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Adding, Subtracting, and Compound Fractions

Like number fractions, you need a common denominator first. Build the least common denominator from factors, combine the numerators carefully, then tackle a fraction stacked inside a fraction.

Building the least common denominator

To add fractions you need a shared bottom. The least common denominator (LCD) is the smallest expression every denominator divides into — the least common multiple of the denominators. Build it by factoring each denominator and taking every distinct factor to the highest power it appears with.

Add   3/(x^2 - x)  +  2/(x - 1)

Factor the denominators:
   x^2 - x = x(x - 1)
   x - 1   = (x - 1)

LCD = x(x - 1)   [every distinct factor, highest power]

Rewrite each fraction over the LCD:
   3/[x(x-1)]  stays as is
   2/(x-1) = 2·x / [x(x-1)] = 2x/[x(x-1)]

Combine numerators over the LCD:
   (3 + 2x) / [x(x - 1)]  =  (2x + 3) / [x(x - 1)]

   x ≠ 0,  x ≠ 1
Multiply each fraction by the missing factors so every bottom becomes the LCD.

Subtraction: distribute the minus sign

Adding and subtracting share one workflow, but subtraction has a famous trap: the minus sign applies to the entire second numerator. Put that numerator in brackets and distribute the negative before combining like terms.

Subtract   (2x + 1)/(x - 3)  -  (x - 4)/(x - 3)

Same denominator already, so combine over (x - 3):
   [(2x + 1) - (x - 4)] / (x - 3)

Distribute the minus across (x - 4):
   (2x + 1 - x + 4) / (x - 3)        <- note -(-4) = +4

Combine like terms:
   (x + 5) / (x - 3),   x ≠ 3
−(x − 4) becomes −x + 4. Forgetting that flip is the most common mistake.

Complex (compound) fractions

A complex fraction is a fraction whose numerator or denominator is itself a fraction — a fraction stacked on a fraction. The cleanest method: simplify the top into one fraction and the bottom into one fraction, then divide by multiplying by the reciprocal.

Simplify the complex fraction   (1 + 1/x) / (1 - 1/x^2)

Top  as one fraction:   1 + 1/x   = (x + 1)/x
Bottom as one fraction: 1 - 1/x^2 = (x^2 - 1)/x^2 = (x-1)(x+1)/x^2

Divide = multiply by the reciprocal of the bottom:
   (x + 1)/x  ·  x^2 / [(x - 1)(x + 1)]

Cancel (x + 1) and one x:
   x / (x - 1),    x ≠ 0,  x ≠ 1,  x ≠ -1
Combine top, combine bottom, then divide by flipping — reuse guide 3's reciprocal trick.