Algebraic Fractions: Examples with Answers

An algebraic fraction is a fraction whose numerator, denominator, or both contain a variable. You handle it with the same ideas as ordinary fractions, plus factoring.

Simplifying

Factor first, then cancel factors shared by the top and bottom. For example, (x² − 9)/(x + 3) = (x − 3)(x + 3)/(x + 3) = x − 3, provided x ≠ −3.

Multiplying and dividing

Multiply straight across: (a/b) × (c/d) = ac/bd. To divide, flip the second fraction and multiply: (a/b) ÷ (c/d) = (a/b) × (d/c).

Adding and subtracting

Find a common denominator first. For example, 1/x + 1/2 = (2 + x)/2x. Always state any value that would make a denominator zero, because those values are excluded.

Worked answers

Simplify (2x + 6)/2: factor the top as 2(x + 3), cancel 2, and you get x + 3. Add x/3 + x/6: the common denominator is 6, so (2x + x)/6 = 3x/6 = x/2.

Learn more

For a video explanation of this topic, see Khan Academy Algebra course.