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Plus, how to find a common denominator for easy problem solving
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When 2 fractions have the same number in the denominator, they’re known as like denominators (also called common denominators). Adding fractions with like denominators is super easy—simply add all the numerators together and keep the denominator the same! The same is true for subtracting fractions that have common denominators. We spoke with math tutor David Jia to learn how to add and subtract like fractions, as well as how to find common denominators when the denominators are different. Read on to find out more!
Adding Fractions With Like Denominators: Simple Steps
To add fractions that have like denominators, just add the numerators together and keep the denominator the same. Then, rewrite the fraction with the sum of the numerators over the denominator. For example, 1/5 + 3/5 = 4/5.
Steps
Section 1 of 3:
Adding Fractions With Common Denominators
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Identify the numerator and denominator of each fraction. There are 2 parts to all fractions: the numerator, which is the number above the line, and the denominator, which is the number below the line. While the denominator tells you how many parts a whole has been broken into, the numerator tells you how many pieces of that whole there are. [1] X Research source
- For example
:
- In the fraction 1/2, the numerator is 1 and the denominator is 2.
- In the fraction 2/5, the numerator is 2 and the denominator is 5.
- In the fractions 3/8, the numerator is 3 and the denominator is 8.
- For example
:
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Check that the denominators are the same in both fractions. Jia says “whenever you're adding…fractions, you have to have a common denominator,” which means the “bottom number has to be the same.” Ensure the fractions in the problem you’re working on share a common denominator. It’s easy to add like fractions together, as the resulting fraction has the same denominator as the original fractions.
- For example
:
- The fractions 3/5 and 2/5 have a common denominator of 5.
- The fractions 3/8, 5/8, and 17/8 have a common denominator of 8.
- If the fractions don’t have a common denominator , don’t worry—we explain how to find common denominators below!
- When 2 or more fractions have common denominators, it means they have the same number in the denominator. In other words, the fractions all represent wholes that are broken into the same number of pieces. [2] X Research source
Advertisement - For example
:
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Add the numerators of the fractions together. To add fractions with like denominators together, simply sum the numerators of each fraction together. In the example of 3/5 + 2/5, add the numerators 3 and 2, which equals 5. 5 is your new numerator. [3] X Research source
- For example
:
- Add 2/7 + 4/7. 2 + 4 = 6, so the numerator is 6.
- Add 3/8 + 5/8 + 17/8. 3 + 5 + 17 = 25, so the numerator is 25.
- For example
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Write the answer with the new numerator and same denominator. Remember to use the same common denominator in your answer, since the number of parts that the whole is divided into remains the same. When adding fractions with like denominators, you only add the numerators (the number of individual pieces). [4] X Research source
- For example
:
- 3/5 + 2/5 = 5/5
- 2/7 + 4/7 = 6/7.
- 3/8 + 5/8 + 17/8 = 25/8
- For example
:
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Simplify the fraction, if you can. A fraction is in its simplest form when both the numerator and the denominator don’t have any common factors they can be divided by. Jia says you simplify a fraction by “dividing the top and the bottom by whatever their common factors are.”
- For example , if your answer is 3/6, both the numerator and denominator have a common factor of 3. This means they can both be divided by 3 to produce a whole number. So, divide both the numerator and denominator by 3: 3 ÷ 3 = 1 and 6 ÷ 3 = 2, so the simplified fraction is 1/2.
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.To simplify fractions, divide the numerator and denominator by a common factor. This creates a new, easier-to-use fraction with smaller components, but it represents the same value. For instance, if you divide both the numerator and denominator of 6/12 by 2, you get 3/6, which is equal to 1/2.
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Solve the fraction if the problem asks you to. Sometimes a fraction can be put into simpler terms, including dividing it to get a whole number or decimal. In the example 3/5 + 2/5, the answer of 5/5 is easy to solve because any fraction where the numerator and denominator are the same is equal to 1. [5] X Research source Think about it like a pie that’s been cut into 3 pieces. If you eat all 3 pieces of the pie, then you’ve eaten 1 whole pie.
- Jia says “fractions [are] just a form of division,” so any fraction can be converted from a fraction by dividing the numerator by the denominator, which often gives you a decimal number as the answer. For instance, 5/8 can also be written as 5 ÷ 8, which equals 0.625.
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Convert improper fractions to mixed numbers if the problem asks. When a fraction has a numerator that’s bigger than the denominator, such as 25/8, it is called an improper fraction (the reverse, which is when the numerator is smaller than the denominator, is called a proper fraction). Improper fractions can be converted into a mixed number, which is a number that has a whole number plus a proper fraction. [6] X Research source
- To convert an improper fraction
, like 25/8, to a mixed number:
- Divide the fraction’s numerator by its denominator. This tells you how many times 8 can go into 25 as a whole number. 8 can go into 25 3 times (8 x 3 = 24).
- Determine what’s left over. 8 doesn’t divide into 25 perfectly, meaning there’s a remainder left over. Simply subtract 8 x 3 = 24 from the original numerator: 25 – 24 = 1. The remainder is 1 and it becomes your new numerator.
- Rewrite the mixed number. The denominator is the same as the denominator in the original improper fraction. So, you can rewrite 25/8 as 3 and 1/8.
