Historically, mathematics is an interesting combination of the practical and the theoretical. Sumerian tablets from 5,000 years ago preserve ancient accounting records, including measures and payments. The measures were added and subtracted to keep track of inventory. That's about as practical as it gets. Ancient mathematicians were fascinated by the interesting properties of mathematics, completely separate from any connection to practical applications. Without practical applications, the methods that these mathematicians developed fit new applications just perfectly.
One example of a fraction that had no practical application was the tangent function. The tangent is the fraction with one of the short sides of a right triangle as the numerator and the other short side in the denominator. The tangent function was originally created just as a mathematical curiosity, with no thought for practical applications. It didn't take long for a practical use to show up: measuring the size of the Earth. By measuring the ratio of the length of a shadow to the length of the stick that cast the shadow, Eratosthenes measured the angle of the sun in Alexandria and figured the circumference of the Earth to be approximately 40,000 kilometers.
Farmers would like to know the optimum conditions for growing corn. Probably the most important factor is the amount of water the corn needs. You might think this would vary as the corn grew through different stages, and you'd be right. It turns out that very young corn needs 3/100" of water per day, going up to as high as 7/20" per day. So a farmer calculating the amount of irrigation necessary will have to use an equation with fractions, because that's just the way it is: the corn needs only a fractional part of an inch of water each day.
When you get a paycheck, there are a variety of deductions taken out to result in your net pay. Luckily, these are all fractions because if they were whole numbers, you'd be paying to work. So you might have a four percent contribution to a health plan, eight percent in income tax and 3.2 percent for social security. Again, these are all fractions. Same thing when you spend some of that paycheck. The sales tax is just a small percentage of the purchase price. These are all situations that people invented and it just makes sense that these numbers would be fractions.
If every triangle was the same size as every other triangle and had every side equal to every other, then you wouldn't need fractions to describe the triangle: you'd just make each side equal one and you'd be done. If every stalk of corn always needed the same amount of water every day, then you could just call that "one cornwater," and you wouldn't need fractions. But there is real variation in the world. And when quantities are different from each other, they are only rarely exactly a whole multiple of some other unit. Fractions naturally occur all throughout the world, and mathematics reflects the world. So at its root, equations with fractions stem from variations in the world.