Division undoes multiplication, so the sign rules you learned for multiplying integers carry straight over.
Idea clave
Divide the absolute values. Same signs give a positive quotient; different signs give a negative one. Also, $-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}$.
Por qué funciona
$-24 \div 6$ asks: what times 6 is $-24$? Since $(-4) \times 6 = -24$, the answer is $-4$. And $-24 \div (-6)$ asks: what times $-6$ is $-24$? Since $4 \times (-6) = -24$, the answer is $4$. Every division fact comes from a multiplication fact, so it follows the same sign rules.
| Multiplication fact | Division facts it gives |
|---|---|
| $(-4) \times 6 = -24$ | $-24 \div 6 = -4$ and $-24 \div (-4) = 6$ |
| $4 \times (-6) = -24$ | $-24 \div (-6) = 4$ and $-24 \div 4 = -6$ |
| $(-4) \times (-6) = 24$ | $24 \div (-6) = -4$ and $24 \div (-4) = -6$ |
Worked examples
Ejemplo resuelto: Different signs
Find $-36 \div 9$.
- $36 \div 9 = 4$.
- Different signs: negative. Check: $(-4) \times 9 = -36$ ✓.
Respuesta: $-4$
Ejemplo resuelto: Same signs
Find $\frac{-56}{-8}$.
- $56 \div 8 = 7$.
- Same signs: positive. Check: $7 \times (-8) = -56$ ✓.
Respuesta: $7$
Ejemplo resuelto: A steady drop
A stock lost \$18 in 6 days, the same amount each day. What was the change each day?
- The total change is $-18$ dollars, split into 6 equal parts.
- $-18 \div 6 = -3$: down \$3 a day.
Respuesta: $-3$ dollars per day
Error común
Thinking that a negative divided by a negative is negative, so $-56 \div (-8) = -7$. Check with multiplication: $-7 \times (-8) = 56$, not $-56$. Every division can be checked this way.
$\frac{-3}{4}$, $\frac{3}{-4}$ and $-\frac{3}{4}$ are the same number: one negative sign anywhere makes the quotient negative. But $\frac{-3}{-4} = \frac{3}{4}$ is positive. And dividing by $0$ is never allowed: no number times $0$ gives $-24$.