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All right, we're going to do a word problem that uses exponents and our logs.
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And in this video, we're going to do this investment problem, seen how long it takes for an investment to triple.
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So first we have to know the compound interest formula, A equals P, times 1 plus, R over N to the NT.
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So here's our problem.
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How many years does it take for an investment of $4,000 to triple to $12,000?
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Now I didn't have to tell you what it would triple to, you would simply take $4,000 times 3, correct?
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And it's put in a savings account earning 6% interest that is compounded quarterly.
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Quarterly means 4 times a year.
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And we're going to round our answer to the nearest tenth of a year.
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All right, so let's see what we know.
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We know that P is the original investment.
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And that was $4,000.
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And we wanted to end up at $12,000.
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We don't know what T is, but R is our interest rate, and that's 6%, so we could write 0.06, or 600, if you want to write it as a fraction, that's fine too.
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And N is how many times it's invested per year.
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So that's how we start the problem.
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So we're just going to fill those numbers in to this formula.
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So A, that's going to be our $12,000, equals $4,000, times 1 plus R, which is.06, divided by 4, to the 4T.
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I don't know T.
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The first thing I'm going to do here is divide both sides by $4,000.
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So on the left hand side, oops, 4,000, here we go.
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I'm just going to get 3.
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Equals, and this is going to cancel, so I just have what's in parentheses.
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So now I have to simplify what's in parentheses if I can.
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You can do that in your calculator if you want.
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I do a lot of stuff in my head first.
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I think of this as 4 over 4 plus.06 over 4.
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So that's going to be 4.06 over 4 to the 4T.
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And then you might want to just do that in your calculator.
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Make sure it comes out evenly.
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And that gives you 1.015 to the 4T.
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Now I want you to think about this problem.
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What if I did not tell you the original amount was 4,000?
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I just said how long would it take for something to triple?
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So what if I said I gave you $1,000?
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I wanted to triple.
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I would have put a 1,000 here, 3,000 here.
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And when I divide by 1,000, I'm going to get 3.
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So anytime you're doubling or tripling, it really doesn't matter
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what the original amount was.
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If you're going to triple it over here, you'll see a 3 when you're done.
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And if you double it, you'll see a 2.
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So the fact is that if somebody else has a different amount to invest
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in the same bank account, it's going to take them just as long
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to triple their money as somebody who invested $4,000.
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So think about that.
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Now how are we going to solve this?
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We have T, which is what we need to solve right at the time,
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as an exponent.
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So that means we want to use logarithms.
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And again, you could always use your log or your natural log.
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This time, because it's a different problem, I'm going to use my natural log.
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So I'm going to take the natural log of both sides of the problems.
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So I'm going to take the natural log of 3.
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And I'm going to take the natural log of 1.015 to the 4T.
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Now once you do that, once you've written it in terms of logs,
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you could take this exponent, put it out in front of the log.
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And that's the key to seeing how cool logarithms are.
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So that'll just be the natural log of 1.015.
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Now remember, natural log of 1.015 is simply a number,
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just like natural log of 3 is some number.
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And if you wanted to use the log instead of natural log,
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in the end, it's not going to make any difference.
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Go ahead and do it twice if you'd like.
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All right, but the trick is I'm trying to solve for the T here.
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So I need to divide both sides by the coefficient of T.
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And four times natural log of 1.015,
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that is the coefficient of T.
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I just sort of wrote the T in the middle.
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But remember, that's just a number, four times the natural log of 1.015.
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So what we want to do is divide both sides by that.
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So if I do that, I'll be able to solve for T.
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So on the left side, I'll have natural log of 3 all over four
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times the natural log of 1.015.
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And that will give me the exact value of T.
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So at this point, you want to get out of calculator
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and very carefully do this calculation.
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All right, with my calculations, I got that that was going to be 18.447,
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you know, years.
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Okay, so there's actually more decimal places here.
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And you know you got something wrong if there's hardly any years,
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like if you got like a.05 years, of course,
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it's going to take a long time to triple your investment.
