Now that we know what positive fractions are, we consider three types of positive fractions: proper fractions, improper fractions, and mixed numbers.

Positive Proper Fraction

Fractions in which the whole number in the numerator is strictly less than the whole number in the denominator are called positive proper fractions. On the number line, proper fractions are located in the interval from 0 to 1. Positive proper fractions are always less than one.

The closed circle at 0 indicates that 0 is included, while the open circle at 1 indicates that 1 is not included.

Some examples of positive proper fractions are

{}1212 size 12{ { {1} over {2} } } {}, 3535 size 12{ { {3} over {5} } } {}, 20272027 size 12{ { {"20"} over {"27"} } } {}, and 106255106255 size 12{ { {"106"} over {"255"} } } {}

Note that 1 < 21 < 2 size 12{"1 "<" 2"} {}, 3 < 53 < 5 size 12{"3 "<" 5"} {}, 20 < 2720 < 27 size 12{"20 "<" 27"} {}, and 106 < 225106 < 225 size 12{"106 "<" 225"} {}.

Positive Improper Fractions

Fractions in which the whole number in the numerator is greater than or equal to the whole number in the denominator are called positive improper fractions. On the number line, improper fractions lie to the right of (and including) 1. Positive improper fractions are always greater than or equal to 1.

Some examples of positive improper fractions are

3232 size 12{ { {3} over {2} } } {}, 8585 size 12{ { {8} over {5} } } {}, 4444 size 12{ { {4} over {4} } } {}, and 1051610516 size 12{ { {"105"} over {"16"} } } {}

Note that 3≥23≥2 size 12{3 >= 2} {}, 8≥58≥5 size 12{8 >= 5} {}, 4≥44≥4 size 12{4 >= 4} {}, and 105≥16105≥16 size 12{"105" >= "16"} {}.

Positive Mixed Numbers

A number of the form

nonzero whole number + proper fraction nonzero whole number + proper fraction size 12{"nonzero whole number "+" proper fraction"} {}

is called a positive mixed number. For example, 235235 size 12{2 { {3} over {5} } } {} is a mixed number. On the number line, mixed numbers are located in the interval to the right of (and includ­ing) 1. Mixed numbers are always greater than or equal to 1.

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 Number = Natural Number + Proper Fraction

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 ⋅)

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

To understand how we might convert an improper fraction to a mixed number, let's consider the fraction, 4343 size 12{ { {4} over {3} } } {}.

43 = 13 + 13 + 13 ︸ 1 + 13 = 1 + 13 = 1 13 43 = 13 + 13 + 13 ︸ 1 + 13 = 1 + 13 = 1 13

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

We can illustrate a procedure for converting an improper fraction to a mixed number using this example. However, the conversion is more easily accomplished by dividing the numerator by the denominator and using the result to write the mixed number.

Converting an Improper Fraction to a Mixed Number

To convert an improper fraction to a mixed number, divide the numerator by the denominator.

To understand how to convert a mixed number to an improper fraction, we'll recall

mixed number = (natural number) + (proper fraction)

and consider the following diagram.

Recall that multiplication describes repeated addition.

Notice that 5353 size 12{ { {5} over {3} } } {} can be obtained from 123123 size 12{1 { {2} over {3} } } {} using multiplication in the following way.

Multiply: 3⋅1=33⋅1=3 size 12{3 cdot 1 - 3} {}

Add: 3+2=53+2=5 size 12{3+2=5} {}. Place the 5 over the 3: 5353 size 12{ { {5} over {3} } } {}

The procedure for converting a mixed number to an improper fraction is illustrated in this example.

Converting a Mixed Number to an Improper Fraction

To convert a mixed number to an improper fraction,

For the following 15 problems, identify each expression as a proper fraction, an improper fraction, or a mixed number.

For the following 15 problems, convert each of the improper fractions to its corresponding mixed number.

For the following 15 problems, convert each of the mixed num­bers to its corresponding improper fraction.

Calculator Problems

For the following 8 problems, use a calculator to convert each mixed number to its corresponding improper fraction.