Subtracting Fractions
Fractions & DecimalsSubtracting 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 top numbers and keep the denominator. If not, find a common denominator first, just like with addition.
Example
$5/8 - 2/8 = 3/8$ (same denominator: just subtract tops). Different: $3/4 - 1/3$. Use $12$: $9/12 - 4/12 = 5/12$.
Key Insight
Subtraction and addition of fractions follow the exact same setup. The only difference is you subtract instead of add the numerators once you have a common denominator.
Definition
Subtract $a/b - c/d$ by finding the LCD, converting both fractions, subtracting numerators, and simplifying. For mixed numbers, convert to improper fractions first (or borrow from the whole number when the fractional part being subtracted is larger).
Example
$4$ and $1/6 - 2$ and $5/6$. The fraction $1/6 < 5/6$, so borrow: rewrite $4$ and $1/6$ as $3$ and $7/6$. Then $3$ and $7/6 - 2$ and $5/6 = 1$ and $2/6 = 1$ and $1/3$.
Key Insight
Borrowing in mixed-number subtraction mirrors borrowing in whole-number subtraction. Both use the idea that $1$ whole $=$ (denominator/denominator), so borrowing $1$ from the whole number adds one full denominator-worth to the fraction.
Definition
Subtraction in $\mathbb{Q}$ is defined as addition of the additive inverse: $a/b - c/d = a/b + (-c/d) = a/b + (-c)/d = (ad-bc)/(bd)$. The additive inverse of $a/b$ is $-a/b$ (or equivalently $a/(-b)$). This confirms $\mathbb{Q}$ is an abelian group under addition, with subtraction derived.
Example
In abstract algebra, subtracting fractions in any field of fractions $\text{Frac}(R)$ uses the same formula. For $R = \mathbb{Z}[x]$ (polynomials), $(x+1)/(x-2) - 3/(x+1) = ((x+1)^2 - 3(x-2))/((x-2)(x+1)) = (x^2+2x+1-3x+6)/((x-2)(x+1)) = (x^2-x+7)/(x^2-x-2)$.
Key Insight
The fact that subtraction is derived from addition (via additive inverse) is a general principle: in any group, subtraction is not a primitive operation but a notational shorthand. This simplifies the axioms of algebraic structures and clarifies why subtraction is "harder" than addition for learners - it is a composite operation.