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

A bakery sells two types of artisanal bread: Sourdough and Brioche. Last week, the bakery sold a total of 1,2001,200 loaves of bread. Did Sourdough account for more than 60%60\% of the total revenue generated from sales of these two types of bread?

(1) The selling price of a loaf of Brioche was 25%25\% greater than the selling price of a loaf of Sourdough.
(2) The number of Sourdough loaves sold was 70%70\% of the total number of loaves sold.

  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.
The correct answer identifies that neither statement alone gives both the price ratio and quantity ratio, but taking both statements together provides PSPB=45\frac{P_S}{P_B} = \frac{4}{5} and NSNB=73\frac{N_S}{N_B} = \frac{7}{3}. Multiplying these yields NSPSNBPB=2815≈1.867\frac{N_S P_S}{N_B P_B} = \frac{28}{15} \approx 1.867, which is greater than 1.51.5 (the threshold required for a 60%60\% revenue share). Thus, both statements together provide a definitive 'Yes' answer.

Step-by-Step Solution

1
Rephrase the question stem target into an algebraic condition.
Let NSN_S and NBN_B be the quantities of Sourdough and Brioche sold, and PSP_S and PBP_B be their respective unit prices. Sourdough revenue is RS=NSPSR_S = N_S P_S and total revenue is Rtotal=NSPS+NBPBR_{total} = N_S P_S + N_B P_B. We need to test if NSPSNSPS+NBPB>0.60\frac{N_S P_S}{N_S P_S + N_B P_B} > 0.60, which simplifies to NSPSNBPB>1.5\frac{N_S P_S}{N_B P_B} > 1.5.
Simplifying the target inequality shows that knowing the combined ratio NSPSNBPB=(NSNB)×(PSPB)\frac{N_S P_S}{N_B P_B} = \left(\frac{N_S}{N_B}\right) \times \left(\frac{P_S}{P_B}\right) is necessary and sufficient.
2
Evaluate Statement (1) independently.
Statement (1) states PB=1.25PS=54PSP_B = 1.25 P_S = \frac{5}{4} P_S, so PSPB=45\frac{P_S}{P_B} = \frac{4}{5}.
Since no information is given about the ratio of quantities NSNB\frac{N_S}{N_B}, the overall revenue ratio cannot be determined. Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently.
Statement (2) states NS=0.70×1,200=840N_S = 0.70 \times 1,200 = 840, which means NB=360N_B = 360. Thus NSNB=840360=73\frac{N_S}{N_B} = \frac{840}{360} = \frac{7}{3}.
Since no information is given about the relative unit prices PSPB\frac{P_S}{P_B}, the overall revenue ratio cannot be determined. Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together.
Combine the price ratio PSPB=45\frac{P_S}{P_B} = \frac{4}{5} and quantity ratio NSNB=73\frac{N_S}{N_B} = \frac{7}{3}. The revenue ratio is NSPSNBPB=(73)×(45)=2815≈1.867\frac{N_S P_S}{N_B P_B} = \left(\frac{7}{3}\right) \times \left(\frac{4}{5}\right) = \frac{28}{15} \approx 1.867.
Since 1.867>1.51.867 > 1.5, Sourdough's revenue share is strictly greater than 60%60\%. This yields a definitive 'Yes', so both statements together are sufficient.

Key Concept

Data Sufficiency evaluation of revenue proportions using combined quantity and price ratios.
Estimated Time:1m 30s
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Topics

Rates, Ratios, Percentages, and Applied Word Problems

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