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Learn to combine expressions using the least common denominator
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Adding fractions with unlike denominators may look tricky, but once you make the denominators the same, addition is a snap. If you're working with improper fractions where the numerators are larger than the denominators, make the denominators the same. Then add the numerators. We talked to pro math tutor David Jia to show you how, plus how to handle improper fractions and mixed numbers with different denominators.
Adding Unlike Fractions
First, find the least common denominator for the fractions you’re adding. Then, convert each fraction so that they all have the same denominator, which will be the least common denominator. Finally, add their numerators together, and simplify the answer.
Steps
Practice Problems
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1Find the least common multiple of both denominators. “Whenever you're adding or subtracting fractions, you have to have a common denominator, meaning that bottom number has to be the same,” Jia says. So before we can add the fractions together, we need them to have the same denominator, or bottom number. To do this, we find the least common multiple (LCM). Find the least common multiple by listing multiples of each number until you find a multiple that they share.
- For example, say we’re trying to add ½ and 2/6. We start by listing multiples of 2: 2, 4, 6, 8, 10, 12. Then, list multiples of 6: 6, 12, 18, 24, 30.
- We can see from our lists that the lowest multiple that both denominators share is 12, so 12 is our least common multiple.
- Multiples are the numbers you get when you keep adding a number to itself, or multiplying a number by each other number.
- A “proper fraction” is a fraction where the numerator (top number) is less than the denominator (bottom number).
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2Rewrite each fraction using the LCM. Now that we have an LCM, Jia tells us, we convert each fraction so that they all have the LCM as their denominator. To do this, we ask, “What number do we multiply the denominator by to get the LCM?” Once we know that, we multiply both the numerator (top number) and denominator by that number to convert the fraction.
- In our example, we have ½ + 2/6. Let’s start with ½. What number multiplied by 2 equals 12? The answer is 6, so we multiply both the numerator and denominator by 6. 1 x 6 = 6, and 2 x 6 =12, so our new fraction is 6/12.
- Then, we do the same for 2/6. 6 x 2 = 12, so multiply both the numerator and denominator by 2. 2 x 2 = 4, and 6 x 2 = 12, so our new fraction is 4/12.
- Now, rewrite the equation with our converted fractions: 6/12 + 4/12 .
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3Add the numerators together. Jia says that once we convert both fractions to have a common denominator, we can now add them together! All we need to do is add straight across to combine the numerators.
- 6/12 + 4/12 = 10/12 .
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4Simplify the answer, if possible. Our answer is 10/12, but we always want to simplify or reduce the fraction down to its smallest form, Jia reminds us. To do this, we divide both the numerator and the denominator by the greatest common factor. To find the greatest common factor, we list the factors of both the numerator and the denominator and find the highest number they share.
- For example, the factors of 10 are 1, 2, 5, 10. The factors of 12 are 1, 2, 3, 4, 6, 12. So we can say the greatest common factor of the two numbers is 2.
- Now, divide both the numerator and denominator by 2 (10 / 2 and 12 / 2) to get â…š, so our final answer is â…š .
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Find the least common multiple (LCM) for the denominators. Because you have to make the denominators the same before you add the fractions, find a common multiple that they share, Jia says. Then choose the lowest one.
- For example, for 9/5 + 14/7, the multiples of 5 are 5, 10, 15, 20, 25, 30, and 35 while the multiples of 7 are 7, 14, 21, 28, and 35. 35 is the least common multiple .
- An “improper fraction” is a fraction where the numerator (top number) is more than the denominator (bottom number).
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Multiply the numerator and denominator to get like denominators. Then, Jia says we need to multiply each entire fraction to make the denominators become the least common multiple.
- For example, multiply 9/5 by 7 to get a denominator of 35. You also multiply the numerator by 7, so the fraction becomes 63/35.
- Then, 7 x 5 = 35, so we multiply 14 x 5 to get 70, and our new fraction is 70/35.
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Add the numerators but leave the denominators the same. Once all of the denominators in your problem are the same, Jia says, add the numerators. Put the answer over the denominator.
- For example, 63 + 70 = 133. Place it over the denominator to get 133/35.
