Question

If Alyssa is twice as old as Brandon, by how many years is Brandon older than Clara?
(1) Four years ago, Alyssa was twice as old as Clara is now.
(2) Alyssa is 8 years older than Clara.

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

Hint:

In data sufficiency, you don't always need to find the values of all the variables. Focus only on the target value or expression the question asks for. In this case, even though we couldn't find the individual ages, we could find the difference between them.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is an age-related word problem presented in a data sufficiency format. We need to find the value of the expression \(B - C\), where B is Brandon's current age and C is Clara's current age. Let A be Alyssa's current age.

Step 2: Key Formula or Approach:
We will translate the information from the question stem and each statement into algebraic equations. Then we will check if we can solve for a unique value of \(B-C\).
From the question stem, we get our first equation:
Equation (i): \(A = 2B\)

Step 3: Detailed Explanation:
Analyzing Statement (1): Four years ago, Alyssa was twice as old as Clara is now.
Four years ago, Alyssa's age was \(A-4\). Clara's current age is C. This statement gives us:
Equation (ii): \(A - 4 = 2C\)
Now we have a system of two equations with three variables: 1. \(A = 2B\) 2. \(A - 4 = 2C\) We want to find \(B-C\). Let's express B and C in terms of A and then subtract. From (1), we can write \(B = \frac{A}{2}\).
From (2), we can write \(C = \frac{A-4}{2}\).
Now, let's compute \(B-C\): \[ B - C = \frac{A}{2} - \frac{A-4}{2} \] \[ B - C = \frac{A - (A-4)}{2} \] \[ B - C = \frac{A - A + 4}{2} \] \[ B - C = \frac{4}{2} = 2 \] We found a unique numerical value for \(B-C\). Therefore, Brandon is 2 years older than Clara. Statement (1) alone is sufficient.
Analyzing Statement (2): Alyssa is 8 years older than Clara.
This statement gives us the equation:
Equation (iii): \(A = C + 8\)
Now we combine this with the equation from the stem: 1. \(A = 2B\) 2. \(A = C + 8\) We want to find \(B-C\). Again, let's express B and C in terms of A. From (1), we have \(B = \frac{A}{2}\).
From (2), we have \(C = A - 8\).
Now, let's compute \(B-C\): \[ B - C = \frac{A}{2} - (A - 8) \] \[ B - C = \frac{A}{2} - A + 8 \] \[ B - C = 8 - \frac{A}{2} \] The value of \(B-C\) depends on the value of A. Since we don't know A's age, we cannot find a unique value for \(B-C\). For example, if A=20, B=10, C=12, and \(B-C = -2\). If A=30, B=15, C=22, and \(B-C = -7\). Statement (2) alone is not sufficient.

Step 4: Final Answer:
Statement (1) is sufficient to find the value of \(B-C\), but statement (2) is not.