Quantitative comparison
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What this topic is really about
The average of 4, 6, and 8 is calculated as (4 + 6 + 8) / 3 = 6, while the median (middle value) of this ordered set is also 6. Because both quantities equal 6, the two quantities are equal. Option A is incorrect because the average does not exceed the median for this symmetric set.
For 0 < x < 1, x² is greater than x³.. For any number between 0 and 1, raising it to a higher exponent makes the value smaller, so x² exceeds x³ in this interval, making Quantity A larger.
See the mechanism
When x is between 0 and 1, raising x to a higher power results in a smaller value. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.
An exam-style question, fully explained
Quantity A: x²; Quantity B: x³, given x is between 0 and 1. Which is greater?
- Identify what the question tests: Quantity A: x²; Quantity B: x³, given x is between 0 and 1..
- For any number between 0 and 1, raising it to a higher exponent makes the value smaller, so x² exceeds x³ in this interval, making Quantity A larger.
- Choice B assumes the larger exponent yields a larger result, a misconception that only holds when x>1, not within the given range.
- Why it matters: When x is between 0 and 1, raising x to a higher power results in a smaller value. This is because as the exponent increases, the fraction becomes smaller. Thus, x² will always be greater than x³ in this interval. The misconception that a larger exponent always yields a larger result only applies when x is greater than 1.
Traps the examiner sets
- Many people incorrectly assume that a higher exponent will always result in a larger value, which is not true for numbers between 0 and 1. This mistake leads to choosing the wrong answer.
- Students often mistake the exponent as a multiplier rather than a power, thinking 10^{4}=4 or adding the exponent to the coefficient, leading to an incorrect comparison.
- Option A is incorrect because the average does not exceed the median for this symmetric set.
Test your recall
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Run a focused 10-question mini-mock on Quantitative comparison and see it stick.
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