What is the formula for converting an improper fraction to a mixed number

Question

There are methods for converting improper fractions to mixed numbers, but I am interested on finding a formula to which I can input the numerator and denominator of an improper fraction and get an expression of the form $$x+\frac{y}{z},$$ which is pretty much how a mixed number looks like.

Answer 1

There is no formula because expressing the number by euclidean division isn't "pretty much how a mixed number looks like" it is exactly what a mixed number looks like. When you write "$1 \frac{1}{2}$", it is simply a shorter way to write "$1 + \frac12$".

So to answer your question, if you have a improper fraction $\frac{x}{y}$ the closest thing you'll get is first doing the euclidean division of $x$ by $y$, which will get you something of the form $x=q\cdot y+r$, where $q$ is the quotient and $r$ the remainder. Divide both sides by $y$ and you get $\frac{x}{y} = q + \frac{r}{y}$. Notice that you have expressed $\frac{x}{y}$ as a whole number plus a fraction, so to express $\frac{x}{y}$ as a mixed fraction, you simply write $q \frac{r}{y}$.