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1.11 Fractions: proper fractions, improper fractions, and mixed  (Page 2/2)

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Thus,

5 3 = 1 2 3 size 12{ { {5} 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. All three parts are shaded.

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

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

Thus,

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 1 1 3 size 12{1 { {1} over {3} } } {} can be expressed as 1 + 1 3 size 12{1+ { {1} over {3} } } {} The fraction 5 7 8 size 12{5 { {7} over {8} } } {} can be expressed as 5 + 7 8 size 12{5+ { {7} over {8} } } {} .

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

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

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

Converting improper fractions to mixed numbers

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

Two rectangles, each divided into three equal parts with vertical bars. Each part contains the fraction, one-third. Under the rectangle on the left is a bracket grouping all three parts together to make one. Under the rectangle on the right is a bracket under only one of the three parts, making one third. The two bracketed segments are added together.

4 3 = 1 3 + 1 3 + 1 3 ︸ 1 + 1 3 = 1 + 1 3 = 1 1 3

Thus, 4 3 = 1 1 3 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.
  1. The whole number part of the mixed number is the quotient.
  2. The fractional part of the mixed number is the remainder written over the divisor (the denominator of the improper fraction).

Sample set a

Convert each improper fraction to its corresponding mixed number.

5 3 size 12{ { {5} over {3} } } {} Divide 5 by 3.

Long division. 5 divided by 3 is one, with a remainder of 2. 1 is the whole number part, 2 is the numerator of the fractional part, and 3 is the denominator of the fractional part.

The improper fraction 5 3 = 1 2 3 size 12{ { {5} over {3} } =1 { {2} over {3} } } {} .

A number line with marks for 0, 1, and 2. In between 1 and 2 is a dot for five thirds, or one and two thirds.

46 9 size 12{ { {"46"} over {9} } } {} . Divide 46 by 9.

Long division. 46 divided by 9 is 5, with a remainder of 1. 5 is the whole number part, 1 is the numerator of the fractional part, and 9 is the denominator of the fractional part.

The improper fraction 46 9 = 5 1 9 size 12{ { {"46"} over {9} } =5 { {1} over {9} } } {} .

A number line with marks for 0, 5, and 6. In between 5 and 6 is a dot showing the location of forty-six ninths, or five and one ninth.

83 11 size 12{ { {"83"} over {"11"} } } {} . Divide 83 by 11.

Long division. 83 divided by 11 is 7, with a remainder of 6. 7 is the whole number part, 6 is the numerator of the fractional part, and 11 is the denominator of the fractional part.

The improper fraction 83 11 = 7 6 11 size 12{ { {"83"} over {"11"} } =7 { {6} over {"11"} } } {} .

A number line with marks for 0, 7, and 8. In between 7 and 8 is a dot showing the location of eighty-three elevenths, or seven and six elevenths.

104 4 size 12{ { {"104"} over {4} } } {} Divide 104 by 4.

Long division. 104 divided by 4 is 26, with a remainder of 0. 26 is the whole number part, 0 is the numerator of the fractional part, and 4 is the denominator of the fractional part.

104 4 = 26 0 4 = 26 size 12{ { {"104"} over {4} } ="26" { {0} over {4} } ="26"} {}

The improper fraction 104 4 = 26 size 12{ { {"104"} over {4} } ="26"} {} .

A number line with marks for 0, 25, 26, and 27. 26 is marked with a dot, showing the location of one hundred four fourths.

Practice set a

Convert each improper fraction to its corresponding mixed number.

9 2 size 12{ { {9} over {2} } } {}

4 1 2 size 12{4 { {1} over {2} } } {}

11 3 size 12{ { {"11"} over {3} } } {}

3 2 3 size 12{3 { {2} over {3} } } {}

14 11 size 12{ { {"14"} over {"11"} } } {}

1 3 11 size 12{1 { {3} over {"11"} } } {}

31 13 size 12{ { {"31"} over {"13"} } } {}

2 5 13 size 12{2 { {5} over {"13"} } } {}

79 4 size 12{ { {"79"} over {4} } } {}

19 3 4 size 12{"19" { {3} over {4} } } {}

496 8 size 12{ { {"496"} over {8} } } {}

62

Converting mixed numbers to improper fractions

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.

Two rectangles, each divided into three equal parts with vertical bars. Each part contains the fraction, one-third. Under the rectangle on the left is a bracket grouping all three parts together to make one. Under the rectangle on the right is a bracket under two of the three parts, making two thirds. The two bracketed segments are added together.

one and two thirds is equivalent to one plus two thirds. One can be expanded to three thirds, making the original number equivalent to the sum of five one-thirds, or five thirds.

Recall that multiplication describes repeated addition.

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

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

one and two thirds, with an arrow drawn from the denominator to the one.

Add: 3 + 2 = 5 size 12{3+2=5} {} . Place the 5 over the 3: 5 3 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,
  1. Multiply the denominator of the fractional part of the mixed number by the whole number part.
  2. To this product, add the numerator of the fractional part.
  3. Place this result over the denominator of the fractional part.

Sample set b

Convert each mixed number to an improper fraction.

5 7 8 size 12{5 { {7} over {8} } } {}

  1. Multiply: 8 ⋅ 5 = 40 size 12{8 cdot 5="40"} {} .
  2. Add: 40 + 7 = 47 size 12{"40 "+" 7 "=" 47"} {} .
  3. Place 47 over 8: 47 8 size 12{ { {"47"} over {8} } } {} .

Thus, 5 7 8 = 47 8 size 12{5 { {7} over {8} } = { {"47"} over {8} } } {} .

A number line showing the location of five and seven eigths, or 47 eights.

