# Fraction Rules

Pronunciation: /ˈfræk ʃən rulz/ ?

 Figure 1: Fraction

Fraction rules are a set of algebraic rules for working with fractions. A fraction has a numerator and a denominator. A fraction represents a division operation. The numerator is the dividend. The denominator is the divisor.

Fraction Rules
OperationEquationsExamplesDescription
Adding two fractions[2]To add fractions, transform each fraction so they have a common denominator. Add the numerators and use the common denominator as the denominator. Reduce the fraction. See Operations on Fractions: Addition and Subtraction.
Subtracting two fractionsTo subtract fractions, transform each fraction so they have a common denominator. Subtract the numerators and use the common denominator as the denominator. Reduce the fraction. See Operations on Fractions: Addition and Subtraction.
Multiplying two fractions[2]To multiply fractions, multiply the numerators and multiply the denominators. Reduce the fraction. See Operations on Fractions: Multiplication.
Multiplying a fraction and a whole number.To multiply a fraction and a whole number, multiply the numerator by the whole number. The denominator remains unchanged. Reduce the fraction if possible.
Dividing two fractions[2]To divide fractions, flip the divisor upside down then multiply by the dividend. Reduce the fraction. See Operations on Fractions: Division.
Dividing a fraction by an integer.To divide a fraction by a whole number, convert the whole number to a fraction, the divide the fractions.
Raising a fraction to a power.See Operations on Fractions: Exponentiation.
Converting a mixed number to an improper fraction.To convert a mixed number to an improper fraction, multiply the whole part by the denominator and add the product to the numerator. The denominator remains unchanged. See How to Convert a Mixed Number to a Fraction.
Converting an improper fraction to a mixed number.To convert an improper fraction to a mixed number, divide the numerator by the denominator using a remainder. The mixed number is the quotient plus the remainder divided by the denominator. See How to Convert a Fraction to a Mixed Number..
Zero numerator.Applying the property of multiplying by zero, a zero numerator with a zero denominator is zero. See Property of Multiplying by 0.
Zero denominator.Since division by zero is undefined, a zero denominator makes the fraction undefined.
One minus sign.Since , apply the associative property of multiplication to get
Two minus signs.Since , apply the associative property of multiplication to get
If a fraction has the same non-zero numerator and denominator, the fraction's value is 1.Anything except 0 divided by itself is 1.
Any integer can be made into a fraction.Since , apply the property of multiplying by 1: . See Property of Multiplying by 1.
Reducing fractions.Given two arbitrary values a and b, and values c, d, and e such that a=c*d and b=c*e, . See Reducing Fractions.
Building fractions.Given a fraction a/b and a number d that is a multiple of d, find e such that b·e=d, then a/b=(a·e)/(b·e).
Operations on complex fractions.Simplify the complex fractions, then use the rules for simple fractions.To manipulate a complex fraction, convert it to a simple fraction, then follow the rules for simple fractions. See Complex Fraction.
Converting a decimal number to a fraction.To convert a decimal to a fraction, change the decimal to a whole number and divide it by 10n where n is the number of digits after the decimal point.
Converting a percentage to a fraction.To convert a percentage to a fraction, use the percentage as the numerator, 100 as the denominator, then simplify.
Comparing fractions with like denominators.To compare fractions with like denominators, compare the numerators. The relationship between the fractions is the same as the relationship between the denominators.
Comparing fractions with unlike denominators.To compare fractions with unlike denominators, either convert them to a decimal or transform them to a common denominator, then compare them.
Table 1

### References

1. 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). http://www.archive.org/stream/thenumbersystemo17920gut/17920-pdf#page/n21/mode/1up/search/fraction.
2. Oberg, Erik. Arithmetic Simplified, pp 21-31. Industrial Press, 1914. (Accessed: 2010-01-22). http://www.archive.org/stream/arithmeticsimpli00oberrich#page/21/mode/1up/search/fraction.
3. Oberg, Erik. Elementary Algebra, pg 23. Industrial Press, 1914. (Accessed: 2010-01-22). http://www.archive.org/stream/elementaryalgebr00oberrich#page/n26/mode/1up/search/fraction.
4. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry, pp 9-11,36-40. International Textbook Company, January 1963. (Accessed: 2010-01-12). http://www.archive.org/stream/algebraandtrigon033520mbp#page/n18/mode/1up.

### Printed Resources

Fraction Rules. 2010-02-05. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/f/fractionrules.html.