First, we need some terminology. When we wish to divide one quantity by another, the quantity that we are dividing by is called the divisor, and the quantity being divided is call the dividend. The result of the operation is sometimes called the quotient (although the word quotient is used in other ways in mathematics as well).
In the first and third examples here, we factored the factors after the multiplication step to check for possible simplification of the result. In the second example, we could see that there was no possibility of simplification, because 5 is obviously not a factor of either 4 or 8.
mixing numerators and denominators in the multiplication step. This gives entirely the wrong answer. You must do the multiplication step just like you would multiply any other pair of fractons together.
when we drop the denominator of 1 (which can always be done because of the property that for any number b). We get exactly the same result as we would have obtained using the original rule for dividing by a fraction. This example illustrates why the invert and multiply rule works for dividing by a fraction.
As has been true of all other arithmetic operations, division with mixed numbers requires you to first convert the mixed numbers to pure fractions and then apply the methods appropriate for fractions.