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Sophie, working at a constant rate, can complete a job working alone in 34 hours. Julien, working at a constant rate, can complete the same job twice as fast as Sophie. Sophie takes a 1 hour break after each 2 hours she works, and Julien takes a 1 hour break after each 3 hours he works. If they start working at the same time and follow these break patterns, after how many hours from the start will the job be completed?
A. 12
B. 13
C. 14
D. 15
E. 16
A. 12
B. 13
C. 14
D. 15
E. 16
05 Mar 2026, 09:03
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Official Solution:
Sophie, working at a constant rate, can complete a job working alone in 34 hours. Julien, working at a constant rate, can complete the same job twice as fast as Sophie. Sophie takes a 1 hour break after each 2 hours she works, and Julien takes a 1 hour break after each 3 hours he works. If they start working at the same time and follow these break patterns, after how many hours from the start will the job be completed?
A. 12
B. 13
C. 14
D. 15
E. 16
Make the whole job 34 units of work.
Then Sophie does 1 unit per hour when working, since she finishes 34 units in 34 hours.
Julien is twice as fast, so he does 2 units per hour when working.
Now use the fact that their break schedules line up every 12 hours, since Sophie’s work plus break cycle is 3 hours and Julien’s is 4 hours, and the common reset is 12 hours.
In 12 hours:
Sophie completes 4 cycles, so she works 8 hours total, producing 8 units.
Julien completes 3 cycles, so he works 9 hours total, producing 18 units.
Together they produce 26 units in 12 hours.
So 8 units remain.
At the start of the next 12 hour block, the first 3 hours always look like this:
Hour 1 and hour 2: both work, so they produce 3 units per hour, total 6 units.
Hour 3: Sophie is on break, Julien works, so they produce 2 units.
That finishes the remaining 8 units in 3 more hours.
Total time is 12 + 3 = 15 hours.
Answer: D
Sophie, working at a constant rate, can complete a job working alone in 34 hours. Julien, working at a constant rate, can complete the same job twice as fast as Sophie. Sophie takes a 1 hour break after each 2 hours she works, and Julien takes a 1 hour break after each 3 hours he works. If they start working at the same time and follow these break patterns, after how many hours from the start will the job be completed?
A. 12
B. 13
C. 14
D. 15
E. 16
Make the whole job 34 units of work.
Then Sophie does 1 unit per hour when working, since she finishes 34 units in 34 hours.
Julien is twice as fast, so he does 2 units per hour when working.
Now use the fact that their break schedules line up every 12 hours, since Sophie’s work plus break cycle is 3 hours and Julien’s is 4 hours, and the common reset is 12 hours.
In 12 hours:
Sophie completes 4 cycles, so she works 8 hours total, producing 8 units.
Julien completes 3 cycles, so he works 9 hours total, producing 18 units.
Together they produce 26 units in 12 hours.
So 8 units remain.
At the start of the next 12 hour block, the first 3 hours always look like this:
Hour 1 and hour 2: both work, so they produce 3 units per hour, total 6 units.
Hour 3: Sophie is on break, Julien works, so they produce 2 units.
That finishes the remaining 8 units in 3 more hours.
Total time is 12 + 3 = 15 hours.
Answer: D
