Interest Suppose 40,000 is borrowed for 5 years at 3 % interest. Find the interes

Interest Suppose 40,000 is borrowed for 5 years at ...

Key Concepts

Effective Annual Rate

The effective annual rate (EAR) is the actual interest rate an investor earns or a borrower pays after accounting for the effects of compounding. It is calculated based on the nominal rate and the compounding frequency, and provides a clear comparison across different compounding schedules by reflecting the true economic cost or yield in one year.

Compound Interest

Compound interest is the process of calculating interest on both the initial principal and the accumulated interest from previous periods. This method leads to interest earnings that increase at an exponential rate over time, as each period's interest calculation takes into account the interest that was added in previous periods.

Compounding Frequency

Compounding frequency refers to how often interest is applied to an investment or loan within a given year. Common frequencies include annually, semiannually, quarterly, and monthly. A higher frequency of compounding generally results in more interest accrued over the same period since interest is calculated and added to the principal more frequently.

Transcript

00:01 In this problem, we're told that $40 ,000 is borrowed for five years at a 3 % interest.

00:07 Find the interest paid over this period if the interest is compounded as follows.

00:11 Annually, semi -annually, quarterly, monthly, and continuously.

00:16 So first, we're going to do parts a through d because they're pretty similar, and then we'll come back to part e at the end.

00:25 So the formula for determining the amount after the total amount of money after some amount has been deposited, or borrowed for a certain amount of time is the formula a equals p times 1 plus r divided by m to the power of t times m, where p is the amount you started with.

00:53 So in our case, the amount that was borrowed at the beginning was $40 ,000, so that's our p.

01:01 The rate is the interest rate, the annual interest rate, which in our case is 3%.

01:09 If we convert that to a decimal, that's 0 .03.

01:13 So that's what we're going to plug in there.

01:16 T is the total times.

01:17 This is the number of years.

01:20 And in our case, we're told that the money was borrowed for five years.

01:25 So t is equal to five.

01:28 And m is how many times per year our interest is compounded.

01:33 A number of times per year, the interest is compounded.

01:45 Which depends on if it's compounded annually or semi -eurore.

01:48 Annually or quarterly.

01:50 So for example, in part a, we're asked annually if it's compounded annually.

01:56 So that means one time per year, so m is equal to one.

02:00 Part b is semi -annually.

02:04 So that's twice a year.

02:06 So m is equal to two.

02:09 For quarterly, that's once a quarter, so four times a year.

02:15 And finally, monthly, that means it happens once a month, which is 12 times a year.

02:24 So now we have all the information that we need to find our total amount after five years.

02:31 But before we actually do a calculation, we should think about what the problem's asking.

02:35 The problem is not asking how much total will be owed after five years.

02:40 We specifically want to know what the interest will be.

02:43 So after we find our answers, so after we plug all this in, that's the number that we get is going to be the total amount that we owe.

02:54 But in this problem, we only are caring about how much interest is paid.

02:58 So what we should do is we should subtract the original amount.

03:02 So subtract $40 ,000 because that was money that we owed anyway because we borrowed the $40 ,000.

03:10 So what we want to know is how much interest we're paying.

03:12 So on top of paying back the $40 ,000, how much interest will we owe? so we should find this for each of these four parts.

03:20 Let's do this on a new page.

03:23 So in part a, we're going to look at the equation...

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