Fraction

Pronunciation: /ˈfræk.ʃən/ Explain

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

The expression on the top is called the numerator and the expression on the bottom is called the 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:

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

Operations on Fractions

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

Operation	Description	Example
Addition	To add 2 fractions: Find the least common denominator Convert each fraction to the least common denominator. Add the numerators, copy the common denominator Reduce the fraction, if needed.
Subtraction	To subtract 2 fractions: Find the common denominator Convert each fraction to the common denominator. Subtract the numerators, copy the common denominator Reduce the fraction, if needed.
Multiplication	To multiply 2 fractions: Multiply the numerators. Multiply the denominators Reduce the fraction, if needed.
Division	To divide 2 fraction: Multiply the first fraction by the reciprocal of the second fraction.
Least common denominator	To find the least common denominator of two fractions: Find the prime factors of both fractions. Take the product of the prime factors, making sure each factor is multiplied once. The product is the least common denominator.
Reduce a fraction	To reduce a fraction: Find the prime factors of both the numerator and the denominator. Cancel common factors. Multiply the remaining factors.

How to convert a mixed number to a fraction.

Original Expression	Operation	Calculation	Description
1		none	Mixed number to convert.
2			Multiply the whole number by the denominator. The product is 10.
3			Add the product and the numerator. The sum is 13.
4			Use the sum as the numerator and copy the denominator from the fraction.

Why this works:

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

How to convert a fraction to a mixed number.

Step	Original Expression	Operation	Description
1		none	Fraction to convert.
2			Divide 23 by 7. This gives three wholes and a remainder of 2/7.
3			Make the mixed number.

Why this works:

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. https://www.allmathwords.org/en/f/fraction.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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.)