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28 Apr 2025, 02:41
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A factory has main machines, all of which operate at the same constant rate, and backup machines, all of which operate at the same constant rate. 30 main machines working together can manufacture a batch of parts in 2 hours. Initially, 20 main machines began working, and after 1.5 hours, 20 backup machines joined them. After the backup machines join, will it take less than 0.75 additional hours to finish manufacturing the batch?
(1) The rate of a backup machine is lower than the rate of a main machine.
(2) 60 backup machines working together can manufacture the same batch of parts in 2 hours.
(1) The rate of a backup machine is lower than the rate of a main machine.
(2) 60 backup machines working together can manufacture the same batch of parts in 2 hours.
28 Apr 2025, 02:41
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Official Solution:
In a factory, 30 identical machines working together can manufacture a batch of parts in 2 hours. Initially, 20 machines began working, and after 1.5 hours, 20 identical backup machines joined them. Will it take less than 0.75 hours for all the machines together to finish manufacturing the batch?
Assuming each main machine completes 1 unit of work in 1 hour, the total work would be 30 * 2 = 60 units. In 1.5 hours, 20 main machines would complete 20 * 1.5 = 30 units of work. In the next 0.75 hours, they would complete 20 * 0.75 = 15 units of work, leaving 60 - 30 - 15 = 15 units. So essentially, the question boils down to whether 20 backup machines can complete 15 units of work in 0.75 hours.
(1) The rate of a backup machine is lower than the rate of a main machine.
We know that 20 main machines can complete 20 * 0.75 = 15 units of work in 0.75 hours. Since the backup machines are slower, 20 backup machines would complete less than 15 units of work in 0.75 hours. This is exactly what we needed to check. Sufficient.
(2) 60 backup machines working together can manufacture the same batch of parts in 2 hours.
This was the only missing piece from the stem. Now, we know everything, and there is no need to calculate anything for this statement. From the stem, we know the rate and number of main machines involved, and the size of the work to be done. From this statement, we know the rate of the backup machines. We also know all the relevant time values given in the question. Therefore, we can surely answer any question about this setup. Sufficient.
Answer: D
In a factory, 30 identical machines working together can manufacture a batch of parts in 2 hours. Initially, 20 machines began working, and after 1.5 hours, 20 identical backup machines joined them. Will it take less than 0.75 hours for all the machines together to finish manufacturing the batch?
Assuming each main machine completes 1 unit of work in 1 hour, the total work would be 30 * 2 = 60 units. In 1.5 hours, 20 main machines would complete 20 * 1.5 = 30 units of work. In the next 0.75 hours, they would complete 20 * 0.75 = 15 units of work, leaving 60 - 30 - 15 = 15 units. So essentially, the question boils down to whether 20 backup machines can complete 15 units of work in 0.75 hours.
(1) The rate of a backup machine is lower than the rate of a main machine.
We know that 20 main machines can complete 20 * 0.75 = 15 units of work in 0.75 hours. Since the backup machines are slower, 20 backup machines would complete less than 15 units of work in 0.75 hours. This is exactly what we needed to check. Sufficient.
(2) 60 backup machines working together can manufacture the same batch of parts in 2 hours.
This was the only missing piece from the stem. Now, we know everything, and there is no need to calculate anything for this statement. From the stem, we know the rate and number of main machines involved, and the size of the work to be done. From this statement, we know the rate of the backup machines. We also know all the relevant time values given in the question. Therefore, we can surely answer any question about this setup. Sufficient.
Answer: D
