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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
This article has been viewed 193,308 times.
To add or subtract fractions with different denominators, you must first find the least common denominator between them. It is the least common multiple of each of the initial denominators in the equation, or the smallest integer that can be divided by each denominator. [1] X Research Source Determining the least common denominator allows you to convert the denominators to the same number so you can add and subtract them.
Steps
List Multiples [2] X Research Sources
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List the multiples of each denominator. List several multiples for each denominator in the equation. Each list should include products where the denominator is multiplied by 1, 2, 3, 4, etc.
- Example: 1/2 + 1/3 + 1/5
- Multiples of 2: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14; etc
- Multiples of 3: 3 * 1 = 3; 3 * 2 = 6; 3 *3 = 9; 3 * 4 = 12; 3 * 5 = 15; 3 * 6 = 18; 3 * 7 = 21; etc
- Multiples of 5: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35; etc
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Determine the least common multiple. Look through each list and mark any multiples that are common among all the original denominators. After determining the common multiples, find the smallest denominator.
- Note that if you still haven’t found a common denominator, you may have to keep writing multiples until you find a common multiple.
- This method is easier to use when the denominators are small numbers.
- In this example, the denominators that have only one in common are 30: 2 * 15 = 30 ; 3 * 10 = 30 ; 5 * 6 = 30
- So least common denominator = 30
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Rewrite the original equation. To change each fraction in the equation so that the fractional value stays the same, you will need to multiply the numerator and denominator by the same factors that you used to multiply the respective denominators when finding the least common denominator. .
- For example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
- New equation: 15/30 + 10/30 + 6/30
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Solve the rewritten problem. After finding the least common denominator and changing the fractions accordingly, you can solve the problem without any difficulty. Remember to simplify the fraction in the last step.
- Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30
Using Greatest Common Factor [3] X Research Source
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List all the factors of each denominator. The factors of a number are all the integers that the number is divisible by. [4] X Research Source Number 6 has four factors: 6, 3, 2, and 1. Every number has a factor of 1 because 1 multiplied by any number equals the same number.
- For example: 3/8 + 5/12.
- The factors of 8: 1, 2, 4, and 8
- The factors of 12: 1, 2, 3, 4, 6, 12
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Determine the greatest common factor between two denominators. After listing all the factors of each denominator, circle all the common factors. The greatest common factor is the factor that will be used to solve the problem.
- In this example, 8 and 12 have the common factors of 1, 2, and 4.
- The greatest common factor is 4.
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Multiply the denominators together. To use the greatest common factor to solve a problem, you must first multiply the two denominators.
- In this example: 8 * 12 = 96
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Divide the result obtained by the greatest common factor. After finding the product of the two denominators, divide that product by the greatest common factor in the previous step. This number is your least common denominator.
- Example: 96 / 4 = 24
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Divide the least common denominator by the original denominator. To find the factor by which the denominators are all equal, divide the least common denominator you found by the original denominator. Multiply the numerator and denominator of each fraction by this number. The hourly denominators will be equal to the least common denominator.
- For example: 24 / 8 = 3; 24/12 = 2
- (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
- 9/24 + 10/24
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Solve the rewritten equation. With the least common denominator found, you can add and subtract fractions in the equation without any difficulty. Remember to reduce the fraction to the final result, if possible.
- Example: 9/24 + 10/24 = 19/24
Analyzing Each Denominator Product of Prime Factors [5] X Research Source
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Split each denominator into primes. Analyze each denominator as a product of prime factors. A prime number is a number that cannot be divisible by any number other than 1 and itself. [6] X Research Source
- Example: 1/4 + 1/5 + 1/12
- Parsing 4 into primes: 2 * 2
- Parsing 5 into primes: 5
- Parse 12 into primes: 2 * 2 * 3
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Count the number of occurrences of each prime number. Calculate the total number of times that each prime appears in each product.
- Example: There are 2 numbers 2 out of 4; no 2 out of 5; 2 numbers 2 in 12
- There is no 3 in 4 and 5; a number 3 in 12
- There is no 5 in 4 and 12; a number 5 in 5
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Get the maximum number of occurrences of each prime number. Determine how many times each prime occurs, and record that number.
- Example: The maximum number of occurrences of 2 is two; of 3 is one; of 5 is one
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Write that prime equal to the number of times you counted in the step above. Write only the number of times they occur most often, not all the times they appear in the denominators.
- Example: 2, 2, 3, 5
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Multiply all the prime numbers in this sequence together. Multiply the primes that we wrote in the step above together. The resulting product is the least common denominator.
- Example: 2 * 2 * 3 * 5 = 60
- Least common denominator = 60
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Divide the least common denominator by the original denominator. To find the factor by which the denominators are all equal, divide the least common denominator you found by the original denominator. Multiply the numerator and denominator of each fraction by this number. The hourly denominators will be equal to the least common denominator.
- For example: 60/4 = 15; 60/5 = 12; 60/12 = 5
- 15 * (1/4) = 15/60; 12 * (1/5) = 12/60; 5 * (1/12) = 5/60
- 15/60 + 12/60 + 5/60
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Solve the rewritten equation. With the least common denominator found, you can add and subtract fractions as usual. Remember to reduce the fraction to the final result, if possible.
- Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
Working with Integers and Mixed Numbers [7] X Research Sources
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Convert each integer and mixed number into a non-canonical fraction. Convert mixed numbers into irregular fractions by multiplying the whole number by the denominator and adding the numerator to the product. Convert an integer to a non-canonical fraction by placing it above the denominator of “1”.
- Example: 8 + 2 1/4 + 2/3
- 8 = 8/1
- 2 1/4; 2 * 4 + 1 = 8 + 1 = 9; 9/4
- Rewrite equation: 8/1 + 9/4 + 2/3
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Find the least common denominator. Use any of the methods above to find the least common denominator. Note that, in this example we will be using the “list multiples” method, where a list of the multiples of each denominator is enumerated and the least common denominator is determined from these lists.
- Note that you don’t need to list multiples for 1 because any number multiplied by 1 equals itself; in other words, every number is a multiple of 1 .
- For example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12 ; 4 * 4 = 16; etc
- 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12 ; etc
- Least common denominator = 12
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Rewrite the original equation. Do not multiply the denominator yourself, you must multiply the whole fraction by the number necessary to change the original denominator to the least common denominator.
- For example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 12/27; (4/4) * (2/3) = 8/12
- 96/12 + 27/12 + 8/12
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Solve the equation. With the least common denominator found and the original equation converted to the form of the least common denominator, you can add and subtract fractions without difficulty. Remember to reduce the fraction to the final result, if possible.
- Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12
Things you need
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This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
This article has been viewed 193,308 times.
To add or subtract fractions with different denominators, you must first find the least common denominator between them. It is the least common multiple of each of the initial denominators in the equation, or the smallest integer that can be divided by each denominator. [1] X Research Source Determining the least common denominator allows you to convert the denominators to the same number so you can add and subtract them.
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