Adding and subtracting fractions  Adding and subtracting fractions with the same, or common, denominator  Adding and subtracting fractions with different denominators  The least common multiple  Shortcuts

Adding or subtracting fractions with the same denominator  Add or subtract the numerators.  Write the result over the common denominator.

Adding or subtracting fractions with the different denominators Rewrite the sum or difference as the sum or difference of fractions with a common denominator.

The common denominator 4, used in the previous example, is called the lowest common denominator. The number 4 is the least common multiple of 2 and 4. Lowest common denominator Least common multiple Multiples of 2 : 2, 4, 6, 8, 10, … Multiples of 4 : 4, 8, 12, 16, …

The least common multiple of two or more numbers is the smallest number that each divides into evenly. The least common multiple of two or more numbers is often abbreviated as LCM. Least common multiple

Lowest common denominator The lowest common denominator of two or more fractions is the least common multiple of all the denominators. The lowest common denominator of two or more fractions is often abbreviated as LCD.

What is similar? What is different?

Similarities  In the first example, 2 and 4 share a factor of 2, and the denominator 4 is the lowest common denominator.  In the third example, 6 and 4 share a common factor of 2, but neither denominator is the lowest common denominator. Differences  In the second example, the two denominators share no common factors except 1; the lowest common denominator is the product of the two denominators.

Find the prime factors of each denominator.

Strategy for finding the least common multiple of two or more numbers.  Find the prime factorization of each number.  List all primes that occur in one or more factorizations.

 For each prime factor, identify the maximum number of times it occurs in any one factorization. This is the number of occurrences in the least common multiple. Strategy for finding the least common multiple of two or more numbers.

Find the least common multiple of 30 and 45.

Adding or subtracting fractions with different denominators  The goal is to rewrite the problem so that the fractions have a common denominator.

Factor each denominator into prime factors. Multiply each fraction in the sum or difference by the appropriate form of 1 to determine the equivalent fraction with the common denominator.  Find the least common multiple of all the denominators to use as the common denominator: Adding or subtracting fractions with different denominators The appropriate form of 1 is found by determining the prime factors needed to complete the common denominator.

 Add or subtract the equivalent fractions with the common denominator. Adding or subtracting fractions with different denominators

In this part of the lesson, and in the lesson on multiplying and dividing fractions, we rewrote numerators and/or denominators as the product of prime factors. The following shortcut is useful when performing arithmetic problems, but the strategies we discussed in these lessons will be the only strategy to use when working with algebraic fractions. Shortcuts