WEBVTT
1
00:00:00.000 --> 00:00:10.400
In this video, we do an application problem that uses exponents and our logs to solve,
2
00:00:10.400 --> 00:00:17.960
and so we do this problem on this video.
3
00:00:17.960 --> 00:00:24.800
Use the compound interest formula to find how many years it takes $1300 invested at 9%,
4
00:00:24.800 --> 00:00:28.080
compounded monthly to increase to $2000.
5
00:00:28.080 --> 00:00:35.680
What this means is if you put $1300 in a bank and the interest rate is 9%, and it's compounded
6
00:00:35.680 --> 00:00:44.360
monthly, which means 12 times a year, when is the value in that account at $2000?
7
00:00:44.360 --> 00:00:51.560
So what we want to do is identify what we do know and use this formula here.
8
00:00:51.560 --> 00:00:58.160
So P stands for the original investment A, how much it is after a certain number of years.
9
00:00:58.160 --> 00:00:59.760
So let's fill that in.
10
00:00:59.760 --> 00:01:10.920
We know that P is our original investment of $1300 and A, we want to find out when it's
11
00:01:10.920 --> 00:01:12.880
grown to $2000.
12
00:01:12.880 --> 00:01:15.880
All right, what about the interest rate R?
13
00:01:15.880 --> 00:01:24.360
Well, that's 9%, we want to write that as a decimal or a fraction, so it's.09, and we
14
00:01:24.360 --> 00:01:26.280
don't know the time, right?
15
00:01:26.280 --> 00:01:33.120
But N is how many times per year the interest is compounded since it says monthly and there's
16
00:01:33.120 --> 00:01:37.840
12 months in a year, N is 12.
17
00:01:37.840 --> 00:01:46.160
So we want to fill all these numbers into this formula and then we will compute and find
18
00:01:46.160 --> 00:01:47.520
out what T is.
19
00:01:47.520 --> 00:01:48.520
So let's do that.
20
00:01:48.520 --> 00:01:55.880
So we have A, well we're going to plug in, 2000.
21
00:01:55.880 --> 00:02:07.640
P is 1,300, and we have 1 plus.09 divided by 12, because that's R over N.
22
00:02:07.640 --> 00:02:13.760
So N is 12, so it's 12T.
23
00:02:13.760 --> 00:02:17.640
So that's the original part of the problem.
24
00:02:17.640 --> 00:02:23.760
Now from here, there's different ways you might simplify this, but I'm going to start
25
00:02:23.760 --> 00:02:30.920
by simplifying inside parentheses and at the same time dividing both sides by 300, I mean
26
00:02:30.920 --> 00:02:41.520
1,300, so if I divide both sides by 1,300, I'm just going to leave that as 2013.
27
00:02:41.520 --> 00:02:48.000
So on the left side, 2013, because it reduces pretty easily, 2013 equals, and now I have
28
00:02:48.000 --> 00:02:52.640
to simplify inside this parentheses, because now I have a 1 in front of here.
29
00:02:52.640 --> 00:02:57.800
So I could write 1 as 12 over 12, everybody with me there?
30
00:02:57.800 --> 00:02:59.040
12 over 12.
31
00:02:59.040 --> 00:03:10.280
So I'd have a common denominator, so that means in the fraction I would have 12.09 over 12
32
00:03:10.280 --> 00:03:12.040
to the 12T.
33
00:03:12.040 --> 00:03:17.680
So this is what I have so far.
34
00:03:17.680 --> 00:03:20.680
Okay.
35
00:03:20.680 --> 00:03:30.120
At this point, if you can divide 12.09 divided by 12 and have it not be a repeating decimal,
36
00:03:30.120 --> 00:03:31.120
then you could do that.
37
00:03:31.120 --> 00:03:35.960
If you get something that 12 doesn't go into evenly, then what you would do is just leave
38
00:03:35.960 --> 00:03:37.120
it as is.
39
00:03:37.120 --> 00:03:47.720
But in this case, if you do, use your calculator, 12.09 divided by 12, you get 1.0075.
