Question

What is the value of x+2y? 
1. $3^x . 9^y = 27^{12}$ 
2. x=2y 
 

• A. EACH statement ALONE is sufficient to answer the question asked
• B. Statement (1) ALONE is sufficient, but statement (2) 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. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• E. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

When you see an equation with different bases that are powers of the same number (like 3, 9, 27 or 2, 4, 8, 16), the first step should always be to convert everything to the smallest base. This often reveals a simple linear equation between the exponents.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a "What is the value?" data sufficiency question. We need to determine if the given statements provide enough information to find a single, unique numerical value for the expression \(x+2y\).

Step 2: Key Formula or Approach:
The key to solving equations with exponents is to express all terms with the same base. Here, the numbers 3, 9, and 27 can all be expressed as powers of 3. - \(9 = 3^2\) - \(27 = 3^3\) The rule of exponents to be used is: \((a^m)^n = a^{mn}\) and \(a^m \cdot a^n = a^{m+n}\).

Step 3: Detailed Explanation:
Analyzing Statement (1): $3^x . 9^y = 27^{12}$
Let's convert all terms to the base 3: \[ 3^x \cdot (3^2)^y = (3^3)^{12} \] Apply the power of a power rule: \[ 3^x \cdot 3^{2y} = 3^{3 \times 12} \] \[ 3^x \cdot 3^{2y} = 3^{36} \] Apply the product of powers rule: \[ 3^{x+2y} = 3^{36} \] Since the bases are equal, the exponents must also be equal. \[ x+2y = 36 \] This statement gives a unique value for the expression \(x+2y\).
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): x = 2y
This statement provides a relationship between \(x\) and \(y\).
Let's substitute \(x=2y\) into the expression we need to find: \[ x+2y = (2y) + 2y = 4y \] The value of the expression is \(4y\). Since we do not know the value of \(y\), we cannot find a unique numerical value for \(x+2y\).
- If \(y=1\), then \(x=2\), and \(x+2y=4\). - If \(y=2\), then \(x=4\), and \(x+2y=8\). The value is not unique.
Therefore, Statement (2) ALONE is not sufficient.

Step 4: Final Answer:
Statement (1) alone is sufficient to find the value of \(x+2y\), but Statement (2) alone is not. The correct option is (B).