Sophie, working at a constant rate, can complete a job working alone

Question

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

M47-01

Bunuel · Accepted Answer

GMAT Club Official Solution:

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.

GMATFE2025 · Answer

Rate of work of Sophie = 1/34
Rate of work for Julien = 1/17 (2*1/34)

Break time overlap, LCM of (3,4) = 12 hour. That means at 12th hour both of them will be at break jointly. Let's evaluate the work done till then;
Out of 12 hours, Sophie works 8 hours, (rest for 4) = 8*(1/34) = 4/17
Out of 12 hours, Juien works for 9 hours (rest for 3) = 9*(1/17) = 9/17

Balance work at 12th hour = 1- (4/17 + 9/17) = 4/17

For 13th and 14th hour both will work = 2*(1/34+1/17) = 3/17
BALANCE = 4/17-3/17 = 1/17

15th hour, Sophie will rest and Julien works for an hour as 1/17. Hence, work accomplished in 15 hour.

Amity007 · Answer

I solved it via options
1. S works for 2 hours and take break for 1, so cycle is of 3 hours. Work done is 2*1/34 =2/34

2. J works for 3 hours and take break for 1, so cycle is for 4 hours, work done is 6/34

Now, considering option A, 12 hours.
S will complete (12-12/3)*1/34 =8/34
And J will complete (12-12/4)*2/34 =9*2/34=18/34
Total work done=24/34, remaining will be 10/34 ——> after 12 hours

So will go to 15 hours option, 
In this case work done by S is, (15-15/3)/34 =10/34
And work done by J is, (15-15/4) in this case will have to consider 3 coz next break time will be 16th hour. So (15-3)*2/34 =24/34
And if we add up we get the work as 34/34. 
So option D.

laborumplaceat · Answer

1. Identify the RatesLet the total work be 34 units.
 Sophie’s Rate: 34 units / 34 hours= 1 unit/hour.
 Julien’s Rate: He is twice as fast, so he finishes in 17 hours.Rate = 34 units / 17 hours = 2 units/hour.

2. Understand the Work-Break CyclesSince they work at different intervals, we need to track their output over time:
 Sophie (2h work, 1h break): In a 3-hour cycle, she works 2 hours.Work done per 3 hours 2 units.
 Julien (3h work, 1h break): In a 4-hour cycle, he works 3 hours.Work done per 4 hours = 3*2 = 6 units

3. We can check the total work done at specific intervals to narrow down the time. 
 Let’s look at 12 hours (a common multiple of their cycles):

Time Elasped -> 12hrs
 Sophie's Work Status -> 4 cycles (2h on, 1h off)
 Sophie's Units = 4*2 = 8 units
 Julien's Work Status -> 3 cycles (3h on, 1h off)
 Julien's Units = 3*6 = 18 units

Total Work = 26 units

4. Step-by-Step Calculation after Hour 12
 Hour 13: 
 Sophie starts a new cycle (Hour 1 of 2): +1 unit
 Julien starts a new cycle (Hour 1 of 3): +2 units
 Total Work: 26 + 3 = 29 units.
 Hour 14:
 Sophie (Hour 2 of 2): +1 unit
 Julien (Hour 2 of 3): +2 units
 Total Work: 29 + 3 = 32 units.
 Hour 15:
 Sophie takes a break: 0 units
 Julien (Hour 3 of 3): +2 units
 Total Work: 32 + 2 = 34 units.

The job is completed exactly at the end of the 15th hour.

Correct Answer: D

Dereno · Answer

Sophie can complete a job in 34 hours.

Julien can complete the job is half the time taken by Sophie = 17 hrs.

The efficiency of Sophie and Julian is 1 units and 2 units respectively.

Let the total work be 34 units.

The cycle of work for:

Sophie is 2 hrs followed by 1 hours of rest.

Julian is 3 hrs of work followed by 1 hours of rest.

So, in one cycle of 3 hrs, Sophie with efficiency of 1 unit Completes 2 units of work.

Similarly, in one cycle of 4 hrs, Julian with efficiency of 2 units completes 6 units of work.

They both meet again at LCM of (3,4)= 12 hrs.

So in 12 hours ,

Sophie work : break = 2:1. That’s 3x =12 and x =4. Thus, Sophie has done 8 hrs of work, and 4 hrs break.

Julian work : Break = 3:1 . That’s 4x =12, x =3. Thus, Julian has done 9 hours and 3 hrs of break.

Work completed = Sophie 8 hrs + Julian 9 hrs

= (8*1)+(9*2) = 8+18 = 26 units.

Remainder work = 8 units.

12 th hour is break.

In the next 2 hours , both Sophie and Julian has completed 2*(1+2) = 6 units of work.

Work completed = 26+6 = 32 units at the end of 14th hour.

15th hour , its break time for Sophie. While Julian puts 2 units of work. Thus, 32+2=34 units.  END OF WORK .

Option D

thuckhang · Answer

S rate: 1/34, J rate: 1/17
S rests for 1 hour every 2 hours: the period is 3 hours => laps: 3 hours
J rests 1 hour every 3 hours: the period is 4 hours => laps: 4 hours 
LCM (3,4) = 12 => minimum hour that they start working together. 
After 12 hours: S has 4 laps, working for 8 hours and J has 3 laps, working for 9 hours. 
The remaining work is: 1 - (8*1/34 + 9*1/17) = 4/17
The next 2 hour (S break): they work: 2*(1/34+1/17) = 3/17 
The remaining 1 hour: J worked for the rest 1/17 
Total hours: 15 hours

srouth040 · Answer

How did you calculate if she works for 8 hours? , like i had to think of it in a clock and check , but if I wanted to find out through a formula , what would that be?

egmat · Answer

When someone has a  work-break pattern , think of it as repeating cycles.

For Sophie: Works 2 hours, then breaks 1 hour
→ One cycle = 2 + 1 =  3 hours 
→ Work hours per cycle =  2 hours

The Formula:

Actual Work Hours = (Complete Cycles × Work Period) + Partial Cycle Work

Where:
• Complete Cycles = Clock Time ÷ Cycle Length (drop the remainder)
• Partial Cycle = Whatever's left over (check if it falls in work time or break time)

Example: How many hours does Sophie work in 12 clock hours?

Step 1: Find complete cycles
 12 ÷ 3 = 4 complete cycles  (no remainder)

Step 2: Calculate work hours
 4 cycles × 2 hours of work per cycle = 8 hours

That's it! Sophie works  8 hours  in 12 clock hours.

Another Example: What about 14 clock hours?

Step 1:  14 ÷ 3 = 4 cycles  with remainder  2

Step 2: Work from complete cycles =  4 × 2 = 8 hours

Step 3: Partial cycle check
• Remainder = 2 hours
• In Sophie's pattern, the  first 2 hours  of any cycle are work time
• So she works all 2 of those remaining hours

Step 4: Total =  8 + 2 = 10 hours

Sophie, working at a constant rate, can complete a job working alone

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Sophie, working at a constant rate, can complete a job working alone

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