Adding Fractions with Different Denominators
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Like fractions are fractions with the same denominator. You can add and subtract like fractions easily - simply add or subtract the numerators and write the sum over the common denominator.
- Before you can add or subtract fractions with different denominators, you must first find equivalent fractions with the same denominator, like this:
- Find the smallest multiple (LCM) of both numbers.
- Rewrite the fractions as equivalent fractions with the LCM as the denominator.
- When working with fractions, the LCM is called the least common denominator (LCD).
- Since only like fractions can be added or subtracted, we first have to convert unlike fractions to equivalent like fractions. We want to find the smallest, or least, common denominator, because working with smaller numbers makes our calculations easier. The least common denominator, or LCD, of two fractions is the smallest number that can be divided by both denominators. There are two methods for finding the least common denominator of two fractions:
Example 2
Method 1:
Write the multiples of both denominators until you find a common multiple.
- The first method is to simply start writing all the multiples of both denominators, beginning with the numbers themselves. Here's an example of this method. Multiples of 4 are 4, 8, 12, 16, and so forth (because 1 × 4=4, 2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…--that's the number we're looking for, 12, because it's the first one that appears in both lists of multiples. It's the least common multiple, which we'll use as our least common denominator.
Method 2:
Use prime factorization.
- For the second method, we use prime factorization-that is, we write each denominator as a product of its prime factors. The prime factors of 4 are 2 times 2. The prime factors of 6 are 2 times 3. For our least common denominator, we must use every factor that appears in either number. We therefore need the factors 2 and 3, but we must use 2 twice, since it's used twice in the factorization for 4. We get the same answer for our least common denominator, 12.
Example 3
prime factorization of 4 = 2 × 2
prime factorization of 6 = 2 × 3
LCD = 2 × 2 × 3 = 12
- Now that we have our least common denominator, we can make equivalent like fractions by multiplying the numerator and denominator of each fraction by the factor(s) needed. We multiply 3/4 by 3/3, since 3 times 4 is 12, and we multiply 1/6 by 2/2, since 2 times 6 is 12. This gives the equivalent like fractions 9/12 and 2/12. Now we can add the numerators, 9 + 2, to find the answer, 11/12.
Example 4
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