Question

A beer company spent $100,000 last year on hops, yeast, and malt. How much of the total expenditure was for hops? 
1. The expenditure for yeast was 20% greater than the expenditure for malt. 
2. The total expenditure for yeast and malt was equal to the expenditure for hops. 
 

• A. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• B. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• C. EACH statement ALONE is 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. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

In data sufficiency problems that can be modeled with equations, count your variables and your independent equations. To find a unique value for a variable, you generally need as many independent equations as variables. Statement (2) provides a key relationship that simplifies the original equation perfectly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem asks for a specific value (expenditure on hops). It is a data sufficiency question involving a system of linear equations. Let H, Y, and M represent the expenditures on hops, yeast, and malt, respectively.
From the question stem, we have the equation:
\[ H + Y + M = 100,000 \] We need to find the value of H.

Step 2: Key Formula or Approach:
We will analyze each statement to see if it provides enough information, when combined with the initial equation, to solve for H. We need to have as many independent linear equations as we have variables to find a unique solution.

Step 3: Detailed Explanation:
Analyzing Statement (1): The expenditure for yeast was 20% greater than the expenditure for malt.
This can be written as an equation:
\[ Y = M + 0.20M = 1.2M \] Now, let's substitute this into our main equation:
\[ H + (1.2M) + M = 100,000 \] \[ H + 2.2M = 100,000 \] This is a single equation with two unknown variables, H and M. We cannot solve for a unique value of H. For example, if M = \$10,000, then H = \$78,000. If M = \$20,000, then H = \$56,000.
Therefore, statement (1) alone is not sufficient.
Analyzing Statement (2): The total expenditure for yeast and malt was equal to the expenditure for hops.
This can be written as an equation:
\[ Y + M = H \] Now, let's substitute this into our main equation:
\[ H + (Y + M) = 100,000 \] \[ H + (H) = 100,000 \] \[ 2H = 100,000 \] \[ H = 50,000 \] This gives us a unique value for H.
Therefore, statement (2) alone is sufficient.

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