Question

A shopping center increased its revenues by 10% between 2010 and 2011. The shopping center's costs increased by 8% during the same period. What is the firm's percent increase in profits over this period, if profits are defined as revenues minus costs?
(1) The firm's initial profit is $200,000.
(2) The firm's initial revenues are 1.5 times its initial costs.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

For percent change problems involving variables (like revenue and cost), you often need the ratio between the variables, not their absolute values. Statement (2) provides this ratio, making it sufficient.

Answer 1

The Correct Option is B

Step 1: Understanding the Question
Let \(R_1, C_1, P_1\) be the initial revenue, costs, and profit for 2010. Let \(R_2, C_2, P_2\) be the final revenue, costs, and profit for 2011. We are given: \(P_1 = R_1 - C_1\) \(R_2 = 1.10 \times R_1\) \(C_2 = 1.08 \times C_1\) \(P_2 = R_2 - C_2 = 1.10 R_1 - 1.08 C_1\) The question asks for the percent increase in profit, which is \( \frac{P_2 - P_1}{P_1} \times 100%\).
Let's express this in terms of \(R_1\) and \(C_1\): \[ \frac{(1.10 R_1 - 1.08 C_1) - (R_1 - C_1)}{R_1 - C_1} = \frac{0.10 R_1 - 0.08 C_1}{R_1 - C_1} \] To find a numerical answer, we need to know the relationship (ratio) between \(R_1\) and \(C_1\).
Step 2: Analysis of Statement (1)
Statement (1) tells us that \(P_1 = \$200,000\). This means \(R_1 - C_1 = 200,000\). Substituting this into our expression: \[ \frac{0.10 R_1 - 0.08 C_1}{200,000} \] We still have two variables, \(R_1\) and \(C_1\), in the numerator. We can write \(R_1 = C_1 + 200,000\), but the expression will still depend on the value of \(C_1\). We cannot find a unique numerical value for the percent increase.
Therefore, Statement (1) ALONE is not sufficient.
Step 3: Analysis of Statement (2)
Statement (2) tells us that \(R_1 = 1.5 C_1\). This gives the relationship between initial revenue and costs. Let's substitute this into the expression for the percent increase: \[ \frac{0.10 (1.5 C_1) - 0.08 C_1}{1.5 C_1 - C_1} = \frac{0.15 C_1 - 0.08 C_1}{0.5 C_1} \] \[ = \frac{0.07 C_1}{0.5 C_1} = \frac{0.07}{0.5} = 0.14 \] The percent increase is \(0.14 \times 100% = 14%\). This is a specific, unique numerical answer.
Therefore, Statement (2) ALONE is sufficient.
Step 4: Final Answer
Since Statement (2) alone is sufficient and Statement (1) alone is not, the correct answer is (B).