Fractions and decimals can be negative too. To multiply or divide them, combine the sign rules from integers with what you know about dividing fractions.
Idea clave
Decide the sign first (same signs: positive; different signs: negative). Then multiply or divide the absolute values the usual way.
Por qué funciona
Rational numbers follow the same properties as integers, including the distributive property that forced $(-1)(-1) = 1$. Any negative number is $-1$ times a positive number, so $$\left(-\frac{2}{3}\right)\left(-\frac{3}{4}\right) = (-1)(-1) \times \frac{2}{3} \times \frac{3}{4} = 1 \times \frac{6}{12} = \frac{1}{2}$$ The two factors of $-1$ make $1$, which is why two negatives give a positive product here too.
Worked examples
Ejemplo resuelto: Fractions
Find $-\frac{3}{4} \times \frac{2}{5}$.
- Different signs: the product is negative.
- $\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$.
Respuesta: $-\frac{3}{10}$
Ejemplo resuelto: Decimals
Find $(-1.5) \times (-0.4)$.
- Same signs: positive.
- $1.5 \times 0.4 = 0.6$.
Respuesta: $0.6$
Ejemplo resuelto: Dividing fractions
Find $-\frac{5}{6} \div \left(-\frac{10}{3}\right)$.
- Same signs: positive.
- Multiply by the reciprocal: $\frac{5}{6} \times \frac{3}{10} = \frac{15}{60} = \frac{1}{4}$.
Respuesta: $\frac{1}{4}$
Error común
Taking the reciprocal of the wrong number (the first one instead of the divisor), or losing the sign along the way. Settle the sign first and write it down, then work with positive numbers. Check the last example by multiplying back: $$\frac{1}{4} \times \left(-\frac{10}{3}\right) = -\frac{10}{12} = -\frac{5}{6} \checkmark$$
With mixed numbers, write them as fractions first: $$-2\frac{1}{2} \times 1\frac{1}{5} = -\frac{5}{2} \times \frac{6}{5} = -\frac{30}{10} = -3$$ With decimals, a quick estimate places the decimal point: $(-1.5)(-0.4)$ is a bit more than $1 \times 0.4$, so $0.6$ makes sense.