To add or subtract fractions with different denominators (the numbers below the fraction line) you must first find the lowest common denominator. In practice, this is the lowest multiple divisible by all denominators. You may have already approached this concept under the name of least common multiple, which generally refers to integers; however, the methods apply to both. Finding the lowest common denominator you can convert the fractions so that they all have the same denominator and then proceed to the subtractions and additions.
Steps
Method 1 of 4: List the Multiples
Step 1. List the multiples of each denominator
Make a list of the various multiples for each denominator in question. In practice, multiply each denominator by 1; 2; 3; 4 and so on and consider the products.
- For example: 1/2 + 1/3 + 1/5.
- Multiples of 2 are: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14 and so on;
- Multiples of 3 are: 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 are: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35 and so on.
Step 2. Identify the least common multiple
Analyze each list and locate each number that is shared by all of the original denominators. Once you have found all the common multiples, identify the minor.
- Know that if you don't find any common multiple, you'll have to keep making lists until you come across a common product.
- This method is simpler when you are dealing with small numbers in the denominator.
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In the previous example, the denominators share a single multiple of 30; in fact: 2 * 15 =
Step 30.; 3 * 10
Step 30.; 5 * 6
Step 30..
- The lowest common denominator is 30.
Step 3. Rewrite the original equation
To convert each fraction so that the initial equation does not lose its truth, you need to multiply the denominator and numerator (the value above the fraction line) by the same factor used to find the corresponding lowest common denominator.
- Example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5);
- The new equation will look like this: 15/30 + 10/30 + 6/30.
Step 4. Fix the rewritten problem
Once you have found the lowest common denominator and converted the fractions accordingly, you can proceed to add or subtract without further difficulty. Remember that you will eventually need to simplify the resulting fraction.
Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 and 1/30
Method 2 of 4: Use the Greatest Common Divider
Step 1. Make a list of all the factors in each denominator
The factors of a number are all integers that can divide it. The number 6 has four factors: 6; 3; 2 and 1. Each number also has "1" among its divisors, because each value can be multiplied by 1.
- For example: 3/8 + 5/12;
- The factors of 8 are: 1; 2; 4 and 8;
- The factors of 12 are: 1; 2; 3; 4; 6; 12.
Step 2. Identify the greatest common divisor of both denominators
When you have written the list of all divisors for each denominator, circle all common ones. The largest factor is the greatest common factor (GCD), which you will need to use to solve the problem.
- In the example we considered earlier, the numbers 8 and 12 share the divisors 1; 2 and 4.
- The largest of the three is 4.
Step 3. Multiply the denominators together
To use the GCD to solve the problem, you must first multiply the denominators.
Continuing in the previous example: 8 * 12 = 96
Step 4. Divide the product obtained by the greatest common factor
Once you find the product of the various denominators, divide it by the GCD calculated earlier. This way, you will get the lowest common denominator.
Example: 96/4 = 24
Step 5. Now divide the lowest common denominator by the original denominator
To find the multiple you need to make all denominators equal, divide the lowest common denominator you found by the denominator of each fraction. Then, multiply the numerator of the fraction by the quotient you calculated. At this point, all denominators should be equal.
- Example: 24/8 = 3; 24/12 = 2;
- (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
- 9/24 + 10/24.
Step 6. Solve the rewritten equation
Thanks to the lowest common denominator, you can add and subtract fractions. In the end, remember to simplify the result if possible.
For example: 9/24 + 10/24 = 19/24
Method 3 of 4: Break down each Denominator into Prime Factors
Step 1. Break each denominator into prime numbers
Reduce each denominator into a series of prime numbers, which when multiplied together give the denominator itself as a product. Prime numbers are numbers divisible only by 1 and by themselves.
- Example: 1/4 + 1/5 + 1/12.
- Prime factorization of 4: 2 * 2;
- Prime factorization of 5: 5;
- Prime factorization of 12: 2 * 2 * 3.
Step 2. Count the number of times each number appears in the decomposition
Add together the number of times each prime appears in each decomposition for each denominator.
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Example: there are two
Step 2. in 4; none
Step 2. in the 5th and du
Step 2. in 12;
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There isn't any
Step 3. in 4 and 5, while there is u
Step 3. in 12;
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There isn't any
Step 5. in 4 and 12, but there is u
Step 5. in 5.
Step 3. For each prime number, choose the greatest number of times it appears
Identify the greatest number of times each prime factor appears in each decomposition and make a note of it.
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Example: the greater number of times
Step 2. is present is two; the greater number of times in cu
Step 3. is present is one and the greater number of times in cu
Step 5. is present is one.
Step 4. Write each prime number as many times as you counted in the previous step
You don't have to write how many times this appears, but repeat the same number as many times as it appears in all the original denominators. Only take into account the highest count, the one found in the previous step.
Example: 2, 2, 3, 5
Step 5. Multiply all the prime factors you rewrote this way
Proceed to multiply them, considering how many times they have appeared in the decomposition. The product you will get is equal to the lowest common denominator of the initial equation.
- Example: 2 * 2 * 3 * 5 = 60;
- Least common denominator = 60.
Step 6. Divide the lowest common denominator by the original denominator
To find the multiple that makes the various denominators all equal, divide the least common denominator by the original one. Then, multiply the numerator and denominator of each fraction by the quotient obtained. Now the denominators are all equal and equal to the lowest common denominator.
- 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.
Step 7. Solve the rewritten equation
Once you have found the lowest common denominator, you can proceed with the subtraction and addition without further difficulty. In the end, remember to simplify the resulting fraction if possible.
Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
Method 4 of 4: Working with Integers and Mixed Numbers
Step 1. Convert every integer and mixed number to an improper fraction
For mixed numbers, you need to multiply the integer by the denominator and add the product to the numerator. To convert integers to improper fractions, write 1 in the denominator.
- For example: 8 + 2 1/4 + 2/3;
- 8 = 8/1;
- 2 1/4; 2 * 4 + 1 = 8 + 1 = 9; 9/4;
- The rewritten equation will be: 8/1 + 9/4 + 2/3.
Step 2. Find the lowest common denominator
Use any of the methods described above to find this value. In the example discussed in this section, the technique of the first method is used, in which the various multiples of the denominators are listed and then the minimum is identified.
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Remember that you don't have to create a series of multiples for the denominator
Step 1., since any number multiplied by pe
Step 1. it is equal to itself; in other words, every number is a multiple d
Step 1..
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Example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 =
Step 12.; 4 * 4 = 16 and so on;
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3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 =
Step 12. etc;
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The lowest common denominator =
Step 12..
Step 3. Rewrite the original equation
Instead of multiplying just the denominator, you need to multiply the entire fraction by the factor necessary to transform the original denominator into the lowest common denominator.
- Example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 27/12; (4/4) * (2/3) = 8/12;
- 96/12 + 27/12 + 8/12.
Step 4. Solve the rewritten equation
Once you have found the lowest common denominator and the equation has been converted to that number, you can proceed to add and subtract without further problems. In the end, remember to simplify the resulting fraction if possible.
