### Compound Interest Formula

#### Lesson Objective

In this lesson, we will learn about the basics behind compound interest formula...

#### About This Lesson

In this lesson, we will learn:

- Learn about compound interest
- See how compound interest works

(1st math video) - Find the formula for compound interest.

(2nd math video)

The study tips and math video will explain more.

### Study Tips

#### Tip #1

The compound interest formula is given below:

Where:

- A is the total amount of money (including interest) after n years
- P is the principal (the amount money borrowed or invested)
- r is the interest rate (per year or per annum)
- n is the loan or investment duration in years

The math videos below will explain more on this formula. You can see the examples on using this formula here.

### Math Video

#### Video on Figuring Compound Interest

Sponsored Links

#### Math Video Transcript

PART 1

00:00:04.070

In this lesson, we will learn about the compound interest formula.

00:00:09.140

Consider a loan of $1000, with the compound interest rate of 10%, and the duration of the loan is 3 years.

00:00:18.000

Calculate the total amount of money that we need to pay back, after 3 years.

00:00:24.070

Before we learn about the compound interest formula, let's see how compound interest works.

00:00:31.030

Let's represent this green bar with the loan amount, $1000.

00:00:36.210

Now, after the first year, the interest will be 10% of the loan amount, $1000.

00:00:44.090

To calculate the interest, I, we multiply $1000, with the interest rate, 10%.

00:00:53.040

Now, before we can calculate, we need to change 10% to a decimal.

00:00:59.110

We can do so, by dividing 10% with 100. This gives 0.1.

00:01:07.240

$1000 multiply with 0.1, gives $100.

00:01:14.070

This is the amount of interest, after the first year.

00:01:18.160

Hence, we can find the amount of money that we owe after the first year, by adding $1000, with $100.

00:01:29.010

This gives $1100. Let's write this down here.

00:01:37.220

After the 2nd year, since the interest is compounding, the interest will be 10% of the current amount, $1100.

00:01:48.030

To calculate the interest, I, we multiply $1100 with 10%.

00:01:55.170

As calculated earlier, 10% converted to decimal is 0.1. $1100 multiply with 0.1, gives $110.

00:02:10.010

This is the loan's interest after the second year.

00:02:14.110

With this, we can find the amount of money that we owe after the 2nd year, by adding $1100, with $110. This gives $1210. Let's write this down here.

00:02:33.080

After the 3rd year, since the interest is compounding, the interest will be 10% of the current amount, $1210.

00:02:43.120

To calculate the interest, I, we multiply $1210, with 10%.

00:02:51.190

10% converted to a decimal is, 0.1. $1210 multiply with 0.1, gives $121.

00:03:05.200

Therefore, the interest is $121, after the 3rd year.

00:03:12.040

With this, we can find the total amount of money to pay back after 3 years, by adding $1210, with $121.

00:03:23.080

This gives, $1331. Let’s write it down here.

00:03:31.100

Hence, for a loan with compound interest, the total amount of money that we need to pay back after 3 years is, $1331.

00:03:43.220

In the second part, we will see how we can find the compound interest formula.

#### Video on Finding the Compound Interest Formula

Sponsored Links

#### Math Video Transcript

PART 2

00:00:04.000

After understanding how compound interest works, let's find the compound interest formula.

00:00:11.040

Consider a loan of P dollars, with the compound interest rate of r%, and the loan duration is 3 years.

00:00:19.080

Find the total amount of money that we need to pay back, after 3 years.

00:00:25.110

Let's start by representing this green bar, with the loan amount, P.

00:00:31.230

Now, after the first year, the loan's interest will be r%, of P.

00:00:38.130

Hence, we can find the interest, I, by multiplying P, with r.

00:00:45.100

With this, we can find the amount of money that we owe, by adding P, with Pr.

00:00:53.040

This gives, P + Pr.

00:00:57.100

Notice that, these two terms have 'P' as a common factor.

00:01:03.100

Hence, we can factorize these 2 terms, by taking out 'P'.

00:01:09.200

By doing so, we have P(1+r).

00:01:15.110

So, this is the amount of money that we owe, after the first year.

00:01:21.180

Next, let's find out the interest at the end of 2nd year.

00:01:27.150

Before doing so, we let Q = P(1+r). Hence, this expression becomes Q.

00:01:39.180

Since this is a compound interest, the interest will be r%, of Q.

00:01:46.230

Hence, we can find the interest, I, by multiplying Q, with r.

00:01:53.240

Now, we can find the amount of money that we owe, by adding Q, with Qr.

00:02:01.060

By doing so, we have Q + Qr.

00:02:06.050

Again, notice that these 2 terms have 'Q' as a common factor.

00:02:12.050

Therefore, we can factorize these terms by taking out Q. This gives Q(1+r).

00:02:21.090

Let's change Q back to P(1+r). By doing so, we have P(1+r)(1+r).

00:02:34.100

Here, P(1+r)(1+r), gives (1+r)^2.

00:02:43.050

So, after the second year, the amount of money that we owe is, P(1+r)^2.

00:02:51.020

Here, we can see something interesting.

00:02:55.080

We know that, P(1+r) is the same as, P(1+r)^1.

00:03:04.140

Observe that, after the first year, we have P(1+r)^1. After the second year, we have P(1+r)^2.

00:03:20.050

From this, we can guess that, after the 3rd year, the amount money that we need to pay back is, P(1+r)^3.

00:03:31.120

Please verify this, to convince yourself.

00:03:35.180

Hence, if the loan duration is n year, we will have, P(1+r)^n.

00:03:45.090

Therefore, the total amount of money that we need to pay back after n years, A = P(1+r)^n.

00:03:56.000

This is the compound interest formula.

00:04:00.030

That is all for this lesson. Try out the practice question to further your understanding.

### Practice Questions & More

#### Multiple Choice Questions (MCQ)

Now, let's try some MCQ questions to understand this lesson better.

You can start by going through the series of questions on compound interest formula or pick your choice of question below.

- Question 1 on calculating the compound interest
- Question 2 on using the compound interest formula

#### Site-Search and Q&A Library

Please feel free to visit the Q&A Library. You can read the Q&As listed in any of the available categories such as Algebra, Graphs, Exponents and more. Also, you can submit math question, share or give comments there.