Let's analyze the given conditional statement and the possible counterexamples step-by-step:
Conditional Statement: "If two positive numbers are multiplied together, then the product will be greater than both of the two positive numbers."
We will check each option to see if it serves as a counterexample, i.e., whether the product is not greater than both of the original numbers.
1. Option 1: [tex]\(2 \times 4\)[/tex]
- Here, both numbers are positive: 2 and 4.
- The product is [tex]\(2 \times 4 = 8\)[/tex].
- Clearly, [tex]\(8\)[/tex] is greater than both [tex]\(2\)[/tex] and [tex]\(4\)[/tex].
- Therefore, [tex]\(2 \times 4\)[/tex] is not a counterexample.
2. Option 2: [tex]\(5 \times (-3)\)[/tex]
- Here, one number is positive (5) and one number is negative (-3).
- The product is [tex]\(5 \times (-3) = -15\)[/tex].
- The product is negative, and thus it is not greater than either of the initial numbers.
- However, since the given statement only concerns the multiplication of two positive numbers, this example does not qualify for consideration.
- Therefore, [tex]\(5 \times (-3)\)[/tex] cannot serve as a valid counterexample.
3. Option 3: [tex]\(\frac{6}{5} \times \frac{10}{3}\)[/tex]
- Both fractions are positive.
- The product is [tex]\(\frac{6}{5} \times \frac{10}{3} \approx 2\)[/tex].
- Clearly, [tex]\(2\)[/tex] is greater than both [tex]\(\frac{6}{5} \approx 1.2\)[/tex] and [tex]\(\frac{10}{3} \approx 3.33\)[/tex].
- Therefore, [tex]\(\frac{6}{5} \times \frac{10}{3}\)[/tex] is not a counterexample.
4. Option 4: [tex]\(\frac{2}{3} \times 9\)[/tex]
- Here, both numbers are positive: [tex]\(\frac{2}{3}\)[/tex] and [tex]\(9\)[/tex].
- The product is [tex]\(\frac{2}{3} \times 9 = 6\)[/tex].
- When comparing [tex]\(6\)[/tex] with [tex]\(\frac{2}{3}\)[/tex] and [tex]\(9\)[/tex], we see that [tex]\(6\)[/tex] is actually less than [tex]\(9\)[/tex].
- Thus, [tex]\(6\)[/tex] is not greater than one of the original numbers ([tex]\(9\)[/tex]).
- Therefore, [tex]\(\frac{2}{3} \times 9\)[/tex] serves as a counterexample to the conditional statement.
Conclusion:
The counterexample for the conditional statement "If two positive numbers are multiplied together, then the product will be greater than both of the two positive numbers" is [tex]\(\frac{2}{3} \times 9\)[/tex].
