powered by Google (TM)
index: click on a letter
A B C D E
F G H  I  J
K L M N O
P Q R S T
U V W X Y
Z A to Z index
index: subject areas
numbers & symbols
sets, logic, proofs
geometry
algebra
trigonometry
advanced algebra
& pre-calculus
calculus
advanced topics
probability &
statistics
real world
applications
multimedia
entries
about mathwords  
website feedback  


Fraction Rules

Algebra rules for combining fractions. These rules apply for both proper fractions and improper fractions. They apply for all rational expressions as well.

 

A. Special Fractions

1. simplifies to b.

2. does not simplify any further.

3. simplifies to 0.

4. is undefined.

Examples

does not simplify.

is undefined. So is .

Special note: Why is it OK to have 0 on top (in the numerator) and not on the bottom (in the denominator)?

Consider for a moment what division means. The reason that is because 2·5 = 10.

The fraction    because 2·0 = 0.

The fraction can't equal anything. There is no number you can multiply by 0 and get 10 as your answer. The fraction is undefined.

What about ? It's undefined, too, but for a slightly different reason. If you multiply the 0 in the denominator by any number at all you get the 0 in the numerator. It seems that can equal any number. As a result we say is indeterminate, which is a special kind of undefined expression.

B. Negative Fractions

1. is the same as and

2. simplifies to  

3. is NOT the same as

Examples

C. Cancellation (a ≠ 0, b ≠ 0, c ≠ 0)

1. cancels to 1

2. cancels to

3. cancels to

4. cancels to

5. cancels to b

6. cancels to b

Examples

D. Addition

1.

2.

3.

Examples

E. Subtraction

1.

2.

3.

4.

Examples

F. Multiplication

1.

2.

3.

Examples

Careful!!

1.

2. Mixed numbers are shorthand for addition and not multiplication. For example, means and NOT .

G. Division

1.

2.

3.

Examples

 

See also

Distributing rules

 


  this page updated 27-aug-12
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes