Fraction Rules

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. $The fraction 1 over b$ does not simplify any further.

3. $The fraction 0 over b, where b is in the denominator.$ simplifies to 0.

4. is undefined.

Examples

$The fraction 7/1 equals 7$

$The fraction 1/10, with 1 in the numerator and 10 in the denominator.$ does not simplify.

$The fraction 0/4 equals 0$

$The fraction 4/0, where 4 is the numerator and 0 is the denominator, illustrating an undefined fraction.$ is undefined. So is $The fraction 0/0, with 0 in the numerator and 0 in the denominator.$ .

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 $10 divided by 2 equals 5, shown as a fraction with 10 in the numerator and 2 in the denominator.$ is because 2·5 = 10.

The fraction because 2·0 = 0.

The fraction $The fraction 10 over 0, written as 10/0, illustrating an undefined fraction with zero in the denominator.$ can't equal anything. There is no number you can multiply by 0 and get 10 as your answer. The fraction $The fraction 10 over 0, written as 10/0, illustrating an undefined fraction with zero in the denominator.$ is undefined.

What about $The fraction 0/0, with 0 in the numerator and 0 in the denominator.$ ? 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 $The fraction 0/0, with 0 in the numerator and 0 in the denominator.$ can equal any number. As a result we say $The fraction 0/0, with 0 in the numerator and 0 in the denominator.$ is indeterminate, which is a special kind of undefined expression.

B. Negative Fractions

1. $Negative fraction: negative a divided by b, written as -a/b$ is the same as $The fraction negative a over b, written as -a/b$ and

2. simplifies to $The fraction a over b, with variable a in the numerator and variable b in the denominator.$

3. $Negative fraction: negative a divided by b, written as -a/b$ is NOT the same as

Examples

$5/-3 = -5/3 = -(5/3), showing three equivalent ways to write a negative fraction.$

$Equation showing -7 divided by -8 equals 7/8, demonstrating that two negatives in a fraction simplify to a positive.$

$-4/11 ≠ -4/-11, showing these two fractions are not equal$

C. Cancellation (a ≠ 0, b ≠ 0, c ≠ 0)

1. $The fraction a/a, where a is a variable in both the numerator and denominator.$ cancels to 1

2. cancels to

3. $The fraction a/b divided by b/c, showing two stacked fractions side by side with variables a, b, and c.$ cancels to $The fraction a over c, written as a/c$

4. cancels to

5. $The fraction a·b/a, where a is both the numerator multiplier and denominator, simplifying to b.$ cancels to b

6. cancels to b

Examples

$The fraction 6/6 equals 1, illustrating that any nonzero number divided by itself simplifies to 1.$

$Equation showing (-10/9) · (9/13) = -10/13 = -10/13, demonstrating multiplication of fractions with cancellation.$

$(6/11) · (5/6) = 5/11, demonstrating fraction multiplication by canceling the common factor 6.$

$4 · (7/4) = 7, showing a whole number multiplied by its reciprocal fraction equals the numerator.$

$Fraction 2/(-3) times (-3) = 2, showing a fraction multiplied by its denominator equals the numerator.$

D. Addition

1. $Fraction addition rule: a/b + c/b = (a+c)/b, showing fractions with a common denominator b can be combined.$

3. $Formula showing fraction addition: a/b + c/d = ad/bd + bc/bd = (ad + bc)/bd$

Examples

$3/4 + 5/4 = 8/4 = 2, demonstrating addition of fractions with like denominators resulting in a whole number.$

$6 + 8/5 = 30/5 + 8/5 = 38/5, showing addition of a whole number and fraction by converting to common denominator.$

$6/7 + 3/4 = 24/28 + 21/28 = 45/28, demonstrating fraction addition by finding a common denominator of 28.$

E. Subtraction

Examples

$2/3 minus 5/3 equals negative 3/3 equals negative 1, demonstrating subtraction of fractions with common denominators.$

$1 − 9/4 = 4/4 − 9/4 = −5/4, demonstrating fraction subtraction by finding a common denominator of 4.$

$15/7 − 2 = 15/7 − 14/7 = 1/7, showing subtraction of a whole number from a fraction by converting to a common denominator.$

$2/3 − 1/2 = 4/6 − 3/6 = 1/6, showing fraction subtraction by finding a common denominator of 6.$

F. Multiplication

3. $Math equation showing (a/b) · c = (a/b) · (c/1) = ac/b, demonstrating multiplication of a fraction by a whole number.$

Examples

$6 times 2/7 equals 12/7, showing multiplication of a whole number by a fraction.$

Careful!!

1. $Fraction rule showing (a/c) · (b/c) ≠ ab/c, illustrating that multiplying fractions with same denominator does not simplify...$

2. Mixed numbers are shorthand for addition and not multiplication. For example, means $The mixed number 2 and 1/3, showing whole number 2 added to fraction with numerator 1 and denominator 3.$ and NOT $2 · (1/3): multiplication of whole number 2 and fraction one-third$ .

G. Division

1. $Math formula showing (a/b)÷(c/d) = (a/b)·(d/c) = ad/bc, demonstrating division of fractions by multiplying by the reciprocal.$

2. $Math equation showing (a/b)/(c/1) = (a/b)·(1/c) = a/(bc), demonstrating division of fractions by multiplying by reciprocal.$

3. Mathematical equation showing (a/b)/(c) = (a/1)/(b/c) = (a/1)·(c/b) = ac/b

Examples

$(3/10) ÷ (4/7) = (3/10) · (7/4) = 21/40, showing division of fractions by multiplying by the reciprocal.$

Math equation showing (2/3)/(6) = (2/3)/(6/1) = 2/3 · 1/6 = 2/18 = 1/9

$Math equation showing (2/7)/(4/7) = (2/1)·(4/7) = 2/1·4/7 = 8/7, demonstrating division of fractions with like denominators.$

See also

Distributing rules