Fractions are a subject in mathematics that are where many people become confused. In the United States, at least, fractions are usually the first topic in mathematics that follow the basic, whole number math that students learn in elementary school. Between fourth and sixths grade, fractions, and how to add, subtract, multiply, divide and reduce them are going to probably be the biggest topic in arithmetic. (Along with the related decimal and percents.) This is also where many students get lost, and get a fear of mathematics.
I used to think that fractions were a stupid thing to teach. Beyond the colloquial fractions used to describe things like tanks of gas, cups of sugar, and Jon Bon Jovi's dependence on prayer to reach his goals, fractions aren't used much in every day life. 13/17th aren't things we use to measure sugar. And dividing fractions is probably the best example of a useless mathematical skill.
That is how I used to think. I teach ABE/GED now, and I have come to understand how important fractions are, how necessary they are to understanding algebra, and how the skills used to manipulate fractions, even the seemingly useless ones, are a prerequisite for understanding beyond arithmetic and into algebra. Because fractions are not about cups of sugar or even about algorithms. Fractions are about the relationships between numbers.
From a concrete point of view, to say that 1/2 is equal to 3/6 is untrue. If Alice has one dollar, Bob has two dollars, Carl has three dollars, and Debbie has six dollars, Alice and Carl do not have the same amount of money. One and three are very different things. But Alice and Carl both have the same amount of money in relation to Bob and Debbie. It seems like a simple point, but the leap from concrete numbers to describing the relationships between numbers is a gigantic leap, and one that is needed to go into algebra and further.
So I now believe that as seemingly useless, frustrating, and discouraging as the process of teaching people how to divide 13/17 by 50/24 is, it is also very necessary in order for people to begin to work on the more abstract forms of mathematics.
I hope that after reading this, the reader has made sense out of at least one part of fifth grade.