If a rational number \(r\) satisfies \(0 < r < 1,\) which one is the minimum among the following choices?

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If a rational number \(r\) satisfies \(0 < r < 1,\) which one is the minimum among the following choices?

A. \(r\)
B. \(-r\)
C. \(\dfrac{-1}{r}\)
D. \(-r^2\)
E. \(\dfrac{-1}{r^2}\)

OA E

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Let's evaluate the answer choices.

A. r : this number is a positive fraction between zero and one. For example r = 1/4
B. -r: this number is a negative fraction between -1 and zero. For example r = -1/4
C. -1/r: this number is a negative number to the left of -1
For example, if r = 1/4, then -1/r = -1/(1/4) = -4
D. - $$r^2$$ : using r = 1/4, - $$r^2$$ =-1/16: This number is a fraction between -1 and 0
E. $$\frac{-1}{r^2}$$ : this fraction will be the most negative number. For example, using r = 1/4, the value of the expression will be -16

In general, when we square a fraction between 0 and 1, we make it smaller. Taking the reciprocal of the result gives us a larger positive number. Negating this gives us a more negative number.

The correct answer is E