Download Finding LCD & Adding/Subtracting Fractions with Unlike Denominators and more Exams Painting in PDF only on Docsity!
2.6 Adding and Subtracting Fractions and Mixed Numbers with
Unlike Denominators
Learning Objective(s) 1 Find the least common multiple (LCM) of two or more numbers. 2 Find the Least Common Denominator 3 Add fractions with unlike denominators. 4 Add mixed numbers 5 Subtract fractions with unlike denominators.. 6 Subtract mixed numbers without regrouping. 7 Subtract mixed numbers with regrouping. 8 Solve application problems that require the subtraction of fractions or mixed numbers.
Finding Least Common Multiples
Sometimes fractions do not have the same denominator. They have unlike
denominators. Think about the example of the house painters. If one painter has 2 3
can of paint and his painting partner has 1 2
can of paint, how much do they have in
total? How can you add these fractions when they do not have like denominators?
The answer is that you can rewrite one or both of the fractions so that they have the same denominator. This is called finding a common denominator. While any common denominator will do, it is helpful to find the least common multiple of the two numbers in the denominator because this will save having to simplify at the end. The least common multiple is the least number that is a multiple of two or more numbers. Least common multiple is sometimes abbreviated LCM.
There are several ways to find common multiples, some of which you used when comparing fractions. To find the least common multiple (LCM), you can list the multiples of each number and determine which multiples they have in common. The least of these numbers will be the least common multiple. Consider the numbers 4 and 6. Some of their multiples are shown below. You can see that they have several common multiples, and the least of these is 12.
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 (^6 12) 18 (^24) 30 (^36) 42 (^48) 54 (^60) 66 68
Objective 1
Example Problem Find the least common multiple of 30 and 50.
30, 60, 90, 120, 150 , 180, 210, 240
List some multiples of 30.
50, 100, 150 , 200, 250 List some multiples of 50. 150 is found on both lists of multiples.
Look for the least number found on both lists. Answer The least common multiple of 30 and 50 is 150.
The other method for finding the least common multiple is to use prime factorization. This is the method you need for working with rational expressions. The following shows how the factor method works with the numeric example, 4 and 6.
Start by finding the prime factorization of each denominator:
4 = 2 • 2 6 = 3 • 2
Identify the greatest number of times any factor appears in either factorization and multiply those factors to get the least common multiple. For 4 and 6, it would be:
3 • 2 • 2 = 12
Notice that 2 is included twice, because it appears twice in the prime factorization of 4. 12 is the least common multiple of 4 and 6.
The next example also shows how to use prime factorization.
Example Problem Find the least common multiple of 28 and 40.
28 = 2 • 2 • 7 Write the prime factorization of 28. 40= 2 • 2 • 2 • 5 Write the prime factorization of 40.
2 • 2 • 2 • 5 • 7= 280 Write the factors the greatest number of times they appear in either factorization and multiply. Answer The least common multiple of 28 and 40 is 280.
Adding Fractions with Unlike Denominators
To add fractions with unlike denominators, first rewrite them with like denominators. Then, you know what to do! The steps are shown below.
Adding Fractions with Unlike Denominators
- Find a common denominator.
- Rewrite each fraction using the common denominator.
- Now that the fractions have a common denominator, you can add the numerators.
- Simplify to lowest terms, expressing improper fractions as mixed numbers.
You can always find a common denominator by multiplying the two denominators together. See the example below.
Example Problem (^2 ) 3 5
- Add.^ Simplify the answer^.
3 • 5 = 15 Since the denominators are not alike, find a common denominator by multiplying the denominators. 2 5 10 3 5 15
Rewrite each fraction with a denominator of 15.
Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified. Answer (^) 2 1 13 3 5 15
You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.
Objective 3
Example Problem (^3 ) 7 21
Add. Simplify the answer.
Multiples of 7: 7, 14, 21 Multiples of 21: 21
Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and
3 3 9 7 3 21
Rewrite each fraction with a denominator of 21.
Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.
Answer (^) 3 2 11 7 21 21
You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.
Example Problem (^3 1 ) 4 6 8
Add. Simplify the answer and write as a mixed number. 4 = 2 • 2 6 = 3 • 2 8 = 2 • 2 • 2 LCM: 2 • 2 • 2 • 3 = 24
Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 4, 6 and 8.
Rewrite each fraction with a denominator of 24.
Add whole numbers. Add fractions.
Write the improper fraction as a mixed number.
Combine whole numbers and fraction to write a mixed number.
