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How Do You Sum Algebraic Fractions?

Published in Algebraic Fractions 3 mins read

To sum algebraic fractions, you need to follow a few key steps to ensure you have a common denominator, then you can combine the numerators. Here’s how to do it:

Steps to Sum Algebraic Fractions

  1. Find the Lowest Common Multiple (LCM) of the Denominators: This is the first and most crucial step. The LCM becomes your common denominator.

    • For example, if your denominators are 2 and 3, the LCM is 6.
    • With algebraic expressions, you'll need to consider the variables and any coefficients.

  2. Adjust Each Fraction: For each fraction, multiply both the numerator and the denominator by a value that will make its denominator equal to the LCM you found in step 1. This process ensures that each fraction has the same denominator.

  • Essentially, you're multiplying the fraction by a form of 1, keeping the value of the fraction unchanged.
  • Example: If you're adding 1/2 + 1/3, you convert 1/2 to 3/6 by multiplying both top and bottom by 3, and you convert 1/3 to 2/6 by multiplying both top and bottom by 2.

  1. Simplify the Numerators and Denominators: Expand and simplify the algebraic expressions in the numerators. This often involves distributing multiplication over sums or differences or applying other simplification techniques. Ensure the denominator is also fully simplified, often it will be the LCM you have already calculated.

  2. Combine the Numerators: Once you have the same denominator for each fraction, you can add (or subtract) the numerators. The resulting numerator will be a sum (or difference) of the individual numerators, all over the common denominator.

    • Example continuing from above: 3/6 + 2/6 = (3+2)/6=5/6

  3. Simplify the Result: Reduce the final fraction to its simplest form if possible. This might involve canceling out common factors in the numerator and the denominator.

    • Example: if the result is (4x + 2)/2, simplify to 2x + 1, dividing top and bottom by 2

Example

Here's an example to demonstrate how to add algebraic fractions with variables:

Let's say we want to add: x/2 + 3x/4

  1. Find the LCM: The LCM of 2 and 4 is 4.
  2. Adjust Each Fraction:
    • The first fraction needs to have its denominator multiplied by 2 so the same is done to the numerator: x/2 becomes 2x/4
    • The second fraction already has the correct denominator.
  3. Combine the Numerators: Add the adjusted numerators over the common denominator: (2x + 3x) / 4
  4. Simplify the Result: 5x/4 The fraction is already in the simplest form

Key Takeaways

  • Always begin by finding the lowest common multiple of the denominators.
  • Make sure to multiply both the numerator and denominator by the same value when adjusting each fraction.
  • Simplify the resulting fraction after combining numerators.

By following these steps, you can accurately sum algebraic fractions and arrive at simplified results.

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