40
00:03:47.720 --> 00:03:52.040
It does come out to an exact number here to the 12T.
41
00:03:52.040 --> 00:03:57.280
But notice, 12, 2013, so if I tried to divide that out, it doesn't come out exact.
42
00:03:57.280 --> 00:04:01.520
So if something doesn't come out exact, leave it in fractional form.
43
00:04:01.520 --> 00:04:02.520
Right.
44
00:04:02.520 --> 00:04:04.320
So here's the problem.
45
00:04:04.320 --> 00:04:11.720
We're now going to try to solve for T, and T is in the exponent, and the key is, anytime
46
00:04:11.720 --> 00:04:15.960
you have a variable in the exponent that you're trying to solve for, you're going to use
47
00:04:15.960 --> 00:04:16.960
logs.
48
00:04:16.960 --> 00:04:21.720
You're going to take the log of both sides or the natural log of both sides.
49
00:04:21.720 --> 00:04:26.920
So we're going to go on to the next page to continue this.
50
00:04:26.920 --> 00:04:28.080
So this is what I have.
51
00:04:28.080 --> 00:04:31.960
And it's up to you whether you want to use your log button or your natural log button.
52
00:04:31.960 --> 00:04:35.480
I'm just going to use the log button, so I'm going to write the log of the left hand
53
00:04:35.480 --> 00:04:39.840
side is equal to the log of the right hand side.
54
00:04:39.840 --> 00:04:45.840
Now if you want, you could put brackets around here, but it's certainly not required.
55
00:04:45.840 --> 00:04:47.440
All right.
56
00:04:47.440 --> 00:04:55.160
So now if I have this 12T, what I could do is, since it's in the exponent, I could put
57
00:04:55.160 --> 00:04:56.400
it out in front of the log.
58
00:04:56.400 --> 00:05:00.080
So really it's kind of confusing seeing these brackets, but I showed it just that I'm saying
59
00:05:00.080 --> 00:05:02.520
I'm doing it to both sides.
60
00:05:02.520 --> 00:05:04.000
So what do we have now?
61
00:05:04.000 --> 00:05:11.120
I've got this log of 2013, so it's okay, don't worry about it.
62
00:05:11.120 --> 00:05:21.480
Keep writing it, and we're going to put in 12T out to the front times the log of 1.0075.
63
00:05:21.480 --> 00:05:22.480
Okay.
64
00:05:22.480 --> 00:05:30.400
Now the trick here is you're trying to solve for T. So remember, the log of 1.0075, that's
65
00:05:30.400 --> 00:05:35.880
just some number, multiply by T, and then I also have 12 by T. So what I've got here is
66
00:05:35.880 --> 00:05:42.480
the log of 2013, and I'm going to write this so you see this as a coefficient, times 12,
67
00:05:42.480 --> 00:05:52.960
and I have log 12 log of 1.0075 times T. So what I want you to note is that this part
68
00:05:52.960 --> 00:06:02.040
here is simply the coefficient of T. So we're just going to divide both sides by that
69
00:06:02.040 --> 00:06:18.360
number. So I'm going to divide the left, the right side, and the left side.
70
00:06:18.360 --> 00:06:22.080
So I don't do any rounding until this very last step.
