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Adding and Subtracting Fractions with Variables Help

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Updated on Jan 28, 2014
numerator denominator
 

In a fraction, the numerator is above the line, and the denominator is below the line. One mneumonic to remember this is "the denominator is downstairs."

So, in the problem 23 + 15 , 2 and 1 are the numerators, and 3 and 5 are the denominators.

Adding and Subtraction Fractions

To add and subtract fractions, they must have a common denominator23 + 15 does not equal (2 + 1) / (3 + 5) = 38 . 

Finding a common denominator

To get the common denominator to add and subtract fractions, you must find the least common multiple of the original denominators.

 23 + 15

Find the least common multiple:

  • Multiples of 3: 3, 6, 9, 12, 15
  • Multiples of 5: 5, 10, 15

Convert each fraction to its new form. 

  • 23 is the same as 46, 69812, and 1015
  • 15 is the same as 210 and 315

To properly convert each fraction, determine what number multiplied by the original denominator equals the least common multiple. Then, multiply that number by the original numerator of each fraction to get the new numerator.

To get 15 in the denominator of  23, you need to multiply 3 by 5 because 3(5) = 15. You can't do something to the denominator without also doing the same thing to the numerator. So, multiply 2 by 5 to get 10 in the numerator. Here, you are essentially multiplying 2 by 55 (which also equals 1) to get a fraction that you can use.

So  23 + 1is the same as 1015 315 = 1315 

The Butterfly Trick Shortcut

When adding or subtracting two fractions, try this shortcut:

  1. Multiply up from the lower left to the upper right and write the product.
step 1
  1. Multiply up from the lower right to the upper left and write the product.
step 2
  1. Multiply across the bottom to get a common denominator.
step 3
  1. Add or subtract the numerators.
step 4
  1. Reduce fraction if needed.

 

Adding Fractions with Variables

The easiest way to add two fractions with variables is to use the Butterfly Trick above.

adding fractions with variables

 

adding fractions with variables

 

However, to add or subtract three or more fractions, you need to find the common denominator for all of them. Often, the least common multiple is just the product of all of the denominators.

Example 1: 

example 1

The least common multiple of 3c, a, and b is 3abc​.

  • Convert each fraction:

least common multiple

least common multiple

least common multiple

  • Add them all to get the solution:

solution

Example 2:

a2 + b3 + c4

The least common multiple of 2, 3, and 4 is 12​.

  • Convert each fraction:

32a (2b2b) = 6b4ab

34b (aa) = 3a4ab

  • Subtract to find solution:

(6b - 3a)4ab

Practice Problems

  1. 34 - 25
  2. 12 - 110
  3. 67 + 18
  4. 23 + 27
  5. 3a5 + a4
  6. a2b - a3b
  7. 4xy + 2yx
  8. mn7 - 2n21 + m3
  9. 6g5 - 7g10 - 2g15
  10. 3r5s + 2sr + 6s5r

Answers

  1. 720
  2. 25
  3. 5556
  4. 2021
  5. 17a20
  6. a6b
  7. (4x^2 + 2y^2)xy
  8. (3mn - 2n + 7m)21
  9. 11g30
  10. (3r^2 + 16s^2)5rs

​​David Travis is the founder and CEO of Prospect Prep, a New York City-based tutoring agency dedicated to helping students earn better grades, higher scores, and acceptance letters from the colleges of their dreams.

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