Question

Find a counterexample to disprove the conjecture: "If the quotient of two numbers is positive, then the two numbers μst both be positive."

a) -3÷ 2 = −1.5
b) 2-2 = −1
c)-4-2 = 2
d) None of the above

Answer 1

Option c) -4 ÷ -2 = 2 is the correct counterexample that disproves the conjecture, showing that even if the quotient of two numbers is positive, the two numbers do not necessarily have to be positive.

Step-by-step explanation:

To disprove the conjecture "If the quotient of two numbers is positive, then the two numbers must both be positive," we need to find an example where the quotient is positive but one or both of the numbers are negative.

Looking at the multiplication rules for signs, we know that when two positive numbers multiply, we get a positive result, and similarly, when two negative numbers multiply, the result is also positive. However, when the two numbers multiplied have opposite signs, their product (or quotient) is negative. So, to find a counterexample, we should find two numbers with opposite signs whose quotient is positive. This would directly contradict the conjecture.

The correct counterexample from the provided options is c) -4 ÷ -2 = 2. Here, both numbers are negative, but their quotient is positive, disproving the conjecture that both numbers must be positive if their quotient is positive.