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Difficulty: MediumRatios, Rates, and Percentages

A digital streaming platform processes high-definition video files using two server clusters, Cluster A and Cluster B.

- Cluster A processes video data at a constant rate of 100 GB/hour100\text{ GB/hour}, and 18%18\% of its processed data requires secondary re-encoding.
- Cluster B processes video data at a constant rate of 200 GB/hour200\text{ GB/hour}, and 8%8\% of its processed data requires secondary re-encoding.

During a processing cycle, the platform processed a total of 3,000 GB3,000\text{ GB} of video data across both clusters, and exactly 12%12\% of the total processed data required secondary re-encoding.

Which of the following correctly pairs the operating time of Cluster A and the operating time of Cluster B during this cycle?

  1. 12 hours for Cluster A, and 9 hours for Cluster BAnswer
  2. B
    9 hours for Cluster A, and 12 hours for Cluster B
  3. C
    15 hours for Cluster A, and 10 hours for Cluster B
  4. D
    10 hours for Cluster A, and 15 hours for Cluster B
  5. E
    12 hours for Cluster A, and 18 hours for Cluster B

Answer

12 hours for Cluster A, and 9 hours for Cluster B
The solution requires calculating the data volume processed by each cluster using a weighted average equation, and then dividing each volume by its respective processing rate. Cluster A processed 1,200 GB over 12 hours (at 100 GB/hour), and Cluster B processed 1,800 GB over 9 hours (at 200 GB/hour).

Step-by-Step Solution

1
Calculate total re-encoded data volume from the combined percentage.
Total re-encoded data = 0.12×3,000 GB=360 GB0.12 \times 3,000\text{ GB} = 360\text{ GB}.
Knowing the total batch volume and overall percentage gives the exact amount of data re-encoded.
2
Formulate equations for total data and re-encoded data in terms of volume processed by Cluster A (DAD_A) and Cluster B (DBD_B).
DA+DB=3,000D_A + D_B = 3,000 and 0.18DA+0.08DB=3600.18 D_A + 0.08 D_B = 360.
Expressing both given constraints as a system of simultaneous linear equations.
3
Solve for DAD_A and DBD_B.
Substitute DB=3,000−DAD_B = 3,000 - D_A into the second equation: 0.18DA+0.08(3,000−DA)=360  ⟹  0.10DA=120  ⟹  DA=1,200 GB0.18 D_A + 0.08(3,000 - D_A) = 360 \implies 0.10 D_A = 120 \implies D_A = 1,200\text{ GB}, so DB=1,800 GBD_B = 1,800\text{ GB}.
Isolating DAD_A determines the exact volume processed by each cluster.
4
Convert data volumes to operating hours using cluster processing rates.
Cluster A operating time = 1,200 GB/100 GB/hour=12 hours1,200\text{ GB} / 100\text{ GB/hour} = 12\text{ hours}; Cluster B operating time = 1,800 GB/200 GB/hour=9 hours1,800\text{ GB} / 200\text{ GB/hour} = 9\text{ hours}.
Dividing data volume by processing rate yields the elapsed operating time for each cluster.

Key Concept

Weighted Average Mixtures and Work Rates
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Topics

Ratios, Rates, and Percentages

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