Subtracting fractions

Identical to adding, except for one sign

Both fractions are rewritten over $bd$, then the tops are combined:

$$\frac{ad}{bd} - \frac{bc}{bd} = \frac{ad - bc}{bd}$$

Unlike addition, the order cannot be swapped. $\tfrac{3}{4} - \tfrac{1}{2}$ and $\tfrac{1}{2} - \tfrac{3}{4}$ differ by a sign.

Mixed numbers may need borrowing

When the fraction being taken away is the bigger one, borrow a whole and rewrite it as $\tfrac{b}{b}$:

$$4\tfrac{1}{5} = 3 + \frac{5}{5} + \frac{1}{5} = 3\tfrac{6}{5}$$

Why borrowing is the same move you already know

Taking $7$ from $52$ borrows ten ones from the tens column, because the ones column has nothing to give. Nothing is created — one ten is rewritten as ten ones.

Here one whole is rewritten as $\tfrac{5}{5}$, in the units the subtraction needs. Same operation, different base. See subtracting fractions.

Worked examples

Work out $\tfrac{3}{4} - \tfrac{1}{6}$.

$$\frac{3 \times 6 - 4 \times 1}{24} = \frac{14}{24} = \frac{7}{12}$$

Using the least common denominator of $12$ reaches $\tfrac{9}{12} - \tfrac{2}{12}$ directly, with no simplifying at the end.

Work out $5\tfrac{1}{4} - 2\tfrac{3}{4}$.

Borrow first: $5\tfrac{1}{4} = 4\tfrac{5}{4}$.

$$4\tfrac{5}{4} - 2\tfrac{3}{4} = 2\tfrac{2}{4} = 2\tfrac{1}{2}$$

Check by adding back

Add your answer to the number you took away. You should land on the number you started with — the fastest check there is on a subtraction.