In arithmetic, when a negative number is divided by another negative number, what is the sign of the resulting quotient?

Introduction / Context:
This is a basic sign rule question from arithmetic and algebra. Questions about the signs of results when multiplying or dividing positive and negative numbers commonly appear in aptitude tests. Knowing these rules saves time and helps prevent errors in longer calculations.

Given Data / Assumptions:
- We consider division of one negative number by another negative number.- Both numerator and denominator are nonzero real numbers.- We are asked only about the sign of the quotient, not the exact value.

Concept / Approach:
The sign rules for multiplication and division are closely related. When multiplying, positive times positive is positive, positive times negative is negative, and negative times negative is positive. Division is the inverse operation of multiplication, so the same sign rules carry over. Specifically, dividing a negative number by another negative number yields a positive result because it corresponds to answering the question: which number multiplied by a negative gives another negative? The answer must be positive.

Step-by-Step Solution:
- Let the two negative numbers be -a and -b, where a and b are positive real numbers.- The quotient is (-a) / (-b).- We can rewrite this as (-1 * a) / (-1 * b).- Group the sign factors: (-1 / -1) * (a / b).- The ratio (-1 / -1) is equal to 1, which is positive.- The ratio a / b is a positive number because a and b are positive.- Therefore, the product of these two positive quantities is positive.- Hence (-a) / (-b) is a positive number.

Verification / Alternative check:
Take a concrete example: (-10) divided by (-2). If the result were x, then x * (-2) should equal -10. The only real number x that satisfies x * (-2) = -10 is x = 5, which is positive. Similar examples such as (-9) / (-3) = 3 also demonstrate that the quotient is positive. This matches the general rule that a negative divided by a negative gives a positive result.

Why Other Options Are Wrong:
- Option -ve: A negative result would correspond to dividing numbers with opposite signs, such as negative by positive or positive by negative, which is not the case here.- Option 0: The quotient is zero only when the numerator is zero, not just because both numbers are negative.- Option Cannot be determined: The sign is completely determined by the signs of numerator and denominator, so there is no ambiguity.- Option Depends on numbers: The magnitude may depend on the numbers, but the sign does not as long as both are negative.

Common Pitfalls:
A common source of mistakes is mixing up the rules for adding negative numbers with those for multiplying or dividing them. While adding two negatives makes the result more negative, multiplying or dividing two negatives gives a positive result. Always remember that for multiplication and division, like signs give positive and unlike signs give negative.

Final Answer:
When a negative number is divided by another negative number, the quotient is positive (+ve).