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# M25-16

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Math Expert
Joined: 02 Sep 2009
Posts: 55265

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16 Sep 2014, 01:23
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5% (low)

Question Stats:

83% (01:16) correct 17% (01:46) wrong based on 378 sessions

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Jack can complete the task in 8 hours. Tom can complete the same task in 12 hours. If Jack works on the task alone for 4 hours and then Tom starts to help him, 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

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Math Expert
Joined: 02 Sep 2009
Posts: 55265

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16 Sep 2014, 01:23
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Official Solution:

Jack can complete the task in 8 hours. Tom can complete the same task in 12 hours. If Jack works on the task alone for 4 hours and then Tom starts to help him, 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

Since Jack can complete the task in 8 hours then in 4 hours that he works alone he does half of the task, so another half is has to be done. Combined rate of Jack and Tom working together is $$\frac{1}{8}+\frac{1}{12}=\frac{5}{24}$$ task/hour, so working together they can complete the task in $$\frac{1}{(\frac{5}{24})}=\frac{24}{5}$$ hours, which means that for half of the task they need another $$\frac{(\frac{24}{5})}{2}=\frac{24}{10}=2.4$$ hours.

Hence, the total time that Jack worked is $$4+2.4=6.4$$ hours.

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Joined: 11 Jun 2013
Posts: 1
Schools: IMD '16

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15 Oct 2014, 07:21
Easier way to look at such problems is to think in terms of capabilities.

jack takes 8 hrs and Tom takes 12 hrs. So Tom has 2/3 the capacity of jack to complete the same work. In First 4 hrs tom completes half the work, remaining half work is done by Tom + Some one who is 2/3 as good as Tom , or in other words Tom +2/3 Tom = 5/3 Toms are working. So it should take (4)/(5/3) hrs to complete the remaining half.

This yields , 12/5 or 2.4 hrs .

So in total it will be 4+2.4 =6.4 hrs.
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Joined: 29 Mar 2013
Posts: 26
Schools: ISB '16

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01 Dec 2014, 20:42
for me it is:

W=Rate *Time
for 1st R=x/8
2nd R=x/4

First 4 only 1 worked.. i.e. W= x/8 *2 = x/2
Next 4 hours W remaining x/2 = x/8 + x/12 => 5x/12
Time = 1/(5/12) = 2.4
Total time 4+2.4 =6.4
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Joined: 26 Nov 2012
Posts: 590

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12 Sep 2017, 11:38
Jack can complete the task in 8 hours. Tom can complete the same task in 12 hours. If Jack works on the task alone for 4 hours and then Tom starts to help him, 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

====================================================================================================

Jack's rate in one hour = 1/8

Tom's rate in one hour = 1/12

It is mentioned that Jack worked for 4 hours and Tom joined then work got completed. Now Tom worked for x hours and the same number of hours even Jack worked to finish the work (as Tom joined both worked for x hours )

Then we get the below as both worked to finish the same job.

$$\frac{x+4}{8}$$+ $$\frac{x}{12}$$ = 1

=> 5x +12 = 24
=> x = 2.4

Now total number of hours Jack worked is 4 + 2.4 = 6.4 hours..
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Joined: 23 May 2018
Posts: 487
Location: Pakistan

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19 Jun 2018, 23:15
Rather than solving it mathematically, I used logic. The figure had to be more than 6. That left only two answers. E could not be the correct answer as that would imply a severe decrease in the speed of Tom. Thus, D is the correct answer.
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Joined: 29 Jun 2017
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07 Sep 2018, 05:28
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.
Math Expert
Joined: 02 Sep 2009
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07 Sep 2018, 06:19
Megha1119 wrote:
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.

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

On other subjects:
ALL YOU NEED FOR QUANT ! ! !
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Re: M25-16   [#permalink] 07 Sep 2018, 06:19
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# M25-16

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