Pronunciation: /ˈfræk ʃən/ ?

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:

3 5/8

1 2/7

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.

AdditionTo 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+6)/30=31/30
SubtractionTo 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
MultiplicationTo multiply 2 fractions:
  1. Multiply the numerators.
  2. Multiply the denominators
  3. Reduce the fraction, if needed.
DivisionTo divide 2 fraction: Multiply the first fraction by the reciprocal of the second fraction.(3/7)/(3/4)=(3/7)*(4/3)=12/21=7/4.
Least common denominatorTo 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.
5/6 and 3/8. 6=2*3, 8=2*2*2,lcd(6,8)=2*2*2*3=24
Reduce a fractionTo reduce a fraction:
  1. Find the prime factors of both the numerator and the denominator.
  2. Cancel common factors.
  3. Multiply the remaining factors.

How to convert a mixed number to a fraction.




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.

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


  1. fraction. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Fine, Henry B., Ph. D.. Number-System of Algebra Treated Theoretically and Historically, 2nd ed., pp 12-15. D. C. Heath & Co., Boston, U.S.A., 1907. (Accessed: 2009-12-19).
  3. Oberg, Erik. Arithmetic Simplified, pp 21-31. Industrial Press, 1914. (Accessed: 2010-01-22).
  4. Oberg, Erik. Elementary Algebra, pg 23. Industrial Press, 1914. (Accessed: 2010-01-22).
  5. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 9-11. International Textbook Company, January 1963. (Accessed: 2010-01-12).

Printed Resources

Cite this article as:

Fraction. 2009-12-19. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2009-12-19: Added "References" (McAdams, David.)
2008-12-29: Changed 'complex fraction' from keyword to vocabulary link (McAdams, David.)
2008-11-28: Added section 'Complex Fractions' (McAdams, David.)
2008-11-26: Changed equations to images (McAdams, David.)
2008-09-16: Added text describing numerator and denominator, changed some expressions to hot_eqn (McAdams, David.)
2007-10-19: Clarified examples in How To section (McAdams, David.)
2007-09-23: Removed alert (McAdams, David.)
2007-08-20: Add this revision history (McAdams, David.)
2007-08-08: Initial version (McAdams, David.)

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