- To convert an improper fraction
, like 25/8, to a mixed number:
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Section 2 of 3:
Subtracting Fractions With Common Denominators
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Locate the numerators and ensure the denominators are the same. Subtracting fractions follows a similar process to adding them. The denominators have to have the same value in order to subtract the numerators from each other. [7] X Research source
- For example
, subtract 12/26 – 4/26 – 1/26.
- The common denominator is 26.
- The numerators are 12, 4, and 1.
- For example
, subtract 12/26 – 4/26 – 1/26.
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Subtract the numerators and keep the same denominator. Like adding fractions with common denominators, you don’t have to worry about doing anything to the denominator. So, simply find the difference between the numerators to solve your problem. [8] X Research source
- For example
, subtract 12/26 – 4/26 – 1/26.
- 12 – 4 – 1 = 7
- Rewrite the fraction with the new numerator. 12/26 – 4/26 – 1/26 = 7/26.
- For example
, subtract 12/26 – 4/26 – 1/26.
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Reduce or solve the fraction if the problem asks you to. Similar to adding fractions, Jia says subtracting fractions can leave you with an answer you can simplify by finding a common denominator, solve through division, or convert from an improper fraction to a mixed number.
- For example
:
- 8/12 - 2/12 = 6/12. 6 and 12 both have a common factor of 6. So, divide the numerator and denominator by 6 to simplify your answer as 1/2.
- 15/16 - 3/16 = 12/16. 12 ÷ 16 = 0.75.
- 22/9 - 11/9 = 11/9. Divide 11 by 9. 9 can go into 11 one whole time and has a remainder of 2 (11 - 9 = 2). So, you can rewrite 11/9 as 1 2/9.
- For example
:
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Section 3 of 3:
Finding a Common Denominator
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Identify the denominators. Fractions don’t always have the same denominators. In order to add or subtract fractions with unlike denominators , you must first find a common denominator . To start, locate the denominators in the fractions you’re dealing with. [9] X Research source
- For example
, add 5/8 + 6/9.
- The denominators are 8 and 9.
- For example
, add 5/8 + 6/9.
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Determine the least common multiple. To find a common denominator, you need to find the least common multiple of the 2 numbers. This is the smallest positive number that’s a multiple (the product of any number and an integer) of both denominators. To find the least common multiple of the denominators, list the multiples of each number by multiplying them by 1, 2, 3, 4, and so on. [10] X Research source
- For example
:
- The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72 , 80, 88, 96, 104, etc.
- The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72 , 81, 90, 99, 108, etc.
- So, the least common multiple of 8 and 9 is 72.
- For example
:
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Multiply the fractions to get the least common multiple in the denominator. Jia says to multiply each denominator by the correct number to get the common denominator. Remember that whatever you do to each denominator, you must also do to its numerator. So, multiply the numerators by the same number. [11] X Research source
- For example
, add 5/8 + 6/9.
- The least common multiple of 8 and 9 is 72. So, 72 is your new denominator.
- For the fraction 5/8, multiply 8 x 9 to get 72. Then, multiply the numerator by 9, which gives you 5 x 9 = 45.
- For the fraction 6/9: multiply 9 x 8 to get 72. Then, multiply the numerator by 8, which gives you 6 x 8 = 48.
- For example
, add 5/8 + 6/9.
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Rewrite the fractions and add or subtract them together. Your fractions now have the new common denominator that you found and the new numerators that you multiplied by the same values to get the denominator. Simply add or subtract the fractions like you normally would. [12] X Research source
- For example
, add 5/8 + 6/9.
- The fraction 5/8 becomes 45/72 and the fraction 6/9 becomes 48/72.
- Add the numerators. 45 + 48 = 93.
- Rewrite your answer. 45/72 + 48/72 = 93/72.
- Don’t forget to reduce, solve, or convert improper fractions to mixed numbers when applicable and necessary.
- In the above example, 93/72 is an improper fraction. You can rewrite it as the mixed number 1 and 21/72 (72 goes into 93 one time and has a remainder of 21).
- For example
, add 5/8 + 6/9.
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Community Q&A
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Question`How do you convert a fraction to a decimal without dividing?I_l1ke_gam3sCommunity AnswerYou don't. You have to divide. The fraction bar means division.
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QuestionWhat do you do if you have to add a bunch of denominators together when they are different numbers?Community AnswerYou are probably asking about adding together a bunch of fractions that have unlike denominators. See Add Fractions With Unlike Denominators .
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QuestionAdd like fraction 16/21 + 4/21 + 1/21Community AnswerAll three denominators are the same, so you can just add the numerators together: 16/21 + 4/21 + 1/21 = 21/21 = 1.
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References
- ↑ https://www.mathsisfun.com/fractions.html
- ↑ https://www.mathsisfun.com/numbers/common-denominator.html
- ↑ https://www.mathsisfun.com/fractions_addition.html
- ↑ https://www.mathsisfun.com/fractions_addition.html
- ↑ https://www.mathgoodies.com/lessons/fractions/equivalent
- ↑ https://youtu.be/-imFslMIN1g?feature=shared&t=8
- ↑ https://www.mathsisfun.com/fractions_subtraction.html
- ↑ https://www.mathsisfun.com/fractions_subtraction.html
- ↑ https://www.mathsisfun.com/fractions.html
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