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Difficulty: Very hardRates, Ratios, Percentages, and Applied Word Problems

An investment fund allocates its capital between two asset classes, Class PP and Class QQ. At the beginning of a given year, the ratio of capital in Class PP to capital in Class QQ was 3:23 : 2. During the year, Class PP earned an annual return of rp%r_p\% and Class QQ earned an annual return of rq%r_q\%. At the end of the year, a management fee of 2%2\% was deducted from the total ending value of Class PP, and a management fee of 5%5\% was deducted from the total ending value of Class QQ. Was the total dollar amount of the management fee deducted from Class PP greater than the total dollar amount of the management fee deducted from Class QQ?

(1) rp−rq=10r_p - r_q = 10
(2) At the end of the year, before management fees were deducted, the total value of Class PP was 60%60\% greater than the total value of Class QQ.

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

Answer

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (2) alone allows us to express both fee amounts in terms of the single variable VQV_Q, the ending value of Class QQ. Because FP=0.02(1.6VQ)=0.032VQF_P = 0.02(1.6 V_Q) = 0.032 V_Q and FQ=0.05VQF_Q = 0.05 V_Q, we know with certainty that FP<FQF_P < F_Q. This provides a definitive 'No' answer to whether FP>FQF_P > F_Q, which makes Statement (2) alone sufficient. Statement (1) alone leaves the relationship dependent on the unconstrained value of rqr_q, so it is not sufficient.

Step-by-Step Solution

1
Algebraically rephrase the question target.
Let initial capital be P0P_0 and Q0Q_0, where P0=1.5Q0P_0 = 1.5 Q_0. Ending values before fees are VP=1.5Q0(1+rp100)V_P = 1.5 Q_0 (1 + \frac{r_p}{100}) and VQ=Q0(1+rq100)V_Q = Q_0 (1 + \frac{r_q}{100}). The fees are FP=0.02VP=0.03Q0(1+rp100)F_P = 0.02 V_P = 0.03 Q_0 (1 + \frac{r_p}{100}) and FQ=0.05VQ=0.05Q0(1+rq100)F_Q = 0.05 V_Q = 0.05 Q_0 (1 + \frac{r_q}{100}). The question 'Is FP>FQF_P > F_Q?' simplifies to 'Is 0.03(100+rp)>0.05(100+rq)0.03(100 + r_p) > 0.05(100 + r_q)?' or 'Is 3rp−5rq>2003 r_p - 5 r_q > 200?'
Rephrased targets clarify the exact mathematical relationship required to answer the Data Sufficiency prompt.
2
Evaluate Statement (1): rp−rq=10r_p - r_q = 10.
Substituting rp=rq+10r_p = r_q + 10 into the inequality yields 3(rq+10)−5rq=30−2rq>2003(r_q + 10) - 5 r_q = 30 - 2 r_q > 200, which simplifies to rq<−85r_q < -85. Since rqr_q can be any real number (e.g., rq=0r_q = 0 yields 'No', while rq=−100r_q = -100 yields 'Yes'), Statement (1) alone is NOT sufficient.
Without specific values for rqr_q, the inequality cannot be answered definitively.
3
Evaluate Statement (2): VP=1.6VQV_P = 1.6 V_Q.
We are given that VP=1.6VQV_P = 1.6 V_Q. Substituting this directly into the fee expressions gives FP=0.02VP=0.02(1.6VQ)=0.032VQF_P = 0.02 V_P = 0.02 (1.6 V_Q) = 0.032 V_Q. Since FQ=0.05VQF_Q = 0.05 V_Q and portfolio value VQ>0V_Q > 0, 0.032VQ0.032 V_Q is strictly less than 0.05VQ0.05 V_Q. Thus, FP>FQF_P > F_Q is definitively FALSE.
A definitive 'No' answer is sufficient in Data Sufficiency decision logic.

Key Concept

Data Sufficiency Yes/No decision logic and algebraic target rephrasing for percentage adjustments and fee calculations.
Estimated Time:2m 0s
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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