Question:
If \( 0<a<1<b \), which of the following is true about the reciprocals of \( a \) and \( b \)?
If \( 0<a<1<b \), which of the following is true about the reciprocals of \( a \) and \( b \)?
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If a number is between 0 and 1, its reciprocal is greater than 1. If a number is greater than 1, its reciprocal is between 0 and 1.
Updated On: Oct 7, 2025
- \( 1<\frac{1}{a}<\frac{1}{b} \)
- \( \frac{1}{a}<1<\frac{1}{b} \)
- \( \frac{1}{a}<\frac{1}{b}<1 \)
- \( \frac{1}{b}<1<\frac{1}{a} \)
- \( \frac{1}{b}<\frac{1}{a}<1 \)
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The Correct Option is D
Solution and Explanation
Step 1: Analyze \( a \).
Since \( 0<a<1 \), the reciprocal \( \frac{1}{a}>1 \). Step 2: Analyze \( b \).
Since \( b>1 \), the reciprocal \( \frac{1}{b}<1 \). Step 3: Compare.
Thus, we have the inequality: \[ \frac{1}{b}<1<\frac{1}{a}. \] Step 4: Conclusion.
Therefore, the correct ordering is choice (D). \[ \boxed{\frac{1}{b}<1<\frac{1}{a}} \]
Since \( 0<a<1 \), the reciprocal \( \frac{1}{a}>1 \). Step 2: Analyze \( b \).
Since \( b>1 \), the reciprocal \( \frac{1}{b}<1 \). Step 3: Compare.
Thus, we have the inequality: \[ \frac{1}{b}<1<\frac{1}{a}. \] Step 4: Conclusion.
Therefore, the correct ordering is choice (D). \[ \boxed{\frac{1}{b}<1<\frac{1}{a}} \]
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