Buzz
When adding or subtracting fractions with different denominators, the first step is to find the least common denominator (LCD) between them. This is the least common multiple (LCM) of the denominators in the equation, or the smallest integer that can be divided evenly by each of the denominators. Identifying the LCD allows you to convert all fractions to have the same denominator, making it possible to add or subtract them.
Steps
List the Multiples
List the multiples of each denominator. Create a list of several multiples for each denominator in the equation. Each list should include the products of the denominator multiplied by 1, 2, 3, 4, etc.
- For 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.
Identify the Least Common Multiple. Review each list and mark any multiples that appear in all the original denominators. Once you have found the common multiples, select the smallest one.
- Note that if you haven't yet identified a common multiple, you may need to continue listing multiples until you find one that is common to all.
- This method is easier when the denominators are small numbers.
- In this example, the only common multiple is 30: 2 * 15 = 30; 3 * 10 = 30; 5 * 6 = 30
- Thus, the least common denominator = 30
Rewrite the original equation. To change each fraction in the equation while keeping the value the same, multiply both the numerator and denominator by the same factor that was used to multiply the denominator when finding the least common denominator.
- Example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
- New equation: 15/30 + 10/30 + 6/30
Solve the rewritten problem. Once you have found the least common denominator and adjusted the fractions accordingly, you can solve the problem without difficulty. Be sure to simplify the fraction at the final step.
- Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30
Use the Greatest Common Factor
List all factors of each denominator. A factor of a number is any integer that divides that number evenly. The number 6 has four factors: 6, 3, 2, and 1. Every number has 1 as a factor, because 1 multiplied by any number equals that number itself.
- Example: 3/8 + 5/12.
- Factors of 8: 1, 2, 4, and 8
- Factors of 12: 1, 2, 3, 4, 6, 12
Determine the greatest common factor between two denominators. After listing all factors for each denominator, circle the common factors. The greatest common factor is the one used to solve the problem.
- In this example, 8 and 12 have common factors of 1, 2, and 4.
- The greatest common factor is 4.
Multiply the denominators together. To use the greatest common factor to solve the problem, first multiply the two denominators.
- In this example: 8 * 12 = 96
Divide the result by the greatest common factor. After finding the product of the two denominators, divide that product by the greatest common factor from the previous step. This result is the least common denominator.
- Example: 96 / 4 = 24
Divide the least common denominator by the original denominators. To find the factor that, when multiplied, makes the denominators equal, divide the least common denominator by each original denominator. Multiply both the numerator and denominator of each fraction by this factor. The denominators will now be equal to the least common denominator.
- Example: 24 / 8 = 3; 24 / 12 = 2
- (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
- 9/24 + 10/24
Solve the equation now written. With the least common denominator found, you can now add and subtract fractions in the equation without difficulty. Remember to simplify the fraction in the final result, if possible.
- Example: 9/24 + 10/24 = 19/24
Factorize Each Denominator into Prime Factors
Break down each denominator into its prime factors. Prime factorization involves expressing each denominator as a product of prime numbers. A prime number is one that can only be divided by 1 and itself.
- Example: 1/4 + 1/5 + 1/12
- Factor 4 into prime numbers: 2 * 2
- Factor 5 into prime numbers: 5
- Factor 12 into prime numbers: 2 * 2 * 3
Count how many times each prime number appears. Calculate the total occurrences of each prime factor in the factorizations.
- Example: There are 2 factors of 2 in 4; no 2’s in 5; 2 factors of 2 in 12.
- No 3’s in 4 and 5; one 3 in 12.
- No 5’s in 4 and 12; one 5 in 5.
Identify the highest frequency for each prime number. Determine the maximum number of times each prime factor appears and record that number.
- Example: The highest frequency of 2 is two; for 3 it is one; and for 5 it is one.
Write the prime factors according to their highest frequency. Only write the prime numbers as many times as they appear most frequently, not all occurrences in the factorizations.
- Example: 2, 2, 3, 5
Multiply all prime numbers in this sequence together. Multiply the prime numbers we identified in the previous step. The product you get is the least common denominator.
- For example: 2 * 2 * 3 * 5 = 60
- The least common denominator = 60
Divide the least common denominator by the original denominators. To find the factor that, when multiplied, makes all denominators equal, divide the least common denominator you found by each original denominator. Then, multiply both the numerator and denominator of each fraction by that factor. This will make all fractions have the same 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
Solve the equation in its rewritten form. Now that you have the least common denominator, you can add and subtract fractions as usual. Remember to simplify the fraction in the final result, if possible.
- For example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
Working with Integers and Mixed Numbers
Convert each integer and mixed number into an improper fraction. To convert mixed numbers into improper fractions, multiply the whole number by the denominator and add the numerator to the product. To convert an integer to an improper fraction, place the integer over a denominator of "1".
- For example: 8 + 2 1/4 + 2/3
- 8 = 8/1
- 2 1/4; 2 * 4 + 1 = 8 + 1 = 9; 9/4
- Rewritten equation: 8/1 + 9/4 + 2/3
Find the least common denominator. Use any of the methods mentioned above to find the least common denominator. Note that, in this example, we will use the “listing multiples” method, where a list of multiples for each denominator is created, and the least common denominator is determined from these lists.
- Note: You don’t need to list the multiples for 1 because any number multiplied by 1 equals itself; in other words, every number is a multiple of 1.
- Example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12; 4 * 4 = 16; and so on.
- 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12; and so on.
- Least common denominator = 12
Rewrite the original equation. Do not multiply the denominator by itself, instead, multiply the entire fraction by the necessary number to convert the original denominator to the least 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
Solve the equation. With the least common denominator found and the original equation rewritten with the least common denominator, you can now add and subtract the fractions with ease. Remember to simplify the final fraction, if possible.
- Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12
Things you will need
- Pencil
- Paper
- Calculator (optional)
- Ruler
