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Fraction Rules — Definition, Formula & Examples

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. b1\displaystyle \frac{b}{1} simplifies to b.

2. 1b\displaystyle \frac{1}{b} does not simplify any further.

3. 0b\displaystyle \frac{0}{b} simplifies to 0.

4. b0\displaystyle \frac{b}{0} is undefined.

Examples

71=7\frac{7}{1} = 7

110\displaystyle \frac{1}{10} does not simplify.

04=0\frac{0}{4} = 0

40\displaystyle \frac{4}{0} is undefined. So is 00.\displaystyle \frac{0}{0}\text{.}

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 102=5\displaystyle \frac{10}{2} = 5 is because 2·5 = 10.

The fraction  02=0\displaystyle \frac{0}{2} = 0  because 2·0 = 0.

The fraction 100\displaystyle \frac{10}{0} can't equal anything. There is no number you can multiply by 0 and get 10 as your answer. The fraction 100\displaystyle \frac{10}{0} is undefined.

What about 00?\displaystyle \frac{0}{0}\text{?} 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 00\displaystyle \frac{0}{0} can equal any number. As a result we say 00\displaystyle \frac{0}{0} is indeterminate, which is a special kind of undefined expression.

B. Negative Fractions

1. −ab\displaystyle -\frac{a}{b} is the same as −ab\displaystyle \frac{-a}{b} and a−b\displaystyle \frac{a}{-b}

2. −a−b\displaystyle \frac{-a}{-b} simplifies to  ab\displaystyle \frac{a}{b}

3. −ab\displaystyle -\frac{a}{b} is NOT the same as −a−b\displaystyle \frac{-a}{-b}

Examples

5−3=−53=−53\frac{5}{-3} = \frac{-5}{3} = -\frac{5}{3}

−7−8=78\frac{-7}{-8} = \frac{7}{8}

−411≠−4−11-\frac{4}{11} \neq \frac{-4}{-11}

C. Cancellation (a ≠ 0, b ≠ 0, c ≠ 0)

1. aa\displaystyle \frac{a}{a} cancels to 1

2. abac\displaystyle \frac{ab}{ac} cancels to bc\displaystyle \frac{b}{c}

3. ab⋅bc\displaystyle \frac{a}{b} \cdot \frac{b}{c} cancels to ac\displaystyle \frac{a}{c}

4. ab⋅ca\displaystyle \frac{a}{b} \cdot \frac{c}{a} cancels to cb\displaystyle \frac{c}{b}

5. a⋅ba\displaystyle a \cdot \frac{b}{a} cancels to b

6. ba⋅a\displaystyle \frac{b}{a} \cdot a cancels to b

Examples

66=1\frac{6}{6} = 1

1228=3⋅47⋅4=37\frac{12}{28} = \frac{3 \cdot 4}{7 \cdot 4} = \frac{3}{7}

−109⋅913=−1013=−1013\frac{-10}{9} \cdot \frac{9}{13} = \frac{-10}{13} = -\frac{10}{13}

611⋅56=511\frac{6}{11} \cdot \frac{5}{6} = \frac{5}{11}

4⋅74=74 \cdot \frac{7}{4} = 7

2−3 (−3)=2\frac{2}{-3}\,(-3) = 2

D. Addition

1. ab+cb=a+cb\displaystyle \frac{a}{b} + \frac{c}{b} = \frac{a+c}{b}

2. a+bc=acc+bc=ac+bc\displaystyle a + \frac{b}{c} = \frac{ac}{c} + \frac{b}{c} = \frac{ac+b}{c}

3. ab+cd=adbd+bcbd=ad+bcbd\displaystyle \frac{a}{b} + \frac{c}{d} = \frac{ad}{bd} + \frac{bc}{bd} = \frac{ad+bc}{bd}

Examples

34+54=84=2\frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2

6+85=305+85=3856 + \frac{8}{5} = \frac{30}{5} + \frac{8}{5} = \frac{38}{5}

67+34=2428+2128=4528\frac{6}{7} + \frac{3}{4} = \frac{24}{28} + \frac{21}{28} = \frac{45}{28}

E. Subtraction

1. ab−cb=a−cb\displaystyle \frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}

2. a−bc=acc−bc=ac−bc\displaystyle a - \frac{b}{c} = \frac{ac}{c} - \frac{b}{c} = \frac{ac-b}{c}

