Subtracting Fractions
Subtracting fractions follows the same rules as adding: find a common denominator, then subtract the numerators.
Formula
\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}; \quad \frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}
Definition
To subtract fractions, the denominators must be the same: if they are, subtract the numerators and keep the denominator; if not, find a common denominator first, just as with addition. For mixed numbers, convert to improper fractions first, or borrow from the whole number when the fractional part being subtracted is larger. Formally, subtraction in $\mathbb{Q}$ is defined as addition of the additive inverse: $a/b - c/d = a/b + (-c/d) = (ad-bc)/(bd)$, confirming that $\mathbb{Q}$ is an abelian group under addition, with subtraction derived rather than primitive.
Example
$5/8 - 2/8 = 3/8$ with the same denominator; for $3/4 - 1/3$, use $12$: $9/12 - 4/12 = 5/12$. For mixed numbers, $4$ and $1/6 - 2$ and $5/6$ requires borrowing since $1/6 < 5/6$: rewrite $4$ and $1/6$ as $3$ and $7/6$, giving $3$ and $7/6 - 2$ and $5/6 = 1$ and $2/6 = 1$ and $1/3$. The same subtraction formula extends to any field of fractions $\text{Frac}(R)$: for $R = \mathbb{Z}[x]$, $(x+1)/(x-2) - 3/(x+1) = ((x+1)^2 - 3(x-2))/((x-2)(x+1)) = (x^2-x+7)/(x^2-x-2)$.
Key Insight
Subtraction and addition of fractions follow the exact same setup, the only difference is subtracting instead of adding the numerators once you have a common denominator. Borrowing in mixed-number subtraction mirrors borrowing in whole-number subtraction: both use the idea that $1$ whole equals (denominator/denominator), so borrowing $1$ from the whole number adds one full denominator-worth to the fraction. This reflects a general principle in abstract algebra, that in any group subtraction is not a primitive operation but a notational shorthand for adding the inverse, which clarifies why subtraction feels "harder" than addition for learners: it is a composite operation.