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Difficulty: MediumRatios, Rates, and Proportions

An industrial bakery uses two commercial mixers, Mixer X and Mixer Y, to prepare large batches of dough. Working continuously at its own constant rate, Mixer X can complete 1 full batch of dough in 4 hours. Mixer Y, working continuously at its own constant rate, can complete 1 full batch of dough in 6 hours.

If Mixer X works alone for 1 hour and is then joined by Mixer Y to complete the rest of the batch together, which of the following statements must be true? Select all such statements.

  1. The remaining 34\frac{3}{4} of the batch takes 1 hour and 48 minutes for both mixers working together to complete.Answer
  2. Mixer X completes 70%70\% of the total batch of dough.Answer
  3. The ratio of the total amount of dough prepared by Mixer X to that prepared by Mixer Y is 7:37 : 3.Answer
  4. D
    Both mixers working together complete the remaining 34\frac{3}{4} of the batch in 3 hours and 45 minutes.
  5. E
    Mixer Y completes 40%40\% of the total batch of dough.

Answer

The correct statements are those indicating that the remaining work takes 1 hour and 48 minutes, Mixer X completes 70% of the batch, and the ratio of work done by Mixer X to Mixer Y is 7 : 3.
The rate of Mixer X is 1/4 batch per hour and Mixer Y is 1/6 batch per hour. In the first hour, Mixer X completes 1/4 of the batch, leaving 3/4 of the batch. Working together, their combined rate is 1/4 + 1/6 = 5/12 batch per hour. The time needed for the remaining 3/4 batch is (3/4) / (5/12) = 1.8 hours, which equals 1 hour and 48 minutes. Mixer X works for a total of 2.8 hours, completing 2.8 / 4 = 0.7 (or 70%) of the batch. Mixer Y works for 1.8 hours, completing 1.8 / 6 = 0.3 (or 30%) of the batch. The ratio of dough prepared by Mixer X to Mixer Y is 0.7 : 0.3 = 7 : 3. Therefore, the statements asserting 1 hour and 48 minutes remaining time, 70% completed by Mixer X, and a 7 : 3 work ratio are all correct.

Step-by-Step Solution

1
Determine individual work rates and initial work done
Mixer X rate = 14\frac{1}{4} batch/hr, Mixer Y rate = 16\frac{1}{6} batch/hr. In 1 hour, Mixer X completes 14\frac{1}{4} of the batch, leaving 1−14=341 - \frac{1}{4} = \frac{3}{4} of the batch remaining.
Establishing the individual rates allows calculation of remaining work after Mixer X operates alone.
2
Calculate combined rate and time to finish remaining work
Combined rate = 14+16=3+212=512\frac{1}{4} + \frac{1}{6} = \frac{3+2}{12} = \frac{5}{12} batch/hr. Time required = 3/45/12=34×125=95=1.8\frac{3/4}{5/12} = \frac{3}{4} \times \frac{12}{5} = \frac{9}{5} = 1.8 hours = 1 hour 48 minutes.
When both mixers operate together, their rates add.
3
Calculate total work contributions of each mixer
Mixer X total time = 1+1.8=2.81 + 1.8 = 2.8 hours →\rightarrow Work by X = 2.8×14=0.72.8 \times \frac{1}{4} = 0.7 (70%70\%). Mixer Y total time = 1.81.8 hours →\rightarrow Work by Y = 1.8×16=0.31.8 \times \frac{1}{6} = 0.3 (30%30\%). Work ratio X : Y = 0.7:0.3=7:30.7 : 0.3 = 7 : 3.
Multiplying each mixer's active operating time by its rate determines its share of the total batch.

Key Concept

Combined Work Rates and Ratio of Work Completed
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Topics

Ratios, Rates, and Proportions

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