**Introduction to Reducing Fractions**

When working with fractions, you are usually asked to “reduce the fraction to lowest terms” or to “write the fraction in lowest terms” or to “reduce the fraction.” These phrases mean that the numerator and denominator have no common factors. For example, is reduced to lowest terms but is not. Reducing fractions is like fraction multiplication in reverse. We will first use the most basic approach to reducing fractions. In the next section, we will learn a quicker method.

First write the numerator and denominator as a product of prime numbers. Refer to the Appendix if you need to review how to find the prime factorization of a number. Next collect the primes common to both the numerator and denominator (if any) at beginning of each fraction. Split each fraction into two fractions, the first with the common primes. Now the fraction is in the form of “1” times another fraction.

**Examples**

Find practice problems and solutions at Reducing Fractions Practice Problems - Set 1.

**Greatest Common Divisor (GCD)**

Fortunately there is a less tedious method for reducing fractions to their lowest terms. Find the largest number that divides both the numerator and the denominator. This number is called the *greatest common divisor* (GCD). Factor the GCD from the numerator and denominator and rewrite the fraction. In the previous examples and practice problems, the product of the common primes was the GCD.

**Examples**

Find practice problems and solutions at Reducing Fractions Practice Problems - Set 2.

**Reducing Fractions without the GCD**

Sometimes the greatest common divisor is not obvious. In these cases you might find it easier to reduce the fraction in several steps.

**Examples**

Find practice problems and solutions at Reducing Fractions Practice Problems - Set 3.

For the rest of the book, reduce fractions to their lowest terms.

Practice problems for this concept can be found at: Algebra Fractions Practice Test.

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