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.
Key idea
Decide the sign first (same signs: positive; different signs: negative). Then multiply or divide the absolute values the usual way.
Why it works
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
Worked example: Fractions
Find $-\frac{3}{4} \times \frac{2}{5}$.
- Different signs: the product is negative.
- $\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}$.
Answer: $-\frac{3}{10}$
Worked example: Decimals
Find $(-1.5) \times (-0.4)$.
- Same signs: positive.
- $1.5 \times 0.4 = 0.6$.
Answer: $0.6$
Worked example: Dividing fractions
Find $-\frac{5}{6} \div \left(-\frac{10}{3}\right)$.
- Same signs: positive.
- Multiply by the reciprocal: $\frac{5}{6} \times \frac{3}{10} = \frac{15}{60} = \frac{1}{4}$.
Answer: $\frac{1}{4}$
Common mistake
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.