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Simplify or reduce the answer if necessary. If your answer is improper, turn the fraction into a mixed number . To do this, Jia tells us to divide the numerator by the denominator so we get a whole number. Then see how many parts are left and place this number over the denominator. Reduce the fraction if it can be simplified further.
- For example, 133/35 can be simplified to 3 28/35. The fraction can be reduced to 4/5 so the finished answer is 3 4/5.
EXPERT TIPMath TeacherJoseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.Simplifying a fraction just changes the way the fraction is written. To simplify a fraction, you can cancel out the greatest common factor from the numerator and denominator or convert an improper fraction to a mixed number. This doesn't change the inherent value of the fraction.
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Turn the mixed numbers into improper fractions. If you have fractions with whole numbers, changing them into improper fractions will make it easier to add. To do this, multiply the denominator by the whole number, then add it to the numerator. The new numerators of your improper fractions will be larger than their denominators. [1] X Research source
- For example, 6 3/8 + 9 1/24 will turn into 51/8 + 217/24.
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Find the lowest common denominator. Jia reminds us that we need to make the denominators the same. If the denominators are different, write down the multiples of each denominator to find the first one that they have in common. For example, for the problem 51/8 + 217/24, list the multiples of 8 and 24 to find 24.
- Because the multiples of 8 include 8, 16, 24, 32, and 48 and the multiples of 24 include 24, 48, and 72, 24 is the lowest common multiple.
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Make the fraction equivalent if you need to change the denominators. The, Jia says to convert the fractions so that the denominators become the lowest common multiple that you found. Multiply the whole fraction by a number to make the denominator the lowest common multiple. [2] X Research source
- For example, to make the denominator for 51/8 become 24, multiply the whole fraction by 3. You should get 153/24.
- For the fraction 217/24, you won't need to adjust the fraction since it already has the same denominator.
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Add the numerators but leave the denominator the same. Jia tells us to add the numerators once the denominators are the same (or if they were the same from the start). Once you've added the numerators, put the answer over the denominator. Don’t add the denominators.
- So, 153/24 +217/24 = 370/24.
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Simplify your answer. With mixed numbers, the numerator of your answer is larger than the denominator at this point, so you divide the two to get a whole number. To finish making a mixed number, write down how many parts you have left over above the denominator. This will make the numerator which you can put over the same denominator. [3] X Research source
- For example, 370/24 will become 15 10/24 because 24 goes into 370 15 times and has 10 parts of 24 left over. 10/24 can be further reduced to 5/12 for a finished answer of 15 5/12.
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Community Q&A
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QuestionShould I simplify the answers to my math problems?Community AnswerYes, most teachers will want you to show the simplified answer.
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QuestionHow do I determine equivalent fractions that are not in simplest form?Community AnswerMultiply each fraction so that the denominators are the same. Then you can check the numerator to see if the fractions are equal.
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QuestionI followed until #9, but I don't know what a common factor is, what is it?DonaganTop AnswererA common factor is any prime number that will divide evenly into each number being considered. The greatest common factor is the largest whole number that is a common factor. For example, in Step 9 above, 2 is the greatest common factor of 6 and 8, so they are both divided by 2 to arrive at 3 and 4. In another example, to find the greatest common factor of 42 and 315, first find all the factors of each number: 42 = 2 x 3 x 7, and 315 = 3 x 3 x 5 x 7. The common factors are 3 and 7, so the greatest common factor of 42 and 315 is 7.
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Tips
- If you have fractions with unlike signs, such as 3/4 + (-¼), then pretend the unlike sign (in this case the minus or negative sign) applies to the numerator of that fraction. Then, add them like usual. So 3 + (-1) = 2, and the answer is 2/4, or ½.Thanks
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To add fractions with unlike denominators, start by finding the least common multiple for the denominators. Then, divide the least common multiple by the denominator in each fraction. Take the number you get for each fraction and multiply it by the numerator and denominator of that fraction, which will make both denominators equal to the least common multiple. Finally, add the numerators together and leave the denominator as is to get your answer. To learn how to simplify your answer, keep reading!
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