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Compound Interest

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Today we're going to talk about compound interest. So we'll start by looking at how we derive the formula for compound interest, and then we'll do some examples. So let's start by reviewing how interest works.

Let's say I have $5,000 invested into an account that is 3.6% annual interest rate. We can find the value of our account after one year. We're going to assume that the interest is applied only one time per year, and we're also going to assume that we're making no other deposits or withdrawals into or from the account.

So to find the value of the account after one year, we're going to start by converting our interest rate from a percent to a non-percent, just a regular decimal. And we'll do that by taking 3.6 and dividing it by 100, which is going to give me 0.036.

Now that I have my value for my interest, if I wanted to determine the amount of interest that I've earned on the account, I would multiply this value by 5,000. That would tell me how much interest I've earned in dollars.

But if I want to know the total value of the account, including my initial $5,000, I need to add 1 to this value, to account for my original amount. And then I can multiply that by 5,000. So I'm multiplying 5,000 by 1 plus 0.036.

And I can simplify this by simply writing 1.036. And when I multiply that by 5,000, I find that the value in my account after one year is $5,180. So if I wanted to find the value of my account after two years, is I would simply multiply by another factor of 1.036.

I'm applying my interest one more time for the second year. So now I'm multiplying 5,000 by 1.036 and another one. Simplifying this expression, I find that the value of my account after two years is $5,366.48.

Now we can generalize this pattern of multiplying by another factor for every year that your money is being invested. And we can write a formula for the value of an account after any number of years.

That formula looks like this, where A is the value of our account. P is the initial amount, or a principle that we invested. r is our interest rate. and t-- sorry, r is our annual interest rate. And t is our time in years.

So let's look at a formula for compound interest. And we're going to start with the formula we just derived, where interest is applied only one time per year. However, many banks will apply interest throughout the year, and this is called compounding interest. So they take a portion of the annual interest rate and they apply that throughout the year.

So our formula needs to change because we're going to be dividing our annual interest rate by however many times we're compounding the interest. So we define a variable n as the number of times per year that we're kind of compounding the interest.

So this fraction gives us the portion of interest that we're compounding every time. We also need to change our formula because we're going to be multiplying by this factor every time we compound the interest. And that's going to occur n times per year for t years.

So our exponent becomes n times t. So this formula can be used to find-- to solve problems about an account that earns compound interest. So now let's do an example using a compound interest formula.

Let's say I invested $1,000 into an account. And that account is earning 2.4% annual interest rate, and it's being compounded quarterly. So we can use our formula to find the value of our account after let's say three years.

And we're going to round that value to the nearest whole dollar. We're also going to assume that we're making no other withdrawals or deposits into our account. So using our formula we want to know the value of our account, so we want to know A.

We know that our principal, our initial amount, is $1,000. We know that our interest rate, expressed not as a percentage, but as a regular decimal is 0.024. I just divided this value by 100.

And then n however many times compounded in a year is going to be 4. Quarterly means 4. So we'll have 4 in our exponent and our value for t is 3-- 3 years.

So I'm going to start by simplifying this by dividing by my fraction in the parentheses. That's going to give me 0.006. I'll bring down the rest of my terms.

Now I can simplify by adding this becomes 1.006. I'm going to simplify using my exponent operation. So I'm going to rewrite the exponent is 12. 4 times 3 is just 12.

And 1.006 to the 12 power is going to give me approximately 1.074. And it's OK if I'm rounding here because I just want to know the value of the account rounded to the nearest whole dollar.

So now finally multiplying those two values together, I see that the value of my account after 3 years is approximately $1,074. So let's do another example involving our formula for compound interest.

This time I'm investing $500 into an account that earns 1.2% annual interest rate, and is compounded monthly. I want to find how many years, rounded to the nearest year, it's going to take for my initial investment of $500 to grow to $1,000.

So using my formula, my value for A, the amount of money I have in the account, is going to be 1,000. My initial investment was 100-- that's my value for P. My interest rate is 1.2%, and so my value for r is going to be 0.012. We divided by 100.

My value for n is going to be 12. Compounded monthly means 12 times per year. And my value for t I don't know. I'm going to leave that t. I want to know how many years it takes to go from $500 to $1,000.

So to solve for t, I'm going to start by canceling out this 500 in front. I'm going to divide both sides by 500. This will give me 2 is equal to--

Now I'm going to start simplifying. In my parentheses, I'm going to change this fraction to be-- when I divide those two values, I'll be 0.001. Bring down the rest of my terms. Now I can add these two values together.

All right so now I have something that's in exponential form-- an equation in exponential form. I can use a logarithm to solve for the variable that's in the exponent, t. So I'm going to take the log of both sides.

And so that is going to give me log of 2 is equal to log of 1.001 to the 12t. Now I know that there's a property of logarithms that says that this exponent can be turned into a factor multiplied in front of the log.

So this becomes log of 2 is equal to 12t times log of 1.001. So now I'm going to continue isolating my t variable by dividing both sides by log of 1.001.

And now I can use my-- here this will cancel. I can use my calculator to determine this value. Since I'm rounding to the nearest year, I'm going to go ahead and round this value. This is approximately 693.5.

And then finally dividing by 12 to both sides, I find that t is approximately equal to 57.79 years, which if we round to the nearest whole year, it will take approximately 58 years for the value of my account to go from $500 to $1,000.

So let's go of our key points from today. In the compound interest formula, A is the account balance, P is the principal, r is the annual interest rate, and n is the number of times per year that the interest is compounded, and t is the time in years.

Compounding interest means that a portion of the annual interest rate is applied throughout the year. So I hope that these key points and examples helped you understand a little bit more about compound interest. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Compound Interest