Hi, and welcome. My name is Anthony Varela and today we're going to talk about compound interest. So we'll develop a formula to use to solve compound interest problems and then apply it to solve for an account balance and to solve for time. And this is going to require logarithms.
So to develop this formula, we're going to illustrate how $1,000 can grow due to interest alone. So our assumptions are no deposits, withdrawals, purchases, payments, just interest. So this $1,000 grows by 6% every year.
So 6% I'm going to express as a decimal. But I'm going to add this to 1 first. This can be thought of as including 100% of the initial $1,000 plus 6%. So I'm going to write this as 1.06.
Now after another year of gaining interest, I can multiply another 1.06, then after another year adding a factor of 1.06. So I'm going to express this using an exponent to show these repeated multiplication. So I'm going to use the variable t as an exponent. And this represents time in years.
Now a couple of other variables I'd like to introduce are percentage rate, APR. This is expressed as a decimal r, this is added to 1. We also have our principal balance, P, our initial amount. And all of this then equals our account balance, which we can show as A. So, so far we have A equals P times 1 plus r raised to the power of t.
But I haven't addressed compound interest yet. So with compound interest, a portion of the APR is applied n number of times per year. So if interest is compounded quarterly, they'd be four times per year. So we have to divide our interest rates by n to show that we're applying a portion of the interest rates for the whole year.
So, so far this has changed to a P times 1 plus r over n. Now, this also affects our exponent. So for example, if interest is compounded four times per year, after one year, this would happen four times. After two years, this would happen eight times. So we have to adjust our exponent to reflect that.
So our exponent is n times t So this is our compound interest formula, A equals P times 1 plus r over n raised to the power of n times t. Let's go ahead and use this to solve some problems.
So our first situation, we have an investment account that starts with a balance of $4,500. And it's gaining 5.5% interest, which compounds monthly. What is the account balance after three years, assuming no additional deposits or withdrawals were made? So let's narrow this down to our important information, starting with $4,500, 5.5% interest, this compounds monthly, and I need to know the balance of the account after three years.
So lining this up with my variables, my principal amount is $4,500. My interest rate is 0.055. So remember, I always expressed this as a decimal. Compounded monthly means 12 times per year. So n equals 12. And I need to know the account balance after three years. So t equals 3.
So how can we solve then for A, our account balance? So we're going to make our substitutions into our formula. And what should we tackle first? Well, let's take care of what's inside the parentheses, 1 plus 0.055 divided by 12.
So I have rounded my decimal number. Of course, the more digits you use, the more accurate your answer will be. And we need to also put in our exponent 12 times 3. So we're going to raise this to the power of 36.
So apply that exponent before multiplying by 4,500. So we have our large decimal number there raised to the power of 36. And now we're going to multiply that by our principal, $4,500. So A then equals $5,305.27. So after three years my account has gone from $4,500 to $5,305.27 for doing absolutely nothing, that's not bad.
In our second example, we have my credit card which has an annual interest rate of 14.5% and this is compounded monthly. If my current balance is $800, how long will it take for the balance to reach $1,000, also assuming no additional purchases or payments are made?
Now this is our opposite problem. So instead of me making money for doing nothing, I'm owing money for doing nothing. So how long will it take for my $800 purchases to all of a sudden cost me $1,000? So let's narrow this down to our important information, 14.5% interest compounded monthly, so n is 12, current balance is $800, and how long will it take for the balance to reach $1,000?
So fitting this information into our formula, I need to know when the balance will reach $1,000. So A is $1,000. P is $800. r is 14.5% as a decimal dividing this by 12. So my exponent is 12 times t.
So how do I solve for t? Well, let's go ahead and clean up what we see in parentheses. So 1 plus 0.145 over 12. Once again, I'm rounding. Use more decimals will give you a more accurate answer. And now what we're going to do is divide by P before we deal with the exponent.
So $1,000 divided by $800 is 1.25. And now how do we take care of that exponent? Well, this is where we use logarithms. So we're going to apply the log to both sides. And in doing so, you're going to be using a log property.
So if you take the log of an expression with an exponent, that exponent is just multiplied out in front. So we have 12t times the log of this number. So to recap where we're at so far, the log of 1.25 equals 12t times the log of 1.012083.
So now we can solve for t by dividing by 12 times the log of 1.012083. So typing that then into our calculator, we get a decimal number, 1.54824-- so about a year and a half for my credit card balance to jump from $800 to $1,000, assuming my credit card company lets me get away with making no payments for a year and a half.
So let's review our notes on compound interest. Here is our compound interest formula, lots of variables here-- account balance, our principal amount, our annual percentage rate, the number of times per year interest is compounded, and time in years. To solve for A, we can clean up 1 plus r over n, raise it to our exponent n times t, and multiply it by P.
If we're solving for t, we can clean up 1 plus r over n, divide by the principal P, and apply the log, which you're going to be using then a log property. And then you can divide to isolate time. So thanks for watching this tutorial on compound interest. Hope to see you next time.