Question

Is \(x / 8>y / 16\) ? (1) \(x>y\) (2) \(2 x>y\) 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

B. 2 alone, not 1 alone

Step-by-step solution

Understand the Inequality

Given the inequality is \( \frac{x}{8} > \frac{y}{16} \). Simplify it to a common denominator or compare the fractions directly.

Simplify the Inequality

Multiply both sides by 16 to eliminate the fractions: \( 16 \times \frac{x}{8} > 16 \times \frac{y}{16} \). Simplify to get \(2x > y \).

Analyze Statement (1)

Statement (1) says \(x > y\). This does not necessarily mean \(2x > y\), because it might fail for some values of \(x\) and \(y\). For instance, if \(x = 1\) and \(y = 2\), \(x > y\) is false.

Analyze Statement (2)

Statement (2) says \(2x > y\). This is exactly the simplified form of our original inequality \( \frac{x}{8} > \frac{y}{16} \). Therefore, statement (2) is sufficient for the inequality to hold true.

Combine Statements

Combining statement (1) and (2) is not necessary because statement (2) alone is sufficient to prove the inequality \( \frac{x}{8} > \frac{y}{16} \).

Key concepts

inequality comparison

Understanding inequalities is crucial for solving many GMAT problems. In this exercise, we are given the inequality: \(\frac{x}{8} > \frac{y}{16}\). To compare these fractions effectively, we need to simplify them. One of the methods is to find a common denominator. Multiplying both sides by 16 eliminates the fractions, resulting in the simplified inequality: \(2x > y\). It's essential to recognize that simplifying inequalities helps in making direct comparisons. Thus, once simplified, you can clearly see what the original inequality means, making it easier to analyze subsequent statements.

statement analysis

Analyzing statements provided in GMAT questions is a step-by-step process. Let's break down the statements given for our exercise:

• **Statement (1)**: x > y
  We know x > y, but this does not automatically guarantee 2x > y. For example, if x = 1 and y = 2, x > y is false, showing that this statement alone is not sufficient.
• **Statement (2)**: 2x > y
  This statement directly matches the simplified form of our original inequality, indicating it is sufficient.

Statement analysis involves verifying if the provided information (e.g., x > y or 2x > y) meets the conditions required by the problem (e.g., \(2x > y\)). This systematic approach helps us determine the sufficiency of each statement.

sufficiency conditions

In GMAT problem-solving, understanding when information is sufficient is key. For this problem:

• **Sufficiency**: Statement (2) alone is sufficient because it directly translates the inequality \(2x > y\) from the simplified form of the given inequality \( \frac{x}{8} > \frac{y}{16} \).
• **Insufficiency**: Statement (1) alone is not sufficient. Knowing that x > y does not necessarily indicate 2x > y, as there are counterexamples.

Whenever analyzing statements, it’s crucial to check if each statement provides all the information needed to conclude the problem's inequality or condition. This practice ensures clarity, resulting in correct answers and higher GMAT scores.