I WANT TO KNOW WHY
In my quest to know everything, and to know it now, I want to know why two fraction-calculating methods work. I understand how adding and subtracting work with fractions, but multiplying and dividing fractions puzzles me.
When multiplying a fraction, I want to understand why cancellation works. I understand about reducing a fraction to its simplest form. For example 6/8 = 3/4. You reduce the numerator and the denominator of the same fraction. But when you multiply fractions, you reduce the numerator of one fraction and the denominator of a different fraction using a common factor. Why does this work? I could not find any understandable answers online.
When dividing a fraction, you turn the divisor upside down and then multiply it times the dividend. (You can cancel out the numerator of one fraction and the denominator of the other.) Uh? To divide fractions you invert one fraction and multiply them? Who thought that up? Why does it work? Imagine if you could solve all your problems that way. “Yeah, just turn one part upside down and multiply by the other. That should solve it.”
The reasons for why these methods work is likely simple and obvious. That is why I don’t see them. Once I find out the reasons, I am going to scream, “Eureka! Eureka!” Then I will turn one eureka upside down and multiply it by the other. Don’t ask me why.