Loading...
\(-\frac{\sqrt{\frac{1}{3}}}{\frac{1}{3}}\), \(-\frac{\frac{1}{5}}{\sqrt{\frac{1}{5}}}\), \(-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}\), \(-\frac{\sqrt{\frac{1}{5}}}{\frac{1}{5}}\), \(-1\)
If the 5 numbers listed above are denoted \(\mathrm{x_1}\), \(\mathrm{x_2}\), \(\mathrm{x_3}\), \(\mathrm{x_4}\), and \(\mathrm{x_5}\), so that \(\mathrm{x_1} < \mathrm{x_2} < \mathrm{x_3} < \mathrm{x_4} < \mathrm{x_5}\), what is the value of \(\mathrm{x_4}\) ?
Let's understand what we're being asked to do here. We have 5 expressions that all look quite complex, involving fractions and square roots, and they're all negative numbers. We need to arrange them from smallest to largest (remember, smallest means most negative, like -10 is smaller than -2), then find which expression is the fourth one in this order.
Think of it like arranging temperatures: -10°F is colder (smaller) than -2°F. So when we order our 5 negative numbers, we're looking for the one that's fourth from the most negative.
The five expressions are:
Process Skill: TRANSLATE
Now let's make these expressions easier to work with by simplifying them one by one. Remember that dividing by a fraction is the same as multiplying by its reciprocal (flipping the fraction).
Let's work through each one:
First expression: \(-\sqrt{\frac{1}{3}} \div \frac{1}{3}\)
Dividing by \(\frac{1}{3}\) is the same as multiplying by 3.
So: \(-\sqrt{\frac{1}{3}} \times 3 = -3\sqrt{\frac{1}{3}}\)
Since \(\sqrt{\frac{1}{3}} = \frac{1}{\sqrt{3}}\), we get: \(-\frac{3}{\sqrt{3}} = -\frac{3\sqrt{3}}{3} = -\sqrt{3} \approx -1.732\)
Second expression: \(-\frac{1}{5} \div \sqrt{\frac{1}{5}}\)
This becomes: \(-\frac{1}{5} \times \frac{1}{\sqrt{\frac{1}{5}}} = -\frac{1}{5} \times \sqrt{5} = -\frac{\sqrt{5}}{5} \approx -0.447\)
Third expression: \(-\frac{1}{3} \div \sqrt{\frac{1}{3}}\)
This becomes: \(-\frac{1}{3} \times \frac{1}{\sqrt{\frac{1}{3}}} = -\frac{1}{3} \times \sqrt{3} = -\frac{\sqrt{3}}{3} \approx -0.577\)
Fourth expression: \(-\sqrt{\frac{1}{5}} \div \frac{1}{5}\)
Dividing by \(\frac{1}{5}\) is the same as multiplying by 5.
So: \(-\sqrt{\frac{1}{5}} \times 5 = -5\sqrt{\frac{1}{5}} = -\frac{5}{\sqrt{5}} = -\sqrt{5} \approx -2.236\)
Fifth expression: \(-1\)
This is already simplified: \(-1\)
Process Skill: SIMPLIFY
Now let's look at our simplified values and their approximate decimal equivalents:
Since these are all negative numbers, the one with the largest absolute value is actually the smallest number. Think of it on a number line: -2.236 is furthest to the left (smallest), then -1.732, then -1, then -0.577, and finally -0.447 is closest to zero (largest).
Arranging from smallest to largest (most negative to least negative):
Therefore, \(x_4\) corresponds to the third original expression: \(-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}\)
The value of \(x_4\) is \(-\frac{\frac{1}{3}}{\sqrt{\frac{1}{3}}}\), which corresponds to answer choice C.
Students often forget that for negative numbers, the one with the larger absolute value is actually smaller. For example, -3 < -1 because -3 is further from zero on the number line. This is counterintuitive since for positive numbers, 3 > 1. Students may incorrectly think that since |-3| > |-1|, then -3 > -1.
The expressions like \(-\sqrt{\frac{1}{3}}/\frac{1}{3}\) and \(-\frac{1}{5}/\sqrt{\frac{1}{5}}\) can appear intimidating. Students might abandon a systematic approach and try to guess or use rough estimates without proper simplification, leading to incorrect comparisons.
When converting \(-\sqrt{\frac{1}{3}} \div \frac{1}{3}\), students often make errors with the rule "dividing by a fraction equals multiplying by its reciprocal." They might forget to flip the fraction or apply the multiplication incorrectly, getting \(-\sqrt{\frac{1}{3}}/3\) instead of \(-3\sqrt{\frac{1}{3}}\).
When simplifying expressions like \(-\frac{3}{\sqrt{3}}\) to \(-\sqrt{3}\), students frequently make algebraic mistakes. They might incorrectly rationalize the denominator or make sign errors during the manipulation of square root expressions.
When calculating approximate values like \(\sqrt{3} \approx 1.732\) or \(\sqrt{5} \approx 2.236\), students may use incorrect values or make arithmetic mistakes that throw off their entire comparison. Small errors here lead to incorrect ordering of the final values.
After correctly determining the order from smallest to largest, students might select \(x_2\) or \(x_3\) instead of \(x_4\). This happens because they lose track of which position corresponds to which original expression, especially when the ordering differs significantly from the original presentation order.
Students might correctly identify that \(x_4 = -\frac{\sqrt{3}}{3}\), but then look for this simplified form among the answer choices instead of recognizing that it corresponds to the original expression \(-\frac{1}{3}/\sqrt{\frac{1}{3}}\). They need to match their result back to the original notation given in the choices.