously. Let’s
start with the compounded annually problem. We want to find out what interest rate we would have to get if
we invested $4000 compounded annually for 10 years and wanted it to grow to $6800. So the formula we have for compound interest is A equals P times 1 plus r/n raised to the nt power. Now A is the amount of
money that we’re getting back. We want to get back 6800. So we’ll replace A with 6800. And P is the principal, which is the $4000 we
would invest initially. we’ve got 1 plus r/n. r is the interest rate. We don’t know that
so we’ll just write r, and n is the number of compounding
periods per year. Well, if its compounded annually there’s only one compounding period each year. So n would be 1. So this is just (1 + r),
and then we’re raising that to the nt power, where t is time and we want this to happen over a 10-year period. Once again n equals 1. So this is raised to the 10th
power. Now what we have to do is solve for r.
The first thing is to divide both sides of the equation by 4000. So I’ve got 6800 divided by 4000, and that equals 1.7. Now I have 1.7 equals (1 + r) raised to the 10th power. I’d like to get rid
of this exponent. I’ll raise the entire right side of the equation to another power, 1/10, and then the 10 and the
1/10 will cancel each other out. I have to do the same thing to the left
side of the equation. So the left side become 1.7 raised to the 1/10. This means I’ll have 1.7 raised to the one-tenth power and that will equal 1 + r. Let’s finish getting the r all by itself. I’ll just subtract 1 from both
sides of the equation. So it’s 1.7 raised to the 1/10 – 1. I’ll use a calculator. I want 1.7 raised to the 1/10. When I get an answer I subtract 1 from
that, and what I end up with is .05449, and that it equals r. So we’ve got to round this and turn it into a percent. I’ll multiply
by 100. that will give me 5.45%. So the answer to the first part of the
problem is we have to invest this money at 5.45%. Now let’s go on to the next part. This is
where the money is invested and compounded monthly. So we’ll have that same
formula again, A equals P times 1 plus r/n raised to the nt power. A, once again, is 6800, and P is 4000. We multiply that times 1 plus r/n — we don’t know what r is, but n, the number of compounding periods a year, would be 12, because we’re compounding monthly. That’s raised at to the nt power. n is 12, and t is the number of years,
that’s 10. And now we want to solve for r. I’ll divide both sides by 4000.
When I did that last time I got 1.7. So 1.7 equals 1 plus r/12 raised to the 12 times
10, which is 120. I want to get rid of the exponent. So I’ll raise both sides of the equation to the 1/120 power. I have 1.7 raised to the 1/120 equals 1 + r/12. I’ll subtract 1 from both
sides of the equation. I have 1.7 to the 1/120 minus 1 equals r over 12. To get r, we’ll just multiply both
sides the equation by 12. So I have 12 times 1.7 raised to the 1/120 -1 equals r. I’ll use the calculator. First I’ll do 1.7 raised to the 1 divided by 120. I’ll subtract 1 from the answer. And now I want to take that, that’s
everything here in parentheses, and multiply it by 12. and I end up with .05318… I’ll turn this into a percent and round it. So it’s going to be 5.32%. This is a little bit lower than what we
got the last time. That was 5.45%, which makes sense because we’re compounding more frequently. And the last one is what happens when we compound
continuously. We have a different formula for that. The formula of compounding continuously
is A equals P times e raised to the rt. So, A once again is 6800. P is 4000, and I’ve got that raised to the rt. I don’t know what r is, but I know t is
10. So that’s the r times 10, or 10r power. I want to solve for r, so I’ll take the
natural log of both sides of this equation……. Oh, I’m sorry, first I should divide both sides by 4000. That’s the samething I’ve done twice already. I get 1.7 equals e to the 10r. Now we’ll take the natural logs. So I
have ln, the natural log, of 1.7 equals the natural log of e raised to the 10r. I can take this exponent, 10r, and using exponent rule for logarithms, make that into a coefficient. So I have the natural log of 1.7 equals 10r times in the natural log of e. But the natural log of e is just 1. So I’ll cross that out. I have the natural log of 1.7 equals 10r. I’ll divide both sides by 10 and I get the rate is the natural log
of 1.7 divided by 10. Let’s use a calculator to find that
natural log of 1.7. And I want to divide that by 10. and I get .05306… for the rate. We’ll round this and turn it
into a percent. It’s going to be 5.31%, a little bit lower than the
previous rate because it’s compounded continuously.
okay And that’s it. Take care. I’ll see you next
time.

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