Fraction

Pronunciation: /ˈfræk.ʃən/ Explain

A fraction is an expression divided by an expression.[1] Examples of fractions are:

2/5

9/4

(x+3)/(x-2)
The expression on the top is called the numerator and the expression on the bottom is called the denominator.

numerator/denominator

Fractions whose values are greater than one are sometimes called improper fractions. Improper fractions are often easier to use than mixed numbers, especially when using computers.

A whole number followed by a fraction is called a mixed number. Two examples of mixed numbers are:

3 5/8

1 2/7

A complex fraction is a fraction with another fraction in the numerator and/or denominator:

(1+3/4)/2.

Operations on Fractions

For a complete list of rules for operations on fractions, see Fraction Rules.

OperationDescriptionExample
Addition To add 2 fractions:
  1. Find the least common denominator
  2. Convert each fraction to the least common denominator.
  3. Add the numerators, copy the common denominator
  4. Reduce the fraction, if needed.
 (5/6)+(2/10)=(25/30)+(6/30)=(25+3)/30=31/30
Subtraction To subtract 2 fractions:
  1. Find the common denominator
  2. Convert each fraction to the common denominator.
  3. Subtract the numerators, copy the common denominator
  4. Reduce the fraction, if needed.
 (5/6)-(2/10)=(25/30)-(6/30)=(25-6)/30=19/30
Multiplication To multiply 2 fractions:
  1. Multiply the numerators.
  2. Multiply the denominators
  3. Reduce the fraction, if needed.
 (3/4)*(2/3)=(3*2)/(4*3)=6/12=1/2
Division To divide 2 fraction: Multiply the first fraction by the reciprocal of the second fraction. (3/7) divided by (3/4) = (3/7)*(4/3)=4/7
Least common denominator To find the least common denominator of two fractions:
  1. Find the prime factors of both fractions.
  2. Take the product of the prime factors, making sure each factor is multiplied once.
  3. The product is the least common denominator.
 5/8, 3/8 ; 6 = 2*3, 8=2*2*2 ; lcd(6,8)=2*2*2*2=24
Reduce a fraction To reduce a fraction:
  1. Find the prime factors of both the numerator and the denominator.
  2. Cancel common factors.
  3. Multiply the remaining factors.
 (4/12) = (2*2)/(2*2*3) = 1/3

How to convert a mixed number to a fraction.

Original
Expression
Operation
Calculation
Description
12 3/5noneMixed number to convert.
22 3/52*5=10Multiply the whole number by the denominator. The product is 10.
32 3/510+3=13Add the product and the numerator. The sum is 13.
42 3/513/5Use the sum as the numerator and copy the denominator from the fraction.

Why this works:

2 3/5 represented as 13/5
Figure 1: Representation of converting 2 3/5 to 13/5.

How to convert a fraction to a mixed number.

StepOriginal
Expression		OperationDescription
123/7noneFraction to convert.
223/723/7 = 3 r 2Divide 23 by 7. This gives three wholes and a remainder of 2/7.
323/73 2/7Make the mixed number.

Why this works:

Representation of converting 23/7 to 3 2/7.
Figure 2: Representation of converting 23/7 to 3 2/7

References

  1. McAdams, David E.. All Math Words Dictionary, fraction. 2nd Classroom edition 20150108-4799968. pg 82. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically. 2nd edition. pp 12-15. www.archive.org. D. C. Heath & Co., Boston, U.S.A.. 1907. Last Accessed 7/11/2018. http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n21/mode/1up/search/fraction. Buy the book
  3. Oberg, Erik. Arithmetic Simplified. pp 21-31. www.archive.org. Industrial Press. 1914. Last Accessed 7/11/2018. http://www.archive.org/stream/arithmeticsimpli00oberrich#page/21/mode/1up/search/fraction. Buy the book
  4. Oberg, Erik. Elementary Algebra. pg 23. www.archive.org. Industrial Press. 1914. Last Accessed 7/11/2018. http://www.archive.org/stream/elementaryalgebr00oberrich#page/n26/mode/1up/search/fraction. Buy the book
  5. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pp 9-11. www.archive.org. International Textbook Company. January 1963. Last Accessed 7/11/2018. http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up. Buy the book

Cite this article as:

 McAdams, David E. Fraction. 4/21/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/f/fraction.html.

Image Credits

Revision History

4/21/2019: Modified equations and expression to match the new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/9/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
12/29/2008: Changed 'complex fraction' from keyword to vocabulary link. (McAdams, David E.)
11/28/2008: Added section 'Complex Fractions'. (McAdams, David E.)
11/26/2008: Changed equations to images. (McAdams, David E.)
9/16/2008: Added text describing numerator and denominator, changed some expressions to hot_eqn. (McAdams, David E.)
10/19/2007: Clarified examples in How To section. (McAdams, David E.)
9/23/2007: Removed alert. (McAdams, David E.)
8/20/2007: Add this revision history. (McAdams, David E.)
8/8/2007: Initial version. (McAdams, David E.)

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