Quantitative Comparison with a Variable Between 0 and 1
Priya is working on a Quantitative Reasoning section and encounters a quantitative comparison item: Quantity A is x^2; Quantity B is x; the given condition is 0 < x < 1. She has 75 seconds. She tests x = 1/2: Quantity A = 1/4, Quantity B = 1/2, so B is larger. She then tests x = 0.9: A = 0.81, B = 0.9, so B is larger again. She is tempted to choose 'Quantity B is greater' immediately, but she remembers that the instructions ask her to compare the quantities, not to find a single value of x, and that a single test case is not a proof. She also recalls that multiplying or dividing an inequality by a positive number preserves its direction, but squaring a number between 0 and 1 changes its size. She must decide whether the relationship holds for every allowed x or whether it can vary.
1. Use the given constraint to compare x^2 and x algebraically. Show why x^2 < x for every x in the interval 0 < x < 1.
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Start with the constraint 0 < x < 1. Since x is positive, we can multiply both sides of x < 1 by x without changing the direction of the inequality: x · x < x · 1, so x^2 < x. Another way is to subtract: x - x^2 = x(1 - x). Because 0 < x < 1, both x and 1 - x are positive, so their product is positive. That means x - x^2 > 0, or x > x^2. In either approach, Quantity B is greater for every allowed value of x. The algebra proves the relationship, while the two test values only suggested it.
2. Priya's two test values both point to B. Explain why testing alone is not sufficient on a quantitative comparison, and describe a case where testing several values could mislead her.
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Testing values can reveal a pattern, but it cannot prove that the pattern holds for every allowed value. A quantitative comparison asks about the entire domain given by the condition, so one counterexample destroys a conclusion. Testing can mislead if the variable behaves differently in different regions. For example, if the condition were x > 1, then x^2 > x, so Quantity A would be greater. If the condition were -1 < x < 0, then x is negative and x^2 is positive, so Quantity A would again be greater. Priya's tests all lie in the interval 0 < x < 1, where the pattern happens to be consistent, but without the constraint and algebraic reasoning she could not know that.
3. Suppose the condition were changed to -1 < x < 0. How would the comparison change? Explain with a test value and with reasoning about signs.
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The comparison would reverse: Quantity A would be greater. Test x = -0.5. Then Quantity A = (-0.5)^2 = 0.25, and Quantity B = -0.5. Since 0.25 > -0.5, A is greater. The reasoning is that any nonzero real number squared is positive, while x itself is negative in this interval. Therefore x^2 > 0 > x, so A > B for every allowed x. The test value confirms the sign reasoning, and the sign reasoning proves the result for the whole interval.