You have seen that $\frac{3}{4} \div \frac{1}{8}$ asks how many eighths fit in three fourths (see What Dividing by a Fraction Means). Renaming with a common denominator always works, and it leads to a quick rule.
Idea clave
To divide by a fraction, multiply by its reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$. Keep the first fraction and flip only the divisor.
Por qué funciona
With a common denominator, $\frac{2}{3} \div \frac{3}{4} = \frac{8}{12} \div \frac{9}{12}$, which counts groups of 9 pieces in 8 pieces: $\frac{8}{9}$. The rule gives the same: $\frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$. This always happens: $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bd} \div \frac{bc}{bd} = \frac{ad}{bc}$, which is exactly $\frac{a}{b} \times \frac{d}{c}$. Another way to see it: dividing by $\frac{c}{d}$ undoes multiplying by $\frac{c}{d}$, and so does multiplying by $\frac{d}{c}$, because $\frac{c}{d} \times \frac{d}{c} = 1$.
Ejemplo resuelto: Two fractions
Find $\frac{3}{4} \div \frac{2}{5}$.
- Flip the divisor: $\frac{3}{4} \times \frac{5}{2}$.
- Multiply: $\frac{15}{8}$, which is $1\frac{7}{8}$.
- Check by multiplying back: $\frac{15}{8} \times \frac{2}{5} = \frac{30}{40} = \frac{3}{4}$.
Respuesta: $\frac{15}{8} = 1\frac{7}{8}$
Ejemplo resuelto: A mixed number
Find $2\frac{1}{2} \div \frac{3}{4}$.
- Write the mixed number as a fraction: $2\frac{1}{2} = \frac{5}{2}$.
- $\frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3}$.
- As a mixed number, $3\frac{1}{3}$. That makes sense: about three $\frac{3}{4}$s fit in $2\frac{1}{2}$.
Respuesta: $3\frac{1}{3}$
Ejemplo resuelto: A whole number
Find $6 \div \frac{3}{5}$.
- $6 \times \frac{5}{3} = \frac{30}{3} = 10$.
- Check: $10 \times \frac{3}{5} = 6$.
Respuesta: 10
Error común
Flipping the wrong fraction, or both. $\frac{3}{4} \div \frac{2}{5}$ is not $\frac{4}{3} \times \frac{5}{2}$: only the divisor, the number you divide by, is flipped. Quick check: multiply your answer by the divisor. You should get the first number back.