3. ab−c=ab−bcb=a−bcb\displaystyle \frac{a}{b} - c = \frac{a}{b} - \frac{bc}{b} = \frac{a-bc}{b}

4. ab−cd=adbd−bcbd=ad−bcbd\displaystyle \frac{a}{b} - \frac{c}{d} = \frac{ad}{bd} - \frac{bc}{bd} = \frac{ad-bc}{bd}

Examples

23−53=−33=−1\frac{2}{3} - \frac{5}{3} = \frac{-3}{3} = -1

1−94=44−94=−541 - \frac{9}{4} = \frac{4}{4} - \frac{9}{4} = -\frac{5}{4}

157−2=157−147=17\frac{15}{7} - 2 = \frac{15}{7} - \frac{14}{7} = \frac{1}{7}

23−12=46−36=16\frac{2}{3} - \frac{1}{2} = \frac{4}{6} - \frac{3}{6} = \frac{1}{6}

F. Multiplication

1. ab⋅cd=acbd\displaystyle \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

2. a⋅bc=a1⋅bc=abc\displaystyle a \cdot \frac{b}{c} = \frac{a}{1} \cdot \frac{b}{c} = \frac{ab}{c}

3. ab⋅c=ab⋅c1=acb\displaystyle \frac{a}{b} \cdot c = \frac{a}{b} \cdot \frac{c}{1} = \frac{ac}{b}

Examples

5−3⋅711=−3533\frac{5}{-3} \cdot \frac{7}{11} = -\frac{35}{33}

6⋅27=1276 \cdot \frac{2}{7} = \frac{12}{7}

95⋅(−3)=−275\frac{9}{5} \cdot (-3) = -\frac{27}{5}

Careful!!

1. ac⋅bc≠abc\displaystyle \frac{a}{c} \cdot \frac{b}{c} \neq \frac{ab}{c}

2. Mixed numbers are shorthand for addition and not multiplication. For example, 213\displaystyle 2\tfrac{1}{3} means 2+13\displaystyle 2 + \frac{1}{3} and NOT 2⋅13.\displaystyle 2 \cdot \frac{1}{3}\text{.}

G. Division

1.   ab  cd=ab⋅dc=adbc\displaystyle \frac{\;\frac{a}{b}\;}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}

2.   ab  c=  ab  c1=ab⋅1c=abc\displaystyle \frac{\;\frac{a}{b}\;}{c} = \frac{\;\frac{a}{b}\;}{\frac{c}{1}} = \frac{a}{b} \cdot \frac{1}{c} = \frac{a}{bc}

3. abc=a1bc=a1⋅cb=acb\displaystyle \frac{a}{\frac{b}{c}} = \frac{\frac{a}{1}}{\frac{b}{c}} = \frac{a}{1} \cdot \frac{c}{b} = \frac{ac}{b}

Examples

31047=310⋅74=2140\frac{\frac{3}{10}}{\frac{4}{7}} = \frac{3}{10} \cdot \frac{7}{4} = \frac{21}{40}

236=2361=23⋅16=218=19\frac{\frac{2}{3}}{6} = \frac{\frac{2}{3}}{\frac{6}{1}} = \frac{2}{3} \cdot \frac{1}{6} = \frac{2}{18} = \frac{1}{9}

274=2174=21⋅47=87\frac{2}{\frac{7}{4}} = \frac{\frac{2}{1}}{\frac{7}{4}} = \frac{2}{1} \cdot \frac{4}{7} = \frac{8}{7}

See also

Distributing rules

Key Formula

Addition: ab+cd=ad+bcbdMultiplication: ab⋅cd=acbdDivision: ab÷cd=ab⋅dc\text{Addition: } \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \text{Multiplication: } \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd} \qquad \text{Division: } \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}
Where:
  • a,ca, c = Numerators of the fractions
  • b,db, d = Denominators of the fractions (must not be zero)
  • dc\frac{d}{c} = The reciprocal of the second fraction, used when dividing