16 2 3 size 12{"16" { {2} over {3} } } {}

  1. Multiply: 3 ⋅ 16 = 48 size 12{"3 " cdot " 16 "=" 48"} {} .
  2. Add: 48 + 2 = 50 size 12{"48 "+" 2 "=" 50"} {} .
  3. Place 50 over 3: 50 3 size 12{ { {"50"} over {3} } } {}

Thus, 16 2 3 = 50 3 size 12{"16" { {2} over {3} } = { {"50"} over {3} } } {}

Practice set b

Convert each mixed number to its corresponding improper fraction.

8 1 4 size 12{8 { {1} over {4} } } {}

33 4 size 12{ { {"33"} over {4} } } {}

5 3 5 size 12{5 { {3} over {5} } } {}

28 5 size 12{ { {"28"} over {5} } } {}

1 4 15 size 12{1 { {4} over {"15"} } } {}

19 15 size 12{ { {"19"} over {"15"} } } {}

12 2 7 size 12{"12" { {2} over {7} } } {}

86 7 size 12{ { {"86"} over {7} } } {}

Exercises

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

3 2 size 12{ { {3} over {2} } } {}

improper fraction

4 9 size 12{ { {4} over {9} } } {}

5 7 size 12{ { {5} over {7} } } {}

proper fraction

1 8 size 12{ { {1} over {8} } } {}

6 1 4 size 12{6 { {1} over {4} } } {}

mixed number

11 8 size 12{ { {"11"} over {8} } } {}

1, 001 12 size 12{ { {1,"001"} over {"12"} } } {}

improper fraction

191 4 5 size 12{"191" { {4} over {5} } } {}

1 9 13 size 12{1 { {9} over {"13"} } } {}

mixed number

31 6 7 size 12{"31" { {6} over {7} } } {}

3 1 40 size 12{3 { {1} over {"40"} } } {}

mixed number

55 12 size 12{ { {"55"} over {"12"} } } {}

0 9 size 12{ { {0} over {9} } } {}

proper fraction

8 9 size 12{ { {8} over {9} } } {}

101 1 11 size 12{"101" { {1} over {"11"} } } {}

mixed number

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

11 6 size 12{ { {"11"} over {6} } } {}

14 3 size 12{ { {"14"} over {3} } } {}

4 2 3 size 12{4 { {2} over {3} } } {}

25 4 size 12{ { {"25"} over {4} } } {}

35 4 size 12{ { {"35"} over {4} } } {}

8 3 4 size 12{8 { {3} over {4} } } {}

71 8 size 12{ { {"71"} over {8} } } {}

63 7 size 12{ { {"63"} over {7} } } {}

9 size 12{9} {}

121 11 size 12{ { {"121"} over {"11"} } } {}

165 12 size 12{ { {"165"} over {"12"} } } {}

13 9 12 size 12{"13" { {9} over {"12"} } } {} or 13 3 4 size 12{"13" { {3} over {"4"} } } {}

346 15 size 12{ { {"346"} over {"15"} } } {}

5, 000 9 size 12{ { {5,"000"} over {9} } } {}

555 5 9 size 12{"555" { {5} over {9} } } {}

23 5 size 12{ { {"23"} over {5} } } {}

73 2 size 12{ { {"73"} over {2} } } {}

36 1 2 size 12{"36" { {1} over {2} } } {}

19 2 size 12{ { {"19"} over {2} } } {}

316 41 size 12{ { {"316"} over {"41"} } } {}

7 29 41 size 12{7 { {"29"} over {"41"} } } {}

800 3 size 12{ { {"800"} over {3} } } {}

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

4 1 8 size 12{4 { {1} over {8} } } {}

33 8 size 12{ { {"33"} over {8} } } {}

1 5 12 size 12{1 { {5} over {"12"} } } {}

6 7 9 size 12{6 { {7} over {9} } } {}

61 9 size 12{ { {"61"} over {9} } } {}

15 1 4 size 12{"15" { {1} over {4} } } {}

10 5 11 size 12{"10" { {5} over {"11"} } } {}

115 11 size 12{ { {"115"} over {"11"} } } {}

15 3 10 size 12{"15" { {3} over {"10"} } } {}

8 2 3 size 12{8 { {2} over {3} } } {}

26 3 size 12{ { {"26"} over {3} } } {}

4 3 4 size 12{4 { {3} over {4} } } {}

21 2 5 size 12{"21" { {2} over {5} } } {}

107 5 size 12{ { {"107"} over {5} } } {}

17 9 10 size 12{"17" { {9} over {"10"} } } {}

9 20 21 size 12{9 { {"20"} over {"21"} } } {}

209 21 size 12{ { {"209"} over {"21"} } } {}

5 1 16 size 12{5 { {1} over {"16"} } } {}

90 1 100 size 12{"90" { {1} over {"100"} } } {}

9001 100 size 12{ { {"9001"} over {"100"} } } {}

300 43 1, 000 size 12{"300" { {"43"} over {1,"000"} } } {}

19 7 8 size 12{"19" { {7} over {8} } } {}

159 8 size 12{ { {"159"} over {8} } } {}

Why does 0 4 7 size 12{0 { {4} over {7} } } {} not qualify as a mixed number?

See the definition of a mixed number.

Why does 5 qualify as a mixed number?

See the definition of a mixed number.

… because it may be written as 5 0 n , where n is any positive whole number.

Exercises for review

( [link] ) Round 2,614,000 to the nearest thousand.

( [link] ) Determine if 41,826 is divisible by 2 and 3.

( [link] ) Find the least common multiple of 28 and 36.

252

( [link] ) Specify the numerator and denominator of the fraction 12 19 size 12{ { {"12"} over {"19"} } } {} .
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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