How Do You Add 5 Fractions with Different Denominators?

To add 5 fractions with different denominators, you need to find a common denominator for all of them, and then convert each fraction to have that common denominator, following this principle: The basic rule for adding fractions with unlike denominators is to find the LCM of the different denominators and convert the given unlike fractions into like fractions. This can be done by changing their denominators equal to the LCM. Once the denominators become the same, the numerators can be added.

Here's a step-by-step guide:

1. Find the Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that is a multiple of all the denominators of the fractions you are trying to add.
You can find the LCM by listing the multiples of each denominator and identifying the smallest one they share, or by using prime factorization.
- Example: If your denominators are 2, 3, 4, 5, and 6.
  - Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36...
  - Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36...
  - Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
  - Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40...
  - Multiples of 6: 6, 12, 18, 24, 30, 36...
  - The LCM in this case is 60.

2. Convert Fractions to Have the LCM as the Denominator

Once you have the LCM, convert each fraction into an equivalent fraction that has the LCM as the new denominator.
To do this, divide the LCM by the original denominator, and then multiply both the original numerator and denominator by that result.
- Example (Continuing with our example LCM of 60):
  - 1/2 becomes (1 x 30)/(2 x 30) = 30/60
  - 1/3 becomes (1 x 20)/(3 x 20) = 20/60
  - 1/4 becomes (1 x 15)/(4 x 15) = 15/60
  - 1/5 becomes (1 x 12)/(5 x 12) = 12/60
  - 1/6 becomes (1 x 10)/(6 x 10) = 10/60

3. Add the Numerators

Now that all fractions have the same denominator, you can simply add their numerators.
- Example (Continuing with our example): 30/60 + 20/60 + 15/60 + 12/60 + 10/60 = (30 + 20 + 15 + 12 + 10)/60 = 87/60

4. Simplify the Result

Finally, simplify the resulting fraction if possible, by dividing the numerator and denominator by their greatest common factor (GCF).
- Example (Continuing with our example): 87/60 can be simplified because both 87 and 60 are divisible by 3, resulting in 29/20.

Example Recap

To add 1/2 + 1/3 + 1/4 + 1/5 + 1/6

Find the LCM of 2, 3, 4, 5, and 6 which is 60.
Convert the fractions to have a denominator of 60: 30/60 + 20/60 + 15/60 + 12/60 + 10/60
Add the numerators: 30 + 20 + 15 + 12 + 10 = 87.
The result is 87/60.
Simplify the result: 29/20.

Therefore, 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 29/20

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