Question

What is the value of x?
(1) The average of x, y, and z is 10.
(2) The sum of y and z is 25.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

In Data Sufficiency, look for opportunities to substitute an entire expression (like \(y+z\)) rather than solving for individual variables. This can save time and simplify the problem.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This Data Sufficiency question asks if we can find a single numerical value for \(x\) using the information from the two statements.
Step 2: Detailed Explanation:
Analyze Statement (1):
"The average of x, y, and z is 10." This translates to the equation:
\[ \frac{x + y + z}{3} = 10 \]
\[ x + y + z = 30 \]
This single equation has three variables, so we cannot solve for \(x\) alone. Statement (1) is not sufficient.
Analyze Statement (2):
"The sum of y and z is 25." This translates to the equation:
\[ y + z = 25 \]
This equation provides no information about \(x\). Statement (2) is not sufficient.
Analyze Statements (1) and (2) Together:
We have a system of two equations:
1) \(x + y + z = 30\)
2) \(y + z = 25\)
We can substitute the value of \((y + z)\) from the second equation directly into the first equation:
\[ x + (25) = 30 \]
\[ x = 30 - 25 \]
\[ x = 5 \]
Since we found a unique value for \(x\), the combination of both statements is sufficient.
Step 3: Final Answer:
Neither statement is sufficient by itself, but together they are sufficient. This corresponds to option (C).