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So just again, be careful, you might want to use parentheses all the way through.
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This is what you might put in your calculator.
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You might press the, if you want to do it all the way on parentheses,
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then natural log, then parentheses around the three,
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and then another parentheses around all that,
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then the division bar sign, which you might have a divide sign or something like this,
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then another parentheses, then four,
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and natural log of 1.015.
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So, and depending on, you know, what kind of calculator you have,
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you might not be able to do all of this,
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or you might actually need another parentheses around the four times the log.
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Okay, so you might even need another parentheses here,
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depending on what kind of calculator you have, it really depends.
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But this is kind of a low interest rate, 5%, so it does take a long time.
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So we get T, and I said to round to the nearest tenth of a year.
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So it's about 18.4 years.
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So it looks like it's going to take, it takes about 18.4 years to triple.
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The investment.
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Now, we never want to be complacent, and assume we always did everything correct.
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So let's go ahead and go back to the original formula,
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put in the 18.4 years, and see if it really does triple our investment.
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Right, so here's the original problem.
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And let's see if it's true if we put in $4,000 at that rate, if after 18.4 years,
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it is triple to $12,000.
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So let's check it out.
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What would, okay, P would be $4,000, and they have 1.R, which was 0.06,
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it was quarterly, 4, and it was to the 4 times 18.4.
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So what we want to do is calculate this, again using your calculator,
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and we want to see if the amount really is $12,000.
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Okay, so we do have 4,000 times, and then what do we get when we did 1 plus 0.06 divided by 4?
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That was 1.015, and then I have to do 4 times 18.4, and that's 73.6.
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All right, so now you can put that in your calculator, and again, my suggestion is to make sure,
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I like to do this first, and then multiply the whole answer by 4,000,
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but your calculator might let you just simply put in 4,000,
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and then the parentheses 1.015, this is what I'm going to do, I'm going to try it,
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to the 73.6.
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And when I did that, so I actually just plugged in this exactly, and my calculator did it.
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I got this number, and actually a little bit more, when I'm rounding to the nearest cent,
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I got $11,966.32, so you know what?
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It did.
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Triple, because remember, I was approximating this 18.4, so my answer seems completely reasonable.
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Now, I'm going to take the same problem, just to it slightly differently, about the check.
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Let's just see if it's true if I put any amount of meni in, would it triple in 18.4 years?
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So, the question is, no matter how much meni I put in there, will it be true that it gets tripled?
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Well, let's say I just put a dollar in the account, let's see what happens.
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Well, if I put a dollar in the account, I just have 1, right?
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Times 1 point, remember, when we plugged all these numbers in, we had 1.015, right?
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And that was to the four times 18.4, so basically the question is, without end up being $3.
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So, we just need to do this computation in the calculator.
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Well, I got $2.99, so yeah, pretty much tripled.
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So, that was my point, if you're trying to triple or double something, it really doesn't matter with the original amount.
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Because, so here's just one last way the problem could have been worded.
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So, I could have just said, how many years does it take for an investment to triple if it's put into an account,
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earning 6% that is compounded quarterly?
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You know what? All you know is that whatever you put in, you could just say is P, then the amount after that amount of time is 3P.
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So, look how the formula works here.
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You have a equals P, 1 plus r over n to the nT.
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Well, if I plug in 3P for A, if I divide both sides by P, look, you get 3 equals 1 plus r over n to the nT.
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So, there's the formula if you want an investment to triple after a certain number of years.
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If you want to double, it would end up being a 2, because instead of 3P, you would put 2P up here divided by 2.
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And this is what it would look like.
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So, just something to think about and consider.
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All right, but for our problem here, the answer was 8.
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I think it was 18.4 years. Is that right?
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There it is.
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It took 18.4 years.
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So, that's actually 1 mind worth 0.2 years.
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Then, this formula will put you to solve problem according to the..6.
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I know that if you wait and this formula will contain some problems with receiving..