00:01
Hey everyone, today we're solving a problem involving compound interests, and this problem appears in chapter 9, section 3 of your textbook, and the following information is given.
00:16
They tell you that a principle of $5 ,000, or your starting value, is invested at 6 % interest.
00:28
So that's a dollar.
00:34
And then you have 0 .06, kind of as your art, value and that's kind of your a sub 1 value if you want to think of it that way.
00:49
And they tell you that you are asked to find the value 10 years after the initial investment, when the money is compounded, and then there's five parts.
01:05
So in the first part, you're asked to find when it's compounded annually.
01:09
And the third part, you're asked to find when it's compounded semi -annually.
01:14
The third part, you're asked to find when it's compounded quarterly.
01:21
The fourth part, you're asked to find when it's compounded monthly, and then the fifth part, you're asked to find when it's compounded daily.
01:30
So five parts to this problem, but once you get the hang of it, and honestly, once you solve the first part, it's not too terrible.
01:37
So you're going to recall that your formula for compound interest and in plugging in these numbers will be to have your initial value of 5 ,000, and it's going to be multiplied by in parentheses, one plus the interest rate, our value of 0 .06, over what would be n, and n is the amount of times that you need to kind of make to get to a year.
02:11
So when it's yearly, it's just over one, but when it's semi -annually, it's over two, because you need two semi -annualies to basically get to one whole year.
02:20
And then it's raised to the power of n times t, where t is at the time after you're kind of measuring the initial investment.
02:30
So we have 10, and then n is that same value that you plugged in here.
02:34
So we have to the 1 times 10 power or to the 10 power for this problem.
02:38
And what that's going to equal, you can honestly just do this in your calculator.
02:41
So you do 1 plus 0 .06 time, raise it to the power of 10 and then multiply it by 5 ,000.
02:49
And i got, about 8, 954.
02:56
I'm going to round to two decimal places.
02:58
They don't specify.
03:03
So this would be 0 .24.
03:05
And again, remember your units, which i forgot to put in here.
03:08
Let me erase that equal sign.
03:11
And again, we're working with dollars here because the initial investment, or any investment, is involving dollars, really.
03:18
So there you have it.
03:20
It would become, in 10 years from now, if compounded annually, $8 ,954, $0 .24.
03:28
All right, moving on swiftly in part b, again, that same initial investment of $5 ,000 is there.
03:37
But now what you're going to multiply by is 1 plus, again, that our value is that the interest rate never changed from before.
03:45
It's still 6%.
03:46
But we want to compound it semi -annually, which means that our end value, which is how many use that monthly, basically, explain, but it's like semi -annually.
04:01
All right, i think i need two times the money every semi -annually to get to an annually.
04:08
So this would be two.
04:11
Basically, two times semi -annually is annually.
04:16
And then you raise it to the n times t power, so you're ready to figure out that n is 2, and t again is 10, so two times 10 is to the 20th power.
04:24
And this is going to equal.
04:28
So i'm going to do that in my calculator.
04:30
0 .06.
04:32
Let me erase that.
04:33
0 .06 divided by 2 plus 1 .1.
04:36
Raised to the 20th power times 5 ,000.
04:40
And i got 9 ,000.
04:42
Let me write it here.
04:46
Again, this is if it's compounded annually.
04:49
After 10 years or a whole decade, my initial investment of $5 ,000 at an interest rate of 6 % would become $9 ,030 .56.
05:03
Approximately.
05:06
All right, moving on to part c, which asks us to find the value...