I understand that this is an extremely basic question and has been asked before, but we just learned this in school and as I am in eighth grade, I don't really understand previous explanations. I am just wondering why dividing the annual interest rate by the number of compounding periods is relevant & how it affects the total interest amount.
I used the same equation: Y=10(0.4/n)^n(1), but made n=1 the first time and n=2 the second time. There was only a 0.4 dollar difference, which begs the question why do we divide by n in the first place and how does it affect our output?
Let's consider an example. Say on January 1, you put (deposit) $100$ units of currency (dollars, euros, whatever) into a bank account. In return, the bank offers to pay you "6% interest per year." But what does this actually mean?
For instance, the bank could pay out the interest at the end of the year (Dec 31). That means on January 1 of the next year, you'd have $106$ in your bank account instead of $100$. But this might not be very enticing to you, since if you were to withdraw the money before a year has passed, you would not receive the interest payment.
So, the bank instead decides to pay the interest to you more frequently, but in smaller amounts. If they divided up the interest into $12$ equal parts, and pay it out at the end of each month, then on February 1st, you'd have $100(0.06/12) = 0.50$ interest added to your account, for a total balance of $100.50$, as long as you don't make any other deposits or withdrawals.
But because your balance is slightly higher at the beginning of February, then at the end of February, the bank would pay $(100.50)(0.06/12) = 0.5025$ in interest to you, and your balance would be $100.50 + 0.5025 = 101.0025$ on March 1. This is a tiny bit more than you would have gotten if all the bank had done was given you $0.50$ each month. This is the effect of compounding interest--i.e., the interest earned each period also earns interest.
If we continue through the rest of the year, the total balance on Dec 31 would be $$100 \left(1 + \frac{0.06}{12}\right)^{12} \approx 106.16778.$$ The total interest paid, as a percentage of the initial balance, is $$\frac{6.16778}{100} = 6.16778\%.$$ We call this the annual percentage yield of the account.
Remember that the purpose of the monthly compounding was so you would still have been paid some fraction of the interest you earned if you had withdrawn early--you wouldn't have to wait the full year to receive anything.
However, notice how this total balance is about $0.16778$ more than the amount that you would have gotten had the interest been paid out only once at the end of the year. Again, this is the effect of compounding.
Still, you'd have to wait a full month to get whatever interest is being paid out monthly. So why couldn't the bank compound interest daily? In fact, this is what most banks do. As in the monthly case, the idea is to divide the annual rate of $6\%$ into $365$ daily pieces. So after the first day, you would earn $(100)(0.06/365) \approx 0.0164384$ in interest. And after a full year, the total balance would be $$100\left(1 + \frac{0.06}{365}\right)^{365} \approx 106.18313.$$ Accordingly, the annual percentage yield if compounding occurred daily, is $6.18313\%$. This is again slightly larger than both the monthly and annual compounding frequency cases. It might not seem that significant--being fractions of a unit of currency--but it would probably make a difference to you if you had deposited $100000$ instead of $100$, or if you kept the money in the account over a period of many years.
Banks and financial professionals use specific terms to describe interest rates. If they say "APR" (annual percentage rate), this is referring to the $6\%$ figure that we used at the beginning of the example. This is also called a nominal interest rate. In order to calculate the actual amount of interest that is earned, we also have to know the compounding frequency, which is also sometimes called the conversion frequency. So if a savings account offers $6\%$ APR convertible monthly, then that means the annual percentage yield, or "APY," is $6.16778\%$ as we illustrated above. Sometimes, the APY is called the effective interest rate.
To help with your understanding, here are some exercises.