Question

Determine the value of t.
1. \(2t + 6s = 8\)
2. \(t/2 - 2 = -3s/4\)

• 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 to answer the question ask
• 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 specific to the problem are needed.

Hint:

In Data Sufficiency, when you see two statements that are both linear equations with the same two variables, quickly check if they are independent. If one equation is just a multiple of the other, they are dependent and won't give a unique solution. Here, the ratios of coefficients (2/2 vs 6/3) are different, so the lines are independent and will intersect at a single point, providing a unique solution.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find a unique value for the variable \( t \), we generally need a system of equations where the number of independent equations is equal to the number of variables. Here, we have two variables, \( t \) and \( s \).
Step 2: Detailed Explanation:
Analyzing Statement (1): \(2t + 6s = 8\)
This is a single linear equation with two unknown variables, \( t \) and \( s \). We can simplify it by dividing by 2:
\[ t + 3s = 4 \] We cannot determine a unique value for \( t \) without knowing the value of \( s \). For example, if \( s=0 \), \( t=4 \). If \( s=1 \), \( t=1 \). Since there are infinite possible values for \( t \), this statement is not sufficient.
Analyzing Statement (2): \(t/2 - 2 = -3s/4\)
This is also a single linear equation with two variables. Let's simplify it to make it easier to analyze.
First, move the constant to the right side:
\[ \frac{t}{2} = 2 - \frac{3s}{4} \] Multiply the entire equation by 4 to eliminate the denominators:
\[ 4 \left( \frac{t}{2} \right) = 4(2) - 4 \left( \frac{3s}{4} \right) \] \[ 2t = 8 - 3s \] \[ 2t + 3s = 8 \] Again, this is one equation with two unknowns. We cannot find a unique value for \( t \). This statement is not sufficient.
Analyzing Statements (1) and (2) Together:
Now we have a system of two independent linear equations:
1. \( 2t + 6s = 8 \)
2. \( 2t + 3s = 8 \)
We can solve this system. Let's use the elimination method. Subtract equation (2) from equation (1):
\[ (2t + 6s) - (2t + 3s) = 8 - 8 \] \[ 2t + 6s - 2t - 3s = 0 \] \[ 3s = 0 \] \[ s = 0 \] Now substitute \( s = 0 \) back into either equation to find \( t \). Using the simplified version of equation (1), \( t + 3s = 4 \):
\[ t + 3(0) = 4 \] \[ t = 4 \] Since we have found a unique value for \( t \), the two statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient to find the value of \( t \).