Question

Which of these two is the greater, \(x\) or \(y\) ? (1) \(x=2 a\) (2) \(y=5 \mathrm{a}\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

Answer is B. 2 alone is sufficient.

Step-by-step solution

Identify Given Equations

Start by identifying the given equations. From statement (1) we have: \( x = 2a \) From statement (2) we have: \( y = 5a \)

Compare the Equations

Compare the two equations to see which is greater.If \( a \) is positive, multiply both by \( a \):Since \( 5a \) (statement 2) is greater than \( 2a \), \( y \) is greater than \( x \).

Conclusion

Statement (2) alone is sufficient to conclude that \( y \) is greater than \( x \). Therefore, the correct answer is: B. 2 alone, not 1 alone

Key concepts

Comparison of Variables

When comparing variables, we often want to determine which one is greater or smaller. This involves understanding the relationship between them. In the given exercise, two expressions are provided:

• \(x = 2a\)
• \(y = 5a\)

To compare \(x\) and \(y\), we need to know the value or relationship of \(a\). Suppose \(a\) is a positive number, then multiplying a larger coefficient by \(a\) gives a larger product. Therefore, \(5a\) will be greater than \(2a\). It follows that \(y\) is greater than \(x\). Simple comparisons like these are common in GMAT problem-solving exercises, where establishing variable hierarchy is key.

Sufficiency of Statements

When tackling GMAT problem-solving questions, it's vital to determine if the given statements provide enough information to answer the question. This is known as evaluating the sufficiency of statements. In our exercise:

• Statement (1): \(x=2a\).
• Statement (2): \(y=5a\).

Evaluating Statement (1) alone tells us the relation of \(x\) to \(a\), but nothing about \(y\). Hence, Statement (1) alone isn’t sufficient. Evaluating Statement (2), we immediately see that since \(y = 5a\), if \(a\) is positive, \(y\) must be larger than \(x = 2a\), making Statement (2) alone enough to conclude the relationship. Through this analysis, we can determine the sufficiency of each statement, which is essential in GMAT problem solving.

Mathematical Reasoning

Mathematical reasoning involves using logical steps and mathematical operations to derive conclusions. In the given problem, we apply mathematical reasoning to determine the relationship between \(x\) and \(y\). Starting with the expressions \(x = 2a\) and \(y =5a\), and given that \(a\) is positive, we:

• Multiply 'a' with its coefficients.
• Compare the resulting values.

Mathematical reasoning requires us to break down the problem into smaller, manageable parts. By comparing \(2a\) and \(5a\), and leveraging our understanding of multiplication and inequalities, we conclude that \(y > x\). Mastering such reasoning is crucial for excelling in GMAT problem-solving, where each step builds on the one before.