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

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.

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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.

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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.

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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.

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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} } } {}

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

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

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

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

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

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

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

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

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

62

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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.

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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} } } {}

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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} } } {}

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

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

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

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

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

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

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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

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

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

proper fraction

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

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

mixed number

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

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

improper fraction

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

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

mixed number

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

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

mixed number

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

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

proper fraction

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

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

mixed number

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

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

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

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

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

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

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

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

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

9 size 12{9} {}

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

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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"} } } {}

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

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

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

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

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

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

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

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

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

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

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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} } } {}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

See the definition of a mixed number.

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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.

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Calculator problems

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

35 11 12 size 12{"35" { {"11"} over {"12"} } } {}

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27 5 61 size 12{"27" { {5} over {"61"} } } {}

1, 652 61 size 12{ { {1,"652"} over {"61"} } } {}

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83 40 41 size 12{"83" { {"40"} over {"41"} } } {}

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105 21 23 size 12{"105" { {"21"} over {"23"} } } {}

2, 436 23 size 12{ { {2,"436"} over {"23"} } } {}

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72 605 606 size 12{"72" { {"605"} over {"606"} } } {}

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816 19 25 size 12{"816" { {"19"} over {"25"} } } {}

20 , 419 25 size 12{ { {"20","419"} over {"25"} } } {}

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708 42 51 size 12{"708" { {"42"} over {"51"} } } {}

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6, 012 4, 216 8, 117 size 12{6,"012" { {4,"216"} over {8,"117"} } } {}

48 , 803 , 620 8, 117 size 12{ { {"48","803","620"} over {8,"117"} } } {}

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Exercises for review

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

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( [link] ) Find the product. 1,004 1,005 size 12{"1,004 " cdot "1,005"} {} .

1,009,020

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

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

252

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( [link] ) Specify the numerator and denominator of the fraction 12 19 size 12{ { {"12"} over {"19"} } } {} .

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Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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