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Let's work through this together:
Step 1: What does the square condition tell us about n?
We're told that \(n^2 < \frac{1}{100}\). Now, think about what this means. Since \(\frac{1}{100} = 0.01\), we need \(n^2 < 0.01\).
Let me test a few values to build intuition:
Notice what's happening here: for \(n^2\) to be less than \(\frac{1}{100}\), n must be between \(-\frac{1}{10}\) and 0 (since we know n is negative).
So we have: \(-\frac{1}{10} < n < 0\)
Step 2: What happens to the reciprocal of a small negative number?
Here's the key insight you need to see: when you take the reciprocal of a small negative number (close to zero), you get a large negative number.
Let me show you with examples:
Do you see the pattern? As n gets closer to zero (but stays negative), \(\frac{1}{n}\) becomes more and more negative—it goes toward negative infinity.
Step 3: What's the range of the reciprocal?
Since we established that \(-\frac{1}{10} < n < 0\):
Therefore: \(\frac{1}{n} < -10\)
Let's verify with a quick example: If \(n = -\frac{1}{50} = -0.02\), then \(n^2 = 0.0004 < 0.01\) ✓, and \(\frac{1}{n} = -50\), which is indeed less than -10 ✓
Answer: A (Less than -10)
Want to master this type of problem systematically?
The complete solution on Neuron goes beyond just solving this problem—it teaches you the systematic framework for handling reciprocals with inequalities, reveals the common traps students fall into (like confusing the range of n with the range of \(\frac{1}{n}\)), and shows you how to recognize these patterns instantly on test day. You can check out the detailed step-by-step solution on Neuron by e-GMAT to understand the underlying principles that apply to all reciprocal inequality problems. You can also explore comprehensive explanations for other GMAT official questions on Neuron with practice quizzes and detailed analytics to identify your specific weaknesses.
Happy learning!
Step 1: What does the square condition tell us about n?
We're told that \(n^2 < \frac{1}{100}\). Now, think about what this means. Since \(\frac{1}{100} = 0.01\), we need \(n^2 < 0.01\).
Let me test a few values to build intuition:
- If \(n = -0.1\), then \(n^2 = 0.01\) (this is our boundary)
- If \(n = -0.05\), then \(n^2 = 0.0025 < 0.01\) ✓
- If \(n = -0.2\), then \(n^2 = 0.04 > 0.01\) ✗
Notice what's happening here: for \(n^2\) to be less than \(\frac{1}{100}\), n must be between \(-\frac{1}{10}\) and 0 (since we know n is negative).
So we have: \(-\frac{1}{10} < n < 0\)
Step 2: What happens to the reciprocal of a small negative number?
Here's the key insight you need to see: when you take the reciprocal of a small negative number (close to zero), you get a large negative number.
Let me show you with examples:
- If \(n = -\frac{1}{10} = -0.1\), then \(\frac{1}{n} = -10\)
- If \(n = -\frac{1}{20} = -0.05\), then \(\frac{1}{n} = -20\)
- If \(n = -\frac{1}{100} = -0.01\), then \(\frac{1}{n} = -100\)
Do you see the pattern? As n gets closer to zero (but stays negative), \(\frac{1}{n}\) becomes more and more negative—it goes toward negative infinity.
Step 3: What's the range of the reciprocal?
Since we established that \(-\frac{1}{10} < n < 0\):
- When n approaches \(-\frac{1}{10}\) from the right (getting closer to zero), \(\frac{1}{n}\) approaches -10 from the left (becoming more negative)
- When n approaches 0 from the left, \(\frac{1}{n}\) approaches negative infinity
Therefore: \(\frac{1}{n} < -10\)
Let's verify with a quick example: If \(n = -\frac{1}{50} = -0.02\), then \(n^2 = 0.0004 < 0.01\) ✓, and \(\frac{1}{n} = -50\), which is indeed less than -10 ✓
Answer: A (Less than -10)
Want to master this type of problem systematically?
The complete solution on Neuron goes beyond just solving this problem—it teaches you the systematic framework for handling reciprocals with inequalities, reveals the common traps students fall into (like confusing the range of n with the range of \(\frac{1}{n}\)), and shows you how to recognize these patterns instantly on test day. You can check out the detailed step-by-step solution on Neuron by e-GMAT to understand the underlying principles that apply to all reciprocal inequality problems. You can also explore comprehensive explanations for other GMAT official questions on Neuron with practice quizzes and detailed analytics to identify your specific weaknesses.
Happy learning!
