A relationship between improper fractions and mixed numbers is suggested by two facts. The first is that improper fractions and mixed numbers are located in the same interval on the number line. The second fact, that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following observations.

Divide a whole quantity into 3 equal parts.

Now, consider the following examples by observing the respective shaded areas.

In the shaded region, there are 2 one thirds, or
2323 size 12{ { {2} over {3} } } {}.

2
1
3
=
2
3
2
1
3
=
2
3
size 12{2 left ( { {1} over {3} } right )= { {2} over {3} } } {}

There are 3 one thirds, or
3333 size 12{ { {3} over {3} } } {}, or 1.

3
1
3
=
3
3
or
1
3
1
3
=
3
3
or
1
size 12{3 left ( { {1} over {3} } right )= { {3} over {3} } ````` ital "or"````1} {}

Thus,

3
3
=
1
3
3
=
1
size 12{ { {3} over {3} } =1} {}

Improper fraction = whole number.

There are 4 one thirds, or
4343 size 12{ { {4} over {3} } } {}, or 1 and
1313 size 12{ { {1} over {3} } } {}.

413=43413=43 size 12{4 left ( { {1} over {3} } right )= { {4} over {3} } `} {}or
1and131and13 size 12{1``` ital "and"``` { {1} over {3} } } {}

The terms 1 and
1313 size 12{ { {1} over {3} } } {} can be represented as
1+131+13 size 12{1+ { {1} over {3} } } {} or
113113 size 12{1 { {1} over {3} } } {}

Thus,

43=11343=113 size 12{ { {4} over {3} } =1 { {2} over {3} } } {}.

Improper fraction = mixed number.

There are 5 one thirds, or
5353 size 12{ { {5} over {3} } } {}, or 1 and
2323 size 12{ { {2} over {3} } } {}.

5
1
3
=
5
3
or
1
and
2
3
5
1
3
=
5
3
or
1
and
2
3
size 12{5 left ( { {1} over {3} } right )= { {5} over {3} } `````````` ital "or"``````````1`` ital "and"`` { {2} over {3} } } {}

The terms 1 and
2323 size 12{ { {2} over {3} } } {} can be represented as
1+231+23 size 12{1+ { {2} over {3} } } {} or
123123 size 12{1 { {2} over {3} } } {}.

Thus,

53=12353=123 size 12{ { {5} over {3} } =1 { {2} over {3} } } {}.

Improper fraction = mixed number.

There are 6 one thirds, or
6363 size 12{ { {6} over {3} } } {}, or 2.

6
1
3
=
6
3
=
2
6
1
3
=
6
3
=
2
size 12{6 left ( { {1} over {3} } right )= { {6} over {3} } =2} {}

Thus,

6
3
=
2
6
3
=
2
size 12{ { {6} over {3} } =2} {}

Improper fraction = whole number.

The following important fact is illustrated in the preceding examples.

Mixed numbers are the *sum* of a natural number and a proper fraction. Mixed number = (natural number) + (proper fraction)

For example
113113 size 12{1 { {1} over {3} } } {} can be expressed as
1+131+13 size 12{1+ { {1} over {3} } } {} The fraction
578578 size 12{5 { {7} over {8} } } {} can be expressed as
5+785+78 size 12{5+ { {7} over {8} } } {}.

It is important to note that a number such as
5+785+78 size 12{5+ { {7} over {8} } } {} does *not* indicate multiplication. To indicate multiplication, we would need to use a multiplication symbol (such as ⋅)

578578 size 12{5 { {7} over {8} } } {} means
5+785+78 size 12{5+ { {7} over {8} } } {} and *not*
5⋅785⋅78 size 12{5 cdot { {7} over {8} } } {}, which means 5 times
7878 size 12{ { {7} over {8} } } {} or 5 multiplied by
7878 size 12{ { {7} over {8} } } {}.

Thus, mixed numbers may be represented by improper fractions, and improper fractions may be represented by mixed numbers.

Comments:"Used as supplemental materials for developmental math courses."