Back in middle school, We learned that if the interest rate given, r, is annual, it must be divided By the number of times compounded per year, n, to get interest rare per period.
I was reviewing compound interest for a standardised test and it got me thinking. Why is an “annual rate” given in the first place? The interest rate per period applies to a different principal every period so you cant just find that rate by dividing the annually rate equally Into n Period , right? What does, say an interest rate of 10% compounded quarterly, even mean? Is hid something just to confuse you?
I mean, you can. You just have to take powers. Using your example:
You have annual interest rate $10%$, compounded quarterly. Suppose your principle is $100$, then, after one year, you have:
$$100 \times (1 + \frac{10}{4\cdot 100})^4 = 100 \times (1 + 0.025)^4 = 110.38.$$
This is standard practice in finance, and you should get used to it, so that you don't get swindled. Mess around with some numbers.
At the end, the definitions are just long-standing conventions in banking.