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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses proper fractions, improper fractions, and mixed numbers. By the end of the module students should be able to distinguish between proper fractions, improper fractions, and mixed numbers, convert an improper fraction to a mixed number and convert a mixed number to an improper fraction.

Section overview

  • Positive Proper Fractions
  • Positive Improper Fractions
  • Positive Mixed Numbers
  • Relating Positive Improper Fractions and Positive Mixed Numbers
  • Converting an Improper Fraction to a Mixed Number
  • Converting a Mixed Number to an Improper Fraction

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

Positive proper fractions

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.

A number line. 0 is marked with a black dot, and 1 is marked with a hollow dot. The distance between the two is labeled, all proper fractions are located in this interval.

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

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

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

Positive improper fractions

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.

A number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive improper fractions.

Some examples of positive improper fractions are

3 2 size 12{ { {3} over {2} } } {} , 8 5 size 12{ { {8} over {5} } } {} , 4 4 size 12{ { {4} over {4} } } {} , and 105 16 size 12{ { {"105"} over {"16"} } } {}

Note that 3 2 size 12{3>= 2} {} , 8 5 size 12{8>= 5} {} , 4 4 size 12{4>= 4} {} , and 105 16 size 12{"105">= "16"} {} .

Positive mixed numbers

Positive mixed numbers

A number of the form

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

is called a positive mixed number . For example, 2 3 5 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 number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive mixed numbers.

Relating positive improper fractions and positive mixed numbers

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.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third.

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

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

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

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

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded.

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

Improper fraction = whole number.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. One part is shaded.

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

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

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

Thus,

4 3 = 1 1 3 size 12{ { {4} over {3} } =1 { {2} over {3} } } {} .

Improper fraction = mixed number.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

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

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
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Professor
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LITNING Reply
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analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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