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

A technology firm sells only two products: Model Alpha and Model Beta. Last month, what percentage of the firm's total revenue was generated from sales of Model Alpha?

(1) Last month, the unit price of Model Alpha was 20%20\% higher than the unit price of Model Beta.
(2) Last month, the firm sold 50%50\% more units of Model Alpha than units of Model Beta.

  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 to answer the question, but neither statement alone is sufficient.
The question asks for the percentage of total revenue coming from Model Alpha, which simplifies algebraically to finding the ratio of Model Alpha's revenue (pAqAp_A q_A) to Model Beta's revenue (pBqBp_B q_B). Neither statement alone provides both the price ratio and quantity ratio needed. Statement (1) gives only the price ratio (pA/pB=1.2p_A/p_B = 1.2), and Statement (2) gives only the quantity ratio (qA/qB=1.5q_A/q_B = 1.5). Together, multiplying these ratios yields pAqA/(pBqB)=1.8p_A q_A / (p_B q_B) = 1.8, which uniquely determines Model Alpha's percentage share of total revenue as 9/149/14 or approximately 64.29%64.29\%. Thus, both statements together are sufficient.

Step-by-Step Solution

1
Define the target variable and rephrase the question algebraically.
Let pAp_A and pBp_B be the unit prices of Model Alpha and Model Beta, respectively. Let qAq_A and qBq_B be the quantities sold. The total revenue generated by Model Alpha is RA=pAqAR_A = p_A q_A, and total revenue generated by Model Beta is RB=pBqBR_B = p_B q_B. Total firm revenue is Rtotal=pAqA+pBqBR_{total} = p_A q_A + p_B q_B. We need to find pAqApAqA+pBqB×100%\frac{p_A q_A}{p_A q_A + p_B q_B} \times 100\%, which simplifies to 11+pBqBpAqA×100%\frac{1}{1 + \frac{p_B q_B}{p_A q_A}} \times 100\%. Thus, we only need the value of the ratio pAqApBqB\frac{p_A q_A}{p_B q_B}.
Rephrasing the target from finding individual revenue components to finding the single ratio of Model Alpha's revenue to Model Beta's revenue simplifies statement evaluation.
2
Evaluate Statement (1) independently.
Statement (1) states pA=1.2pBp_A = 1.2 p_B, which gives pApB=1.2\frac{p_A}{p_B} = 1.2. However, we have no information about the ratio of quantities qAqB\frac{q_A}{q_B}.
Without quantity data, the revenue ratio pAqApBqB\frac{p_A q_A}{p_B q_B} cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Statement (2) states qA=1.5qBq_A = 1.5 q_B, which gives qAqB=1.5\frac{q_A}{q_B} = 1.5. However, we have no information about the ratio of unit prices pApB\frac{p_A}{p_B}.
Without price data, the revenue ratio pAqApBqB\frac{p_A q_A}{p_B q_B} cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) combined.
Combining both statements gives pAqApBqB=(pApB)×(qAqB)=1.2×1.5=1.8\frac{p_A q_A}{p_B q_B} = \left(\frac{p_A}{p_B}\right) \times \left(\frac{q_A}{q_B}\right) = 1.2 \times 1.5 = 1.8. Therefore, Model Alpha's revenue share is 1.8pBqB1.8pBqB+1.0pBqB=1.82.8=914≈64.29%\frac{1.8 p_B q_B}{1.8 p_B q_B + 1.0 p_B q_B} = \frac{1.8}{2.8} = \frac{9}{14} \approx 64.29\%.
Since a unique numerical percentage can be calculated, both statements together are sufficient.

Key Concept

Data Sufficiency Evaluation for Revenue Ratios and Percentages
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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