equals 
and how to switch between improper fractions and mixed numbers.Recall the earlier discussion of why
equals
and how to switch between improper fractions and mixed numbers.
To do this, the numerator needs to be discussed in more detail. The numerator is a description of how many parts you have. The fraction
tells you that you have five parts. But you have some flexibility in how you arrange those five parts. For instance, you already expressed it as
+
, or 1 +
. Essentially, what you did was to split the numerator into two pieces: 4 and 1. If you wanted to express this as a fraction, you could say that
becomes
. This hasn’t changed anything, because 4 + 1 equals 5, so you still have the same number of parts.
Then, as noted above, you can split the fraction into two separate fractions. For instance
becomes
. This is the same as saying that 5 fourths equals 4 fourths plus 1 fourth. So there are several different ways of representing the same fraction. For example, 
. Here is a visual representation:
As a general rule, you can always split the numerator of a fraction into different parts and thus split a fraction into multiple fractions. This is just reversing the process of adding fractions. When you add fractions, take two fractions with the same denominator and combine them into one fraction. Here do the exact opposite—turn one fraction into two separate fractions, each with the same denominator. Now that the fraction 
is split into two fractions, you can take advantage of the fact that fractions, at their essence, represent division. As was discussed earlier,
= 1, and another way to think of
is 4 ÷ 4.
To switch from an improper fraction to a mixed number, figure out how many complete units there are. To do that, figure out the largest multiple of the denominator that is less than or equal to the numerator. For the fraction
, 4 is the largest multiple of 4 that is less than 5. So split the fraction into
and 1/4. Note that
equals 1, so the mixed number is
.
Try it again with the fraction
. This time, the largest multiple of 4 that is less than 15 is 12. So you can split the fraction
into
+
. In other words,
. And
= 3, so the fraction
becomes the mixed number
.
Try one with a different denominator. How do you turn the fraction
into a mixed number? This time you need the largest multiple of 7 that is less than or equal to 16. The number 14 is the largest multiple of 7 that is less than 16, so once again split the fraction
into
and
. In other words,
. Because 14 divided by 7 equals 2, your mixed number is
.
Change the following improper fractions to mixed numbers:
Now that you know how to change a number from an improper fraction to a mixed number, you also need to be able to do the reverse. Suppose you have the mixed number
. How do you turn this number into a fraction?
Remember that you can think of any integer as a fraction. The number 1, for instance, can be thought of many different ways. It can be thought of as
. It can also be thought of as
. In other words, a unit circle can be split into 2 equal pieces, with 2 of those pieces forming a whole unit circle again. 1 can also be written as
,
,
, and so on.
In fact, you can think of the process of turning mixed numbers into improper fractions as simple fraction addition. The fraction
is the same thing as 5 +
, so you can think of it as
. Now you know what to do—change
so that it has a denominator of 3. The way to do that is to multiply
by
, which equals
.
So the mixed number is really
.
Change the following mixed numbers to improper fractions.
