Signs in division

The idea

The quotient of two integers is positive for like signs and negative for unlike signs: for positive integers a and b, –a ÷ b = a ÷ –b = –(a ÷ b) and –a ÷ –b = a ÷ b.

Division, signs and all

We can always check a division by multiplying back. 12 ÷ 3 = 4, because 3 × 4 = 12.

What will (−12) ÷ 3 be?

• Positive: 4
• Negative: −4
• Not possible

Check every division

Four divisions use the same two sizes, with different signs. For each one choose the quotient. The toy checks it by multiplying back.

The same sign rule as multiplication

A division is a multiplication with a missing number, so the signs behave as they did in multiplication.

For example: (−25) × (−2) = 50, so 50 ÷ (−25) = −2.

The quotient of two integers is positive for like signs and negative for unlike signs. For positive integers a and b: −a ÷ b = a ÷ −b = −(a ÷ b) −a ÷ −b = a ÷ b

• Dividend | Divisor | Quotient
• positive | positive | positive
• negative | negative | positive
• negative | positive | negative
• positive | negative | negative

Notes

The quotient of two integers is positive for like signs and negative for unlike signs: −a ÷ b = a ÷ −b = −(a ÷ b) −a ÷ −b = a ÷ b

Check yourself

Find 150 ÷ (−6).

Answer: -25

150 ÷ 6 = 25, and the signs are unlike, so 150 ÷ (−6) = −25. Check: (−6) × (−25) = 150.

Find (−81) ÷ (−9).

Answer: 9

81 ÷ 9 = 9, and the signs are alike, so (−81) ÷ (−9) = 9. Check: (−9) × 9 = −81.

Which quotient is positive?

Mira says (−28) ÷ 7 = 4, because 28 ÷ 7 = 4 and the minus sign can be ignored. What is wrong?

• (−48) ÷ 6. The signs are unlike, so the quotient is negative: −8.
• (−48) ÷ (−6) — correct. Yes! Both are negative: like signs give a positive quotient, 8.
• 48 ÷ (−6). The signs are unlike, so the quotient is negative: −8.
• 7 × 4 = 28, not −28. The signs are unlike, so the quotient is −4 — correct. Yes! Checking by multiplying back catches the mistake: 7 × (−4) = −28.
• Nothing is wrong. Multiply back: 7 × 4 = 28, but the dividend is −28.
• The quotient is −35. That adds 7 and 28. We divide: 28 ÷ 7 = 4, with a negative sign.