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11 Mar 2026, 03:45
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Three machines, working at the same constant rate, Alpha, Beta, and Gamma, started working together to complete a task in a certain number of days. However, the machines were turned off for a few days, causing the task to be completed 4 days later than originally planned. The first machine, Alpha, was turned off for 3 more days than Beta, and Beta was turned off for 3 more days than Gamma. How many days was Gamma turned off?
A. 1 day
B. 2 days
C. 3 days
D. 4 days
E. 5 days
A. 1 day
B. 2 days
C. 3 days
D. 4 days
E. 5 days
11 Mar 2026, 03:45
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Official Solution:
Three machines, working at the same constant rate, Alpha, Beta, and Gamma, started working together to complete a task in a certain number of days. However, the machines were turned off for a few days, causing the task to be completed 4 days later than originally planned. The first machine, Alpha, was turned off for 3 more days than Beta, and Beta was turned off for 3 more days than Gamma. How many days was Gamma turned off?
A. 1 day
B. 2 days
C. 3 days
D. 4 days
E. 5 days
Let the originally planned time be \(N\) days.
Let each machine’s rate be \(r\) jobs per day.
Planned total work \(= 3 * r * N\).
Let the total off days be:
Alpha \(= A\).
Beta \(= B\).
Gamma \(= C\).
Actual completion time is \(N + 4\) days, so actual work done is:
\(r * ((N + 4 - A) + (N + 4 - B) + (N + 4 - C))\).
\(= r * (3(N + 4) - (A + B + C))\).
Set actual work equal to planned work:
\(3 * r * N = r * (3(N + 4) - (A + B + C))\).
\(3N = 3(N + 4) - (A + B + C)\).
\(A + B + C = 12\).
Given:
\(A = B + 3\).
\(B = C + 3\).
So \(A = C + 6\).
Then:
\(A + B + C = (C + 6) + (C + 3) + C = 3C + 9 = 12\).
\(C = 1\).
Answer: A
Three machines, working at the same constant rate, Alpha, Beta, and Gamma, started working together to complete a task in a certain number of days. However, the machines were turned off for a few days, causing the task to be completed 4 days later than originally planned. The first machine, Alpha, was turned off for 3 more days than Beta, and Beta was turned off for 3 more days than Gamma. How many days was Gamma turned off?
A. 1 day
B. 2 days
C. 3 days
D. 4 days
E. 5 days
Let the originally planned time be \(N\) days.
Let each machine’s rate be \(r\) jobs per day.
Planned total work \(= 3 * r * N\).
Let the total off days be:
Alpha \(= A\).
Beta \(= B\).
Gamma \(= C\).
Actual completion time is \(N + 4\) days, so actual work done is:
\(r * ((N + 4 - A) + (N + 4 - B) + (N + 4 - C))\).
\(= r * (3(N + 4) - (A + B + C))\).
Set actual work equal to planned work:
\(3 * r * N = r * (3(N + 4) - (A + B + C))\).
\(3N = 3(N + 4) - (A + B + C)\).
\(A + B + C = 12\).
Given:
\(A = B + 3\).
\(B = C + 3\).
So \(A = C + 6\).
Then:
\(A + B + C = (C + 6) + (C + 3) + C = 3C + 9 = 12\).
\(C = 1\).
Answer: A