Worked Example

Problem: Simplify the expression: 23+54\frac{2}{3} + \frac{5}{4}
Step 1: Find a common denominator. The denominators are 3 and 4, and their least common multiple is 12.
LCD=12\text{LCD} = 12
Step 2: Rewrite each fraction with denominator 12 by multiplying numerator and denominator by the appropriate factor.
23=2×43×4=812,54=5×34×3=1512\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}, \qquad \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}
Step 3: Add the numerators over the common denominator.
812+1512=8+1512=2312\frac{8}{12} + \frac{15}{12} = \frac{8 + 15}{12} = \frac{23}{12}
Step 4: Check whether the result simplifies. Since 23 is prime and does not divide 12, the fraction is already in simplest form.
2312\frac{23}{12}
Answer: 23+54=2312\frac{2}{3} + \frac{5}{4} = \frac{23}{12}

Another Example

This example demonstrates the division rule (multiply by the reciprocal) combined with cross-cancellation before multiplying, which is a different operation from the addition shown in the first example.

Problem: Simplify: 38÷94\frac{3}{8} \div \frac{9}{4}
Step 1: Apply the division rule: multiply by the reciprocal of the second fraction.
38÷94=38⋅49\frac{3}{8} \div \frac{9}{4} = \frac{3}{8} \cdot \frac{4}{9}
Step 2: Before multiplying across, cancel common factors. The 3 in the numerator and the 9 in the denominator share a factor of 3. The 4 in the numerator and the 8 in the denominator share a factor of 4.
38⋅49=3 18 2⋅4 19 3=12⋅13\frac{3}{8} \cdot \frac{4}{9} = \frac{\cancel{3}^{\,1}}{\cancel{8}^{\,2}} \cdot \frac{\cancel{4}^{\,1}}{\cancel{9}^{\,3}} = \frac{1}{2} \cdot \frac{1}{3}
Step 3: Multiply the remaining numerators and denominators.
1×12×3=16\frac{1 \times 1}{2 \times 3} = \frac{1}{6}
Answer: 38÷94=16\frac{3}{8} \div \frac{9}{4} = \frac{1}{6}

Frequently Asked Questions

Why can't you have zero in the denominator of a fraction?
Division asks 'what number times the denominator gives the numerator?' If the denominator is 0, no number (or every number) satisfies this, so the result is undefined. For example, 100\frac{10}{0} has no answer because no number times 0 equals 10. The special case 00\frac{0}{0} is called indeterminate because every number times 0 equals 0.
How do you add fractions with different denominators?
First find a common denominator — typically the least common multiple (LCM) of the two denominators. Rewrite each fraction as an equivalent fraction with that common denominator. Then add the numerators and keep the denominator the same. Finally, simplify if possible.
What is the difference between multiplying and dividing fractions?
To multiply fractions, multiply the numerators together and the denominators together: ab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}. To divide fractions, flip the second fraction (take its reciprocal) and then multiply: ab÷cd=ab⋅dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}. Division is just multiplication by the reciprocal.

Fraction Addition vs. Fraction Multiplication

Fraction AdditionFraction Multiplication
Formulaab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}ab⋅cd=acbd\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}
Common denominator needed?Yes — you must have the same denominator before combiningNo — just multiply straight across
Numerator operationAdd (or subtract) the numerators after rewriting with a common denominatorMultiply the numerators directly
Common mistakeAdding numerators and denominators separately: 12+13≠25\frac{1}{2}+\frac{1}{3} \neq \frac{2}{5}Forgetting to cancel common factors before multiplying, leading to large numbers that need simplifying

Why It Matters

Fraction rules appear throughout algebra, from solving equations with rational expressions to working with rates, proportions, and probability. In more advanced courses like calculus, you routinely add and simplify algebraic fractions when combining partial fractions or simplifying derivatives. Mastering these rules now prevents errors in nearly every branch of mathematics you will study later.

Common Mistakes

Mistake: Adding fractions by adding both numerators and denominators separately, e.g., writing 12+13=25\frac{1}{2} + \frac{1}{3} = \frac{2}{5}.
Correction: You must find a common denominator first. The correct calculation is 12+13=36+26=56\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}. Only numerators are added; the common denominator stays the same.
Mistake: Cancelling terms that are added rather than factors that are multiplied, e.g., writing 3+x3=x\frac{3 + x}{3} = x.
Correction: You can only cancel a factor common to the entire numerator and the entire denominator. Since the numerator 3+x3 + x is a sum, you cannot cancel the 3. The fraction 3+x3\frac{3+x}{3} simplifies to 1+x31 + \frac{x}{3}, not xx.

Related Terms