Lesson 10.1: Quantitative Comparison

Introduction

In this lesson, we explore the Quantitative Comparison question type found in the GRE General Test. Our objectives are to understand how to effectively tackle these questions and improve our score by mastering the various strategies associated with this format. By understanding the specific answer choices and knowing how to compare quantities accurately and efficiently, students will develop the skills needed to succeed on the exam.

Learning Objectives

• Understand the four fixed answer choices in quantitative comparison questions and their implications.
• Learn to compare quantities without performing unnecessary calculations.
• Explore case testing, including special cases like zero, negatives, and fractions.
• Apply the four answer options in quantitative comparison correctly.
• Develop reasoning skills to compare quantities effectively.

Understanding the Answer Choices

Quantitative Comparison questions provide two quantities, labeled as Quantity A and Quantity B, and require you to determine the relationship between them. The possible answer choices are:

1. Quantity A is greater.
2. Quantity B is greater.
3. The two quantities are equal.
4. The relationship cannot be determined from the information given.

Explanation of Each Choice

• Quantity A is greater: This option indicates that Quantity A has a higher value than Quantity B. It is crucial to support this claim with reasoning or evidence from the problem.
• Quantity B is greater: Similar to the previous choice, this indicates that Quantity B exceeds Quantity A.
• The two quantities are equal: This indicates that both quantities have the same value. Proof is required to validate this outcome.
• The relationship cannot be determined: This choice signals that insufficient information is provided to establish a clear relationship between Quantity A and Quantity B.

Worked Example 1

Question:

If $ x = 2 $ and $ y = 3 $, compare Quantity A ($ x^2 $) and Quantity B ($ 2y $).

• Quantity A: $ x^2 = 2^2 = 4 $
• Quantity B: $ 2y = 2(3) = 6 $

Comparing Quantities:

Since $ 4 < 6 $, we conclude that Quantity B is greater. Thus, the answer is option 2.

Techniques for Comparison

When faced with Quantitative Comparison questions, it’s essential to utilize specific strategies that help in analyzing the given quantities without unnecessary computation. Here are a few techniques:

• Substituting Values: If the quantities involve variables, substituting them with numbers (especially simple values) can help identify relationships quickly.
• Analyzing Extremes: Testing extreme values, such as zero or negative numbers, can reveal different outcomes based on the behavior of the function or expression.
• Considering Special Cases: Fractions, zero, and negative numbers often lead to nuanced outcomes, depending on the relationship defined by the quantities.

Worked Example 2

Question:

Let Quantity A be $ 3x - 5 $ and Quantity B be $ 2x - 7 $.

Determine:

1. What is the relationship when $ x = 5 $?
2. What happens when $ x = 0 $?
3. Conclusion?

Calculations:

• For $ x = 5 $:
• Quantity A: $ 3(5) - 5 = 15 - 5 = 10 $
• Quantity B: $ 2(5) - 7 = 10 - 7 = 3 $
• Here, $ 10 > 3 $ leads to Quantity A is greater.

• For $ x = 0 $:
• Quantity A: $ 3(0) - 5 = -5 $
• Quantity B: $ 2(0) - 7 = -7 $
• Here, $ -5 > -7 $ still leads to Quantity A is greater.

Conclusion of Example 2

In both cases, Quantity A was greater. Therefore, if we have established a relationship that holds for the values we tested, we can conclude that Quantity A is greater in general.

Common Misconceptions

1. Focusing on calculations: Many students might think they need to compute the exact values of complex expressions. Instead, they should focus on the relationships between quantities and look for shortcuts.
2. Neglecting special cases: Zero and negative numbers can produce unexpected results; always test these cases thoroughly to avoid miscalculations.
3. Generalizing without proof: Just because two examples yield the same result does not always mean the relationship holds universally; always examine the conditions carefully.

Summary Strategy

1. Identify relationships quickly: Use substitution and intuitive reasoning first.
2. Test special values: Plugging in critical numbers like zero, negative values, and extremes can provide clarity.
3. Choose confidently: Once the relationships are established through deduction, choose the option that directly reflects the findings.

Conclusion

Quantitative Comparison questions on the GRE can seem daunting if not approached with the right mindset. By recognizing the format of the questions and applying strategic reasoning techniques, students can significantly improve their performance. Remember that mastery over the format itself will be as rewarding as the math skills necessary to solve the numerical values.

Study Notes

• The four answer choices for Quantitative Comparison questions are crucial to understanding.
• Practicing substitution and case testing can reveal relationships quickly.
• Always test numbers critically, especially $ 0 $ and negative values.
• Avoid unnecessary calculations when possible; focus on comparing relationships.
• Develop a systematic approach to analyzing the questions to increase efficiency.