Dividing Fractions
Dividing by a fraction might seem tricky, but there's a simple rule: multiply by the reciprocal.
The "Keep, Change, Flip" Method
This is the easiest way to remember how to divide fractions:
Keep, Change, Flip
- Keep the first fraction exactly as it is.
- Change the division sign (÷) to a multiplication sign (×).
- Flip the *second* fraction (find its reciprocal). The reciprocal is found by swapping the numerator and denominator.
Example: $\frac{a}{b} \div \frac{c}{d}$ becomes $\frac{a}{b} \times \frac{d}{c}$
After you "Keep, Change, Flip", you just follow the rules for multiplying fractions!
The Steps for Dividing Fractions
- Convert (if needed): Change any mixed numbers to improper fractions. Rewrite any integers as fractions over 1 (e.g., $5 = \frac{5}{1}$).
- Keep, Change, Flip: Apply the rule described above.
- Multiply: Follow the steps for multiplying fractions (simplify before or after multiplying numerators and denominators).
- Simplify: Reduce the final answer to lowest terms. Convert to a mixed number if appropriate.
Reminder: Sign Rules for Division
The rules are the same as for multiplication:
- Positive ÷ Positive = Positive (+)
- Negative ÷ Negative = Positive (+)
- Positive ÷ Negative = Negative (-)
- Negative ÷ Positive = Negative (-) Same signs = positive result. Different signs = negative result.
Examples with Step-by-Step Solutions
Example 1: Calculate $\frac{1}{2} \div \frac{3}{4}$
Apply Keep, Change, Flip:
$$ \begin{align*} & \frac{1}{2} \div \frac{3}{4} \\ = & \boxed{\frac{1}{2} \times \frac{4}{3}} \quad & \text{(Keep } \frac{1}{2}, \text{ Change } \div \text{ to } \times, \text{ Flip } \frac{3}{4} \text{ to } \frac{4}{3} \text{)} \\ & \quad & \text{(Now multiply - simplify first if possible)} \\ & \quad \text{Common factor 2 and 4: } \frac{1}{\cancel{2}^1} \times \frac{\cancel{4}^2}{3} \\ = & \frac{1}{1} \times \frac{2}{3} \\ = & \frac{\boxed{1 \times 2}}{\boxed{1 \times 3}} \\ = & \frac{2}{3} \end{align*} $$Result: $\frac{2}{3}$
Example 2: Calculate $\left(-\frac{7}{10}\right) \div \left(\frac{2}{5}\right)$
Apply Keep, Change, Flip (watch the signs):
$$ \begin{align*} & \left(-\frac{7}{10}\right) \div \left(\frac{2}{5}\right) \\ = & \boxed{\left(-\frac{7}{10}\right) \times \left(\frac{5}{2}\right)} \quad & \text{(Keep, Change, Flip)} \\ & \quad & \text{(Simplify before multiplying)} \\ & \quad \text{Common factor 5 and 10: } (-\frac{7}{\cancel{10}^2}) \times (\frac{\cancel{5}^1}{2}) \\ = & \left(-\frac{7}{2}\right) \times \left(\frac{1}{2}\right) \\ = & \frac{\boxed{-7 \times 1}}{\boxed{2 \times 2}} \\ = & -\frac{7}{4} \end{align*} $$Result: $-\frac{7}{4}$ or $-1\frac{3}{4}$
Example 3: Calculate $\frac{5}{6} \div (-3)$
Rewrite integer as fraction, then Keep, Change, Flip:
$$ \begin{align*} & \frac{5}{6} \div \boxed{(-3)} \quad & \text{(Rewrite -3 as } -\frac{3}{1} \text{)} \\ = & \frac{5}{6} \div \left(-\frac{3}{1}\right) \\ = & \boxed{\frac{5}{6} \times \left(-\frac{1}{3}\right)} \quad & \text{(Keep, Change, Flip)} \\ & \quad & \text{(No common factors to simplify)} \\ = & \frac{\boxed{5 \times (-1)}}{\boxed{6 \times 3}} \\ = & -\frac{5}{18} \end{align*} $$Result: $-\frac{5}{18}$