Question

What is the value of \(j+k\)?
1. \(mj+mk = 2m\)
2. \(5j + 5k = 10\)

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

Hint:

Be extremely careful when dividing by a variable in algebra. You can only do so if you are certain the variable is not zero. In data sufficiency, if a variable's value is unknown, you must consider the case where it might be zero. This is a common trap.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The question asks for the value of the expression \(j+k\). This is a value-based data sufficiency question. A statement is sufficient if it allows us to determine a single, unique numerical value for \(j+k\).

Step 2: Key Formula or Approach:
The approach is to use algebraic manipulation, specifically factoring, to simplify the equations in each statement and see if we can isolate the term \(j+k\).

Step 3: Detailed Explanation:
Analyzing Statement (1): \(mj+mk = 2m\).
We can factor out 'm' from the left side of the equation:
\[ m(j+k) = 2m \] Our goal is to solve for \(j+k\). We might be tempted to divide both sides by 'm'.
If we assume \(m \neq 0\), we can divide by m:
\[ j+k = 2 \] However, the statement does not specify that \(m \neq 0\). If \(m = 0\), the equation becomes:
\[ 0(j+k) = 2(0) \] \[ 0 = 0 \] This is true for any values of j and k. In this case, \(j+k\) could be any number. For example, j=1, k=4 gives j+k=5. Or j=2, k=0 gives j+k=2.
Since we cannot determine a unique value for \(j+k\) (it could be 2, or it could be any other number if m=0), statement (1) alone is not sufficient.
Analyzing Statement (2): \(5j + 5k = 10\).
We can factor out 5 from the left side of the equation:
\[ 5(j+k) = 10 \] Since 5 is a non-zero constant, we can safely divide both sides by 5:
\[ j+k = \frac{10}{5} \] \[ j+k = 2 \] This gives a single, unique value for \(j+k\).
Therefore, statement (2) alone is sufficient.

Step 4: Final Answer:
Statement (2) alone is sufficient, but statement (1) alone is not sufficient.