Question

What is the value of x?
Statement I: x+y=20
Statement II: y=8

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Both statements together are sufficient
• D. Neither statement is sufficient

Hint:

In Data Sufficiency, do not spend extra time fully solving the equations once you establish that a unique solution is mathematically guaranteed.
As soon as you see two independent linear equations for two variables, you can confidently conclude they are sufficient together.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is of the Data Sufficiency type.
The main objective is to determine if the given statements (I and II) provide sufficient information to find a unique value for the variable $x$.
We must analyze each statement individually first, and only combine them if neither is sufficient on its own.

Step 2: Key Formula or Approach:
To find a unique solution for variables in linear algebraic equations, the number of independent equations must be equal to the number of unknown variables.
We have two variables, $x$ and $y$. Therefore, we generally require two independent equations to solve for them uniquely.

Step 3: Detailed Explanation:
$\bullet$ Let us analyze Statement I alone:
$\bullet$ Statement I gives us the equation: $x + y = 20$.
$\bullet$ This is a single equation with two unknown variables, $x$ and $y$.
$\bullet$ It has infinitely many solutions (e.g., if $y=1$, then $x=19$; if $y=10$, then $x=10$, etc.).
$\bullet$ Therefore, Statement I alone is not sufficient to determine a unique value for $x$.
$\bullet$ Now, let us analyze Statement II alone:
$\bullet$ Statement II gives us: $y = 8$.
$\bullet$ This statement provides the value of $y$ but gives no information about $x$ or how $x$ relates to $y$.
$\bullet$ Therefore, Statement II alone is not sufficient to determine the value of $x$.
$\bullet$ Now, let us combine both Statement I and Statement II:
$\bullet$ From Statement I, we have $x + y = 20$.
$\bullet$ From Statement II, we have $y = 8$.
$\bullet$ Substituting the value of $y$ from Statement II into the equation from Statement I:
\[ x + 8 = 20 \]
\[ x = 20 - 8 = 12 \]
$\bullet$ This gives a single, unique value for $x$, which is 12.
$\bullet$ Hence, both statements together are sufficient to answer the question, but neither statement alone is sufficient.

Step 4: Final Answer:
Both statements together are sufficient to find the value of $x$.
Therefore, the correct option is (C).