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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. Fraction b over 1, where b is the numerator and 1 is the denominator. 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. Fraction with numerator b and denominator 0 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  0 divided by 2 equals 0  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 Fraction with numerator 'a' and denominator '-b

2. Fraction with numerator negative a and denominator negative b: (-a)/(-b) 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 Fraction with numerator negative a and denominator negative b: (-a)/(-b)

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. Fraction with numerator ab and denominator ac, illustrating simplification by canceling the common factor a. cancels to Fraction b over c, where b is the numerator and c is the denominator.

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. Fraction multiplication: (a/b) · (c/a) cancels to Fraction c over b, where c is the numerator and b is the denominator.

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

6. (b/a) · a cancels to b

Examples

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

12/28 = (3·4)/(7·4) = 3/7, showing simplification by canceling common factor 4

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.

2. Math equation showing: a + b/c = ac/c + b/c = (ac + b)/c

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

1. a/b - c/b = (a-c)/b

2. Equation showing a minus b/c equals ac/c minus b/c equals (ac minus b)/c

3. Math equation showing: a/b - c = a/b - bc/b = (a - bc)/b

4. Fraction subtraction rule: a/b − c/d = ad/bd − bc/bd = (ad − bc)/bd

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

1. Fraction multiplication rule: (a/b) · (c/d) = ac/bd

2. Math equation showing: a · (b/c) = (a/1) · (b/c) = ab/c

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

Examples

5 divided by -3, multiplied by 7/11, equals -35/33

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

9/5 · (−3) = −27/5

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, Mixed number 2 and 1/3 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

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