Answer (^) 5 4 5 8 7 16 6 9 18
Self Check D
3 3 14 5 9
- Add. Simplify the answer and write as a mixed number.
Subtracting Fractions with Unlike Denominators
If the denominators are not the same (they have unlike denominators ), you must first rewrite the fractions with a common denominator. The least common denominator , which is the least common multiple of the denominators, is the most efficient choice, but any common denominator will do. Be sure to check your answer to be sure that it is in simplest form. You can use prime factorization to find the least common multiple (LCM), which will be the least common denominator (LCD). See the example below.
Objective 5
Example Problem (^1 ) 5 6
Subtract. Simplify the answer.
5 • 6 = 30 The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together. 1 6 6 5 6 30
Rewrite each fraction as an equivalent fraction with a denominator of 30.
Subtract the numerators. Simplify the answer if needed. Answer (^) 1 1 1 5 6 30
The example below shows using multiples to find the least common multiple, which will be the least common denominator.
Example Problem (^5 ) 6 4
Subtract. Simplify the answer.
Multiples of 6: 6, 12, 18, 24 Multiples of 4: 4, 8 12, 16, 20 12 is the least common multiple of 6 and 4.
Find the least common multiple of the denominators – this is the least common denominator.
Rewrite each fraction with a denominator of 12.
Subtract the fractions. Simplify the answer if needed.
Answer (^) 5 1 7 6 4 12
Subtracting Mixed Numbers with Regrouping
The regrouping approach shown in the last section will also work with unlike denominators.
Example Problem (^1 ) 7 3 5 4
Subtract. Simplify the answer and write as a mixed number. Multiples of 5: 5, 10, 15, 20, 25 Multiples of 4: 4, 8, 12, 16, 20, 24
1 4 4 5 4 20
⋅ = and 1 5 5 4 5 20
Find a least common denominator. 20 is the least common multiple, so use it for the least common denominator. Rewrite each fraction using the common denominator.
Write the expression using the mixed numbers with the like denominator.
Since the second fraction part, 5 20
, is
larger than the first fraction part, 4 20
, regroup one of the whole numbers and write it as 20 20
.
Rewrite the subtraction expression using the equivalent fractions.
Subtract the whole numbers, subtract the fractions.
Combine the whole number and the fraction.
Answer (^) 1 1 7 3 5 4
Objective 7
Self Check G
6 1 − 25 2 6
Subtract. Simplify the answer and write as a mixed number.
Adding and Subtracting Fractions to Solve Problems
Example Problem A cake recipe requires 2 1 4
cups of milk and
11 2
cups of melted butter. If these are the only liquids, how much liquid is in the recipe?
2 1 11 4 2
Find the total amount of liquid by adding the quantities.
2 1 1 1 4 2
- Group the whole numbers andfractions to make adding easier.
Add fractions. Recall that 1 2 2 4
Combine whole number and fraction.
Answer There are 3 3 4
cups of liquid in the recipe.
Self Check H
What is the total rainfall in a three-day period if it rains 3 1 4
inches the first day, 3 8
inch
the second day, and 2 1 2
inches on the third day?
Objective 8
Example Problem Mike and Jose are painting a room. Jose used 2 3
of a can of
paint and Mike used 1 2
of a can of paint. How much more paint did Jose use? Write the answer as a fraction of a can. 2 1 3 2
Write an expression using subtraction to describe the situation.
2 • 2 4 3 • 2 6 1• 3 3 2 • 3 6
Rewrite the fractions using a common denominator.
Subtract. Check that the fraction is simplified.
Answer (^) Jose used 1 6
of a can more paint than Mike.
Self Check I
Mariah’s sunflower plant grew 18 2 3
inches in one week. Her tulip plant grew 3 3 4
inches
in one week. How many more inches did the sunflower grow in a week than the tulip?
Summary
To adding or subtracting fractions with unlike denominators, first find a common denominator. The least common denominator is easiest to use. The least common multiple can be used as the least common denominator.
2.6 Self Check Solutions
Self Check A Find the least common multiple of 12 and 80.
240 2 • 2 • 2 • 2 • 3 • 5 = 240.
Self Check B
Find the least common denominator of 3 4
and 1 6
. Then express each fraction using the
least common denominator.
LCD: 12
3 3 9 4 3 12
Self Check C
2 4 1 3 5 12
- Add. Simplify the answer and write as a mixed number.
Self Check D
3 3 14 5 9
- Add. Simplify the answer and write as a mixed number.
.
Self Check E 2 1 3 6
− Subtract and simplify the answer.
