If 0 < r < s < t which of the following CANNOT be true?

Rationale

rt < rs cannot be true when 0 < r < s < t.

Given that r, s, and t are positive numbers and r is less than s, multiplying r by t (which is also greater than s) will yield a product smaller than multiplying s by r. Thus, the inequality rt < rs cannot hold true under these conditions.

A (r ^ 2 < s ^ 2 < t ^ 2) YOUR ANSWER

This statement can be true since squaring positive numbers maintains their order. If 0 < r < s < t, then r^2 will be less than s^2, and s^2 will be less than t^2, satisfying the inequality.

B (rt < rs) CORRECT

This statement cannot be true. Since r is less than s and both are positive, when multiplying r by t (which is greater than s), the product rt will be less than rs. Therefore, this inequality is not possible under the given constraints.

C (1/t < 1/s < 1/r) YOUR ANSWER

This statement can be true as it involves the reciprocals of positive numbers. Since t > s > r, it follows that 1/t < 1/s < 1/r, preserving the order when taking reciprocals of positive values.

D (s/r < t/r) YOUR ANSWER

This statement can also be true. Dividing both s and t by r (which is positive) maintains the inequality. Since s > r and t > r, s/r will be less than t/r.

E (r/t < r/s) YOUR ANSWER

This statement is true as well. Since t > s, the ratios r/t and r/s will maintain the order, with r/t being less than r/s because dividing by a larger number (t) yields a smaller fraction than dividing by a smaller number (s).

Conclusion

In the context of the given inequalities involving positive numbers r, s, and t, the statement rt < rs is the only one that cannot be true. The nature of multiplication with respect to the values of r, s, and t directly leads to this conclusion, while all other options maintain their validity under the constraints provided.

If 0 < r < s < t which of the following CANNOT be true?

rt < rs cannot be true when 0 < r < s < t.

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