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Question

Difficulty: MediumRates, Ratios, Percentages, and Applied Word Problems

A fitness club offers two annual membership tiers: Standard and Premium. Last year, what percentage of the fitness club's total annual membership revenue came from Premium memberships?

(1) Last year, the annual price of a Premium membership was 50%50\% greater than the annual price of a Standard membership.

(2) Last year, the ratio of the number of Standard memberships sold to the number of Premium memberships sold was 33 to 22.

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

Answer

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the question stem shows that finding the percentage of total revenue from Premium memberships depends solely on the product of the quantity ratio NSNP\frac{N_S}{N_P} and the price ratio PSPP\frac{P_S}{P_P}. Statement (1) gives PSPP=23\frac{P_S}{P_P} = \frac{2}{3} and Statement (2) gives NSNP=32\frac{N_S}{N_P} = \frac{3}{2}. Neither statement alone provides both ratios, but combining both statements gives a product of 11, which uniquely determines that Premium memberships generated exactly 50%50\% of total revenue. Therefore, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the target question algebraically.
Let NSN_S and NPN_P be the number of Standard and Premium memberships sold, and PSP_S and PPP_P be their respective prices. Total revenue R=NSPS+NPPPR = N_S P_S + N_P P_P. The fraction of revenue from Premium memberships is NPPPNSPS+NPPP=1(NSNP)(PSPP)+1\frac{N_P P_P}{N_S P_S + N_P P_P} = \frac{1}{\left(\frac{N_S}{N_P}\right)\left(\frac{P_S}{P_P}\right) + 1}. Thus, knowing the product of ratios (NSNP)(PSPP)\left(\frac{N_S}{N_P}\right)\left(\frac{P_S}{P_P}\right) is sufficient.
Simplifying the stem target shows that only the relative overall ratio of Standard revenue to Premium revenue is needed, not individual numerical values.
2
Evaluate Statement (1) independently.
Statement (1) states PP=1.5PS=32PSP_P = 1.5 P_S = \frac{3}{2} P_S, which gives PSPP=23\frac{P_S}{P_P} = \frac{2}{3}. However, NSNP\frac{N_S}{N_P} is unknown.
Without the quantity ratio, the overall percentage of revenue cannot be calculated. Statement (1) alone is NOT sufficient.
3
Evaluate Statement (2) independently.
Statement (2) states NSNP=32\frac{N_S}{N_P} = \frac{3}{2}. However, PSPP\frac{P_S}{P_P} is unknown.
Without the price ratio, the overall percentage of revenue cannot be calculated. Statement (2) alone is NOT sufficient.
4
Evaluate Statements (1) and (2) together.
Using both statements, (NSNP)(PSPP)=(32)(23)=1\left(\frac{N_S}{N_P}\right)\left(\frac{P_S}{P_P}\right) = \left(\frac{3}{2}\right)\left(\frac{2}{3}\right) = 1. Substituting this into the rephrased expression gives 11+1=12=50%\frac{1}{1 + 1} = \frac{1}{2} = 50\%.
A single numerical answer (50%50\%) is uniquely determined. Both statements together are sufficient.

Key Concept

Data Sufficiency Rephrasing for Weighted Revenue Ratios
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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