Q. a negative divided by a positive equals

Answer and detailed explanation

Short answer: A negative number divided by a positive number is a negative number.

1. Set up variables.

   Let a be a negative number and b be a positive number. In symbols,

   \(a < 0\) and \(b > 0\).

2. Define the quotient and relate it to multiplication.

   Let \(x\) denote the quotient \(a \div b\), so

   \(x = \dfrac{a}{b}\).

   By the definition of division, this is equivalent to the multiplication statement

   \(b \cdot x = a\).

3. Use the sign of b to determine the sign of x.

   Because \(b > 0\), multiplying a number by \(b\) preserves the sign of that number. Suppose for contradiction that \(x \ge 0\).

   If \(x > 0\) then \(b \cdot x > 0\). If \(x = 0\) then \(b \cdot x = 0\). In either case \(b \cdot x \ge 0\).

   But \(b \cdot x = a\) and we know \(a < 0\). This is a contradiction. Therefore the assumption \(x \ge 0\) is false, so \(x < 0\).

4. Conclude the sign of the quotient.

   We have shown \(x = \dfrac{a}{b} < 0\). Thus a negative divided by a positive is negative.

5. Practical procedure (what to do separately when computing).
   1. Ignore the signs and divide the absolute values: compute \(\dfrac{|a|}{|b|}\).
   2. Determine the sign of the result: negative divided by positive yields a negative sign.
   3. Combine sign and magnitude to get the final answer: \(\dfrac{a}{b} = -\dfrac{|a|}{|b|}\).
6. Examples.

   \(\dfrac{-6}{3} = -2\) because \(3 \cdot (-2) = -6\).

   \(\dfrac{-7}{2} = -3.5\) because \(2 \cdot (-3.5) = -7\).

Final statement: If \(a < 0\) and \(b > 0\), then \(\dfrac{a}{b} < 0\); a negative divided by a positive equals a negative.