Question

Is \( \frac{5^{x+2}}{25}<1 \)?
(1) \( 5^x<1 \)
(2) \( x<0 \)

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

Always try to simplify the question in a Data Sufficiency problem before evaluating the statements. Often, the simplified question is identical or directly related to one of the statements, making the evaluation much faster.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is an inequality problem involving exponents. The best approach is to simplify the inequality in the question first.
Step 2: Key Formula or Approach:
We will use the rules of exponents: 1. \( \frac{a^m}{a^n} = a^{m-n} \) 2. \( a^0 = 1 \) for any non-zero \(a\). 3. For a base \(b>1\), the inequality \( b^p<b^q \) is equivalent to \( p<q \).
Step 3: Detailed Explanation:
First, let's simplify the question's inequality: \[ \frac{5^{x+2}}{25}<1 \] Since \(25 = 5^2\), we can rewrite this as: \[ \frac{5^{x+2}}{5^2}<1 \] Using the exponent rule for division: \[ 5^{(x+2)-2}<1 \] \[ 5^x<1 \] Since \(1 = 5^0\), the inequality becomes: \[ 5^x<5^0 \] Because the base (5) is greater than 1, we can compare the exponents directly: \[ x<0 \] So, the question "Is \( \frac{5^{x+2}}{25}<1 \)" is equivalent to asking "Is \( x<0 \)?".
Analyze Statement (1): \( 5^x<1 \).
As shown in the simplification above, this is equivalent to \( 5^x<5^0 \), which means \( x<0 \).
This directly answers our rephrased question with a "Yes". Statement (1) is sufficient.
Analyze Statement (2): \( x<0 \).
This is the rephrased question itself. It gives a definitive "Yes". Statement (2) is sufficient.
Step 4: Final Answer:
Each statement alone is sufficient to answer the question.