71
00:06:22.080 --> 00:06:29.200
And keep in mind, this is the log of 2013 divided by 12 times the log of this, you can't
72
00:06:29.200 --> 00:06:36.600
combine this 2013 with this 1.0075. These are two separate computations you're going
73
00:06:36.600 --> 00:06:42.440
to have to make to solve for T. Now we put that in the calculator very carefully with
74
00:06:42.440 --> 00:06:48.680
parentheses, et cetera. So in the calculator, I would recommend you put like a bracket
75
00:06:48.680 --> 00:06:54.080
around the log here, and then a parentheses around the 2013's another bracket, then a
76
00:06:54.080 --> 00:06:58.840
division sign, and then another bracket, 12 times the log of this. You want to make sure
77
00:06:58.840 --> 00:07:03.440
that you're very careful how you enter this in your calculator. So go ahead and try entering
78
00:07:03.440 --> 00:07:08.920
this in your calculator and see what you get. It might take you a little while to enter
79
00:07:08.920 --> 00:07:14.720
all that. And this is what I got in my calculator. So I'm going to put approximately because
80
00:07:14.720 --> 00:07:20.680
now it's not going to be exact. It does go on and on. I got 4.804, 4, you know, and it
81
00:07:20.680 --> 00:07:26.960
kind of goes on and on. But it did say round to the nearest tenth of a year. So I'm going
82
00:07:26.960 --> 00:07:34.600
to say that T is approximately 4.8 years. In the back, I'm going to, since it's a word
83
00:07:34.600 --> 00:07:50.600
problem, it takes about 4.8 years, which was the question. It says how long did it take,
84
00:07:50.600 --> 00:08:05.160
how many years? Okay, wow. Not an easy problem. Now at this point, I would suggest actually
85
00:08:05.160 --> 00:08:12.680
checking to see if the seems correct. So let's go back to the original problem. So here's
86
00:08:12.680 --> 00:08:24.200
the original problem. We are identified PR and T. But what I want to do is see if actually
87
00:08:24.200 --> 00:08:34.160
you invested this $1300 at 9% at, you know, compounded monthly for 4.8 years, isn't really
88
00:08:34.160 --> 00:08:39.240
$2,000. So that's what I mean by checking. So I'm going to try that. So I'm going to
89
00:08:39.240 --> 00:08:43.520
say, well, what is it? And the question is, is it going to come out to $2,000? That's
90
00:08:43.520 --> 00:08:50.880
what we would hope. So what would P be? Well, that's $1300. I'm going to have 1 plus my
91
00:08:50.880 --> 00:08:58.760
Rs.09 divided by 12 again to the NT. So it's 12 times. Now I'm going to actually plug in
92
00:08:58.760 --> 00:09:08.200
that number 4.8. Okay? So we're just going to simply compute what happens if you put $1300
93
00:09:08.200 --> 00:09:18.200
in for 4.8 years at this interest rate compounded monthly. So again, we want to simplify inside
94
00:09:18.200 --> 00:09:31.960
this parentheses, which was I think 1.0075. And 12 times 4.8 is 57.6. So that's how many times
95
00:09:31.960 --> 00:09:38.480
it actually gets compounded. And now we're going to use our calculator to do the 1.0075
96
00:09:38.480 --> 00:09:45.960
to the 57.6. And then multiply it by 1,300. Or if you're careful, you can go 1,300 times
97
00:09:45.960 --> 00:09:51.000
and make sure you just parentheses to do this. So go ahead now and just see what happens.
98
00:09:51.000 --> 00:09:56.760
We need to simplify that. Be careful. The order of operations. I usually go ahead in my
99
00:09:56.760 --> 00:10:05.120
calculator just to make sure I do this first. And so I can put my calculator 1.0075 raised
100
00:10:05.120 --> 00:10:10.320
at 57.6. That you do that according to your calculator. And then I press Enter. And if
101
00:10:10.320 --> 00:10:15.840
you do that, you should have something about 1.537854, etc. And then I just go ahead and
102
00:10:15.840 --> 00:10:30.560
multiply it by 1,300 next. And that gives me 1999.21. Okay. Well, that's not exactly
103
00:10:30.560 --> 00:10:35.200
$2,000, right? We want to see if it's about $2,000. And of course, it's not going to be
104
00:10:35.200 --> 00:10:42.000
because this 4.8 was rounded. But it checks because that's pretty darn close, right? So
105
00:10:42.000 --> 00:10:45.280
we must have, when we rounded, we rounded under a little bit. So that's why when we
106
00:10:45.280 --> 00:10:52.160
collected it, and we got a little bit smaller number. So now we verified that that does make
107
00:10:52.160 --> 00:11:17.720
sense about 4.8 years. So it takes about 4.8 years was the answer to this problem.