M25-16

Question

Jack can complete a task in 8 hours, while Tom can do so in 12 hours. If Jack starts and works on the task by himself for 4 hours before Tom joins in, how many hours in total will Jack have worked by the time the task is finished?

A. 5.0
B. 5.2
C. 5.6
D. 6.4
E. 7.2

Show Answer

Official Answer

D

A. 5.0
B. 5.2
C. 5.6
D. 6.4
E. 7.2

Bunuel · Accepted Answer

Official Solution:

Jack can complete a task in 8 hours, while Tom can do so in 12 hours. If Jack starts and works on the task by himself for 4 hours before Tom joins in, how many hours in total will Jack have worked by the time the task is finished?

A. 5.0
B. 5.2
C. 5.6
D. 6.4
E. 7.2

Show Answer

Official Answer

D

A. 5.0
B. 5.2
C. 5.6
D. 6.4
E. 7.2

Since Jack can complete the task in 8 hours, in the 4 hours he works alone he completes half of the task. This leaves another half yet to be finished. When Jack and Tom work together, their combined rate is  \frac{1}{8} + \frac{1}{12} = \frac{5}{24}  of the task per hour. This means they can complete the entire task together in  \frac{24}{5}  hours. Hence, for half of the task, they will need  \frac{(\frac{24}{5})}{2}=\frac{24}{10}=2.4  hours.
 
Therefore, the total duration Jack has worked is  4 + 2.4 = 6.4  hours.

Answer: D

Megha1119 · Answer

Combined rate of Jack and Tom working together is 1/8+1/12=5/24 task/hour, so working together they can complete the task in 1/(5/24)=24/5 hours.

Why divide 1 by 5/24? Please elaborate.

Bunuel · Answer

Because time is a reciprocal of rate.

There are several important things you should know to solve work problems:

1. Time, rate and job in work problems are in the same relationship as time, speed (rate) and distance in rate problems.

time*speed=distance  <-->  time*rate=job \ done . For example when we are told that a man can do a certain job in 3 hours we can write:  3*rate=1  -->  rate=\frac{1}{3}  job/hour. Or when we are told that 2 printers need 5 hours to complete a certain job then  5*(2*rate)=1  --> so rate of 1 printer is  rate=\frac{1}{10}  job/hour. Another example: if we are told that 2 printers need 3 hours to print 12 pages then  3*(2*rate)=12  --> so rate of 1 printer is  rate=2  pages per hour;

So, time to complete one job = reciprocal of rate. For example if 6 hours (time) are needed to complete one job --> 1/6 of the job will be done in 1 hour (rate).

2. We can sum the rates.

If we are told that A can complete one job in 2 hours and B can complete the same job in 3 hours, then A's rate is  rate_a=\frac{job}{time}=\frac{1}{2}  job/hour and B's rate is  rate_b=\frac{job}{time}=\frac{1}{3}  job/hour. Combined rate of A and B working simultaneously would be  rate_{a+b}=rate_a+rate_b=\frac{1}{2}+\frac{1}{3}=\frac{5}{6}  job/hour, which means that they will complete  \frac{5}{6}  job in one hour working together.

3. For multiple entities:  \frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+...+\frac{1}{t_n}=\frac{1}{T} , where  T  is time needed for these entities to complete a given job working simultaneously.

For example if:
Time needed for A to complete the job is A hours;
Time needed for B to complete the job is B hours;
Time needed for C to complete the job is C hours;
...
Time needed for N to complete the job is N hours;

Then:  \frac{1}{A}+\frac{1}{B}+\frac{1}{C}+...+\frac{1}{N}=\frac{1}{T} , where T is the time needed for A, B, C, ..., and N to complete the job working simultaneously.

For two and three entities (workers, pumps, ...):

General formula for calculating the time needed for two workers A and B working simultaneously to complete one job:

Given that  t_1  and  t_2  are the respective individual times needed for  A  and  B  workers (pumps, ...) to complete the job, then time needed for  A  and  B  working simultaneously to complete the job equals to  T_{(A&B)}=\frac{t_1*t_2}{t_1+t_2}  hours, which is reciprocal of the sum of their respective rates ( \frac{1}{t_1}+\frac{1}{t_2}=\frac{1}{T} ).

General formula for calculating the time needed for three A, B and C workers working simultaneously to complete one job:

T_{(A&B&C)}=\frac{t_1*t_2*t_3}{t_1*t_2+t_1*t_3+t_2*t_3}  hours.

17. Work/Rate Problems 
     Theory   
 Work Rate Problems - All in One Topic! 
 Work Word Problems Made Easy 
 How to Solve Relative Rate of Work Questions on the GMAT 
     Questions   
 DS Questions 
 PS Questions

On other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

M25-16

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards