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Fractions. Rules. Reduction to lowest terms.

Def. Rational algebraic fraction. A fraction whose numerator and denominator are both polynomials.

Examples.

Rules of fractions. The rules for the manipulation of algebraic fractions are the same as for fractions in arithmetic.

Rule 1. The value of a fraction is unchanged if its numerator and denominator are both multiplied or divided by the same (non-zero) quantity.

Example.

Two fractions are said to be equivalent if one can be obtained from the other by multiplying or dividing numerator and denominator by the same quantity. The two fractions in the example are equivalent fractions.

Reduction of a fraction to its lowest terms. A fraction is said to be in its lowest terms if its numerator and denominator contain no common factor other than 1. To reduce a fraction to its lowest terms:

1) factor both the numerator and denominator into their constituent prime factors

2) divide both numerator and denominator by all of their common factors.

Problem. Reduce the following fraction to lowest terms:

Solution.

where we divided numerator and denominator by the common factor 3ab3(4x - 3).

The operation of dividing out common factors in the numerator and denominator is called canceling. Cancellation is sometimes indicated by a sloped line. See Fig. 1. The process of reducing a fraction to its lowest terms is called simplifying it.

Rule 2. Signs. There are three signs associated with a fraction: the sign of the numerator, the sign of the denominator, and the sign of the entire fraction. Any two of these signs can be changed without changing the value of the fraction. If any one of these signs is changed, the sign of the fraction changes. If there is no sign before a fraction, a plus sign is implied.

Example.

Rule 3. Addition or subtraction of fractions.

Case 1. Common denominator. The sum and difference of two fractions with a common denominator are given by the formulas

Example.

Case 2. Different denominators. To add or subtract two fractions with different denominators we first multiply the numerators and denominators of both fractions by such quantities as will make their denominators equal, thus changing the fractions into equivalent fractions with the same denominator. We then add or subtract them by the rule for adding or subtracting fractions with a common denominator.

The usual procedure for changing the fractions into equivalent fractions with the same denominator is as follows:

1) Decompose the denominators of both fractions into their prime factors

2) Find the least common multiple of the two denominators. This will be the common denominator to be used. It is called the Least Common Denominator (or L.C.D.).

3) Multiply the numerator and denominator of each fraction by whatever factors are needed to create the Least Common Denominator in that fraction.

Example.

where we multiplied the numerator and denominator of the first fraction by (x - 1) and the numerator and denominator of the second fraction by x to create the Least Common Denominator of x(x-1)(x-2) in each fractions.

Rule 4. Multiplication of fractions. The product of two fractions is given by the following formula:

In other words, the product of two fractions is the product of the numerators divided by the product of the denominators.

Example.

Rule 5. Division of fractions. The quotient of two fractions is given by the following formula:

In other words, to divide one fraction by another, we invert the divisor and multiply.

Example.

Complex fractions. A simple fraction contains no fraction in either its numerator or denominator. A complex fraction is a fraction containing a fraction in either it numerator or denominator or both.

Example. The following is a complex fraction:

To simplify a complex fraction:

1) Reduce the numerator and denominator to simple fractions

2) Divide the two resulting fractions

Example.

References

Hawks, Luby, Touton. Second-Year Algebra

Murray R. Spiegel. College Algebra

Raymond W. Brink. A First Year of College Mathematics