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How to add fractions with different denominators in algebra?

Published in Adding Fractions 2 mins read

Adding fractions with different denominators in algebra requires finding a common denominator before you can perform the addition. Here's a step-by-step guide:

Finding a Common Denominator

The core principle is to convert the fractions into equivalent fractions that share a common denominator. The most efficient common denominator to use is the least common multiple (LCM) of the original denominators.

Steps to Add Fractions with Different Denominators:

  1. Find the Least Common Multiple (LCM): Determine the LCM of the denominators. This will be your common denominator.

  2. Convert Fractions: Rewrite each fraction with the LCM as the new denominator. To do this, determine what number you need to multiply the original denominator by to get the LCM. Then, multiply both the numerator and denominator of the fraction by that same number.

  3. Add the Numerators: Once the fractions have the same denominator, add the numerators together. Keep the common denominator.

  4. Simplify (if possible): Reduce the resulting fraction to its simplest form.

Example

Let's say we want to add 1/2 and 1/3.

  • Step 1: Find the LCM. The LCM of 2 and 3 is 6.
  • Step 2: Convert Fractions.
    • To convert 1/2 to a fraction with a denominator of 6, we multiply both the numerator and denominator by 3: (1 3) / (2 3) = 3/6
    • To convert 1/3 to a fraction with a denominator of 6, we multiply both the numerator and denominator by 2: (1 2) / (3 2) = 2/6
  • Step 3: Add the Numerators. 3/6 + 2/6 = (3 + 2) / 6 = 5/6
  • Step 4: Simplify. 5/6 is already in its simplest form.

Therefore, 1/2 + 1/3 = 5/6. As stated in the reference, rewriting the fractions as 3/6 and 2/6 allows us to add the numerators together, resulting in a sum of 5/6.

Table Summarizing the Process

Step Description Example (1/2 + 1/3)
1. Find the LCM Determine the least common multiple of the denominators. LCM of 2 and 3 is 6
2. Convert Fractions Rewrite each fraction using the LCM as the new denominator. 1/2 = 3/6; 1/3 = 2/6
3. Add the Numerators Add the numerators, keeping the common denominator. 3/6 + 2/6 = 5/6
4. Simplify (if necessary) Reduce the fraction to its simplest form. 5/6 (already simplified)

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