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08 Jan 2025, 03:45
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In a factory, Machine A, working alone, can complete a task in 50 hours, Machine B, working alone, can complete the same task in 60 hours, and Machine C, working alone, can complete it in 75 hours. All three machines started working together on the task. However, Machine A stopped working after 10 hours, and Machine B stopped 15 hours before the task was completed. How many total hours were required to finish the task?
A. 25
B. 30
C. 35
D. 40
E. 45
A. 25
B. 30
C. 35
D. 40
E. 45
08 Jan 2025, 03:45
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Official Solution:
In a factory, Machine A, working alone, can complete a task in 50 hours, Machine B, working alone, can complete the same task in 60 hours, and Machine C, working alone, can complete it in 75 hours. All three machines started working together on the task. However, Machine A stopped working after 10 hours, and Machine B stopped 15 hours before the task was completed. How many total hours were required to finish the task?
A. 25
B. 30
C. 35
D. 40
E. 45
From the stem:
All three machines worked together for the first 10 hours.
Machines B and C worked together for the next \(x\) hours.
Only Machine C worked during the final 15 hours.
This gives the equation:
\(10 * (\frac{1}{50} + \frac{1}{60} + \frac{1}{75}) + x * (\frac{1}{60} + \frac{1}{75}) + 15 * (\frac{1}{75}) = 1\)
\(10 * (\frac{1}{20}) + x * (\frac{3}{100}) + (\frac{1}{5}) = 1\)
\(x = 10\)
Thus, the total time required is (first 10 hours) + (next \(x\) = 10 hours) + (last 15 hours) = 35 hours.
Answer: C
In a factory, Machine A, working alone, can complete a task in 50 hours, Machine B, working alone, can complete the same task in 60 hours, and Machine C, working alone, can complete it in 75 hours. All three machines started working together on the task. However, Machine A stopped working after 10 hours, and Machine B stopped 15 hours before the task was completed. How many total hours were required to finish the task?
A. 25
B. 30
C. 35
D. 40
E. 45
From the stem:
All three machines worked together for the first 10 hours.
Machines B and C worked together for the next \(x\) hours.
Only Machine C worked during the final 15 hours.
This gives the equation:
\(10 * (\frac{1}{50} + \frac{1}{60} + \frac{1}{75}) + x * (\frac{1}{60} + \frac{1}{75}) + 15 * (\frac{1}{75}) = 1\)
\(10 * (\frac{1}{20}) + x * (\frac{3}{100}) + (\frac{1}{5}) = 1\)
\(x = 10\)
Thus, the total time required is (first 10 hours) + (next \(x\) = 10 hours) + (last 15 hours) = 35 hours.
Answer: C
20 Aug 2025, 16:13
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I like the solution - it’s helpful. I calculated the time A worked as 10, B worked as 20 and C worked as 35. But I made the mistake of adding all 3 to get 65 hours. When the amount C worked is the total amount of time it took since some of these times overlap. How do I avoid making this error.
