How to Combine Like Terms With Fractions

You can only add fractions if they have a common denominator.

Combining like terms with fractions requires more steps than combining terms with whole-number coefficients because you must first find the lowest common denominator of all like terms. Organize the algebraic expression with like terms together, then rewrite fractions as their equivalent fractions in the lowest common denominator so that you can add or subtract the fractional coefficients as you would fractional numbers.

Instructions

1. Rearrange the expression so that all like terms are together, ordered from highest degree to lowest degree. This will prevent you from missing a like term or accidentally combining unlike terms. For example, rewrite the expression (2/3)x^2 - (4/5)x + 1/2 - 2x + (3/2)x^2 as (2/3)x^2 + (3/2)x^2 - (4/5)x - 2x + 1/2.

2. Find the lowest common denominator (LCD) between like terms in the expression so that you can combine the fractions. The LCD of two fractions is the least common multiple of the denominators. In the example, the LCD of 2/3 and 3/2 is 6 and the LCD of 4/5 and 2 (or 2/1) is 5.

3. Multiply both the numerator and denominator of each fraction by the number that will turn the denominator into the LCD you found in Step 2. In the example, multiply 2/3 by 2/2 to get the equivalent fraction 4/6, multiply 3/2 by 3/3 to get 9/6 and multiply 2 by 5/5 to get 10/5 so that all like terms have common denominators.

4. Rewrite the expression by replacing the original fractions with the equivalent fractions you obtained in Step 3. In the example, the expression becomes (4/6)x^2 + (9/6)x^2 - (4/5)x - (10/5)x + 1/2.

5. Combine like terms by adding or subtracting the fractions from Step 4 using fraction addition or subtraction, adding or subtracting the numerators without changing the denominator. In the example, 4/6 + 9/6 = 13/6 and -4/5 - 10/5 = -14/5.

6. Write the equation with the like terms rewritten as the single term with the fraction you found in Step 5 as the new coefficient. In the example, the simplified expression is (13/6)x^2 - (14/5)x + 1/2.

References

• Oswego City School District Regents Exam Prep Center: Adding and Subtracting Fractions