It would take one machine 4 hours to complete a large production order

Question

It would take one machine 4 hours to complete a large production order and another machine 3 hours to complete the same order. How many hours would it take both machines, working simultaneously at their respective constant rates, to complete the order?

(A) \(\frac{7}{12}\)

(B) \(1 \frac{1}{2}\)

(C) \(1 \frac{5}{7}\)

(D) \(3 \frac{1}{2}\)

(E) 7

(A)  \frac{7}{12}

(B)  1 \frac{1}{2}

(C)  1 \frac{5}{7}

(D)  3 \frac{1}{2}

(E) 7

Bunuel · Accepted Answer

The rate of the first machine is 1/4 job per hour;
The rate of the second machine is 1/3 job per hour;

Thus, the combined rate of the machines is  \frac{1}{4}+\frac{1}{3}=\frac{7}{12}  job per hour, which means that it takes  \frac{1}{(\frac{7}{12})}=\frac{12}{7}=1 \frac{5}{7}  hours both machines to do the job.

Answer: C.

JeffTargetTestPrep · Answer

We can classify this problem as a “combined worker” problem. To solve this type of problem we should use the formula:

Work (done by worker 1) + Work (done by worker 2) = Total Work Completed

It takes machine one 4 hours to complete a job, so the rate of machine one is ¼. It takes machine two 3 hours to complete a job, so the rate of machine two is 1/3. Since we know they are both working together to complete the job, we can label this unknown time as “t” for each machine during the time that both machines are working together. Since rate x time = work, we can multiply to get the work completed for each machine.

https://i.imgur.com/NlMVzyz.png

Finally, we can plug our two work values into the combined work formula and determine t. Since the job is completed, the total work completed is 1.

Work (done by worker 1) + Work (done by worker 2) = Total Work Completed

(1/4)t + (1/3)t = 1

Multiplying the entire equation by 12 gives us:

3t + 4t = 12

7t = 12

t = 12/7 = 1 5/7

Answer is C.

iNumbv · Answer

Suppose we change the wording of the problem to "It would take 2 machines 4 hours together to complete a large production" How would you solve for that ? I'm trying to better understand the nature of the problem. Thanks. My logic says the rate is still the same since it still is 4 hours. If so than what does it change?

arpana10july · Answer

how can the answer be C
in options dere is no option of 12/7

blueseas · Answer

there is a typo in choice.

machine A takes 4 hr
machine B takes 3 hr
then together they will take = product /sum = (4*3)/(4+3)

runningguy · Answer

12/7 is equivalent to 1 and 5/7

BrentGMATPrepNow · Answer

Another approach is to assign the ENTIRE job a  certain number of units . 
The least common multiple of  4  and  3  is  12 . 
So, let's say the ENTIRE production order consists of  12  widgets.

It would take one machine  4  hours to complete a large production...   
Rate = output/time 
So, this machine's  rate  =  12 / 4  =  3  widgets per hour

...and another machine 3 hours to complete the same order.   
Rate = units/time 
So, this machine's  rate  =  12 / 3  =  4  widgets per hour

So, their COMBINED rate =  3 + 4  =   7   widgets per hour.

Working simultaneously at their respective constant rates, to complete the order ? 
Time = output/rate 
=  12 /  7   hours

Answer:  C

adkikani · Answer

JeffTargetTestPrep   Bunuel

https://i.imgur.com/NlMVzyz.png

When we are given t as 3 and 4 hours why are we still assigning an unknown variable?

EMPOWERgmatRichC · Answer

Hi boomgoesthegmat,

In "work" questions, there are 2 common ways to get to the correct answer (although there are several different ways to "do the math"):

1) When there are 2 entities (people, machines, etc.) working on a task together, use the Work Formula. 
2) Convert the individual rates of the 2 (or more) entities, combine and be sure to answer the question that's asked.

In this question, here's how you could use the two options mentioned:

We're told... 
Machine A = 4 hours to complete an order 
Machine B = 3 hours to complete the same order

We're asked how long it would take the two machines, WORKING TOGETHER, to complete the order.

1) Using the Work Formula: (A)(B)/(A+B).....

(4)(3)/(4+3) = 12/7 hours to complete the job

2) Using the individual rates:

Machine A: 
4 hours to do the entire job --> 1 hour to do 1/4 of the job 
Machine B: 
3 hours to do the entire job --> 1 hour to do 1/3 of the job

In 1 hour, 1/4 + 1/3 = 7/12 of the job is done

**Note: this calculation tells you the FRACTION of the JOB that is complete in 1 HOUR**

Since there is 1 job to complete.....1/(7/12) = 12/7 hours to complete the job

Either way, the Final Answer is... C

GMAT assassins aren't born, they're made, 
Rich

pushpitkc · Answer

Hi  adkikani

The reason we are assigning t(an unknown variable) for time, because we don't
know how much time will be take to complete the entire work, when working together.

When we are given 3 and 4 hours as time, it means that the individual machines
will complete the entire work(working alone) in 3 and 4 hours respectively.

Alternate approach

Since it takes one machine 4 hours to complete the order and the second machine takes 
3 hours to complete the order, we can assume the work done to be 60 units.

The individual rate of the first machine to complete the work will be 15 units/hour 
and the rate of the second machine is 20 units/hour. So, together they will complete 35 units in an hour.

Therefore, the time taken for both the machines(combined) is  \frac{60}{35} = \frac{12}{7} =1\frac{5}{7}  (Option C)

Hope that helps!

dave13 · Answer

Hi Bunuel

can you help to understand the logic behind this \frac{1}{(\frac{7}{12})}=\frac{12}{7}=1 why do you flip \frac{7}{12} in order to know how much time will it take both to complete

pushpitkc · Answer

Hi  dave13

Basically, If  \frac{1}{x}  of the work takes place in 1 hour, the time taken to complete the work will be x!

Here in this question, x =  \frac{7}{12}

So, when we finish  \frac{1}{(\frac{7}{12})}  of the work in 1 hour, the time taken to complete the work is  \frac{12}{7}

Hope that helps you!

eakabuah · Answer

Ta=4hrs, Tb=3hrs
Ra=1/Ta =1/4 and Rb=1/Tb=1/3
Rate when the two machines work together, Rab, can be gotten as follows:
1/Rab =1/Ra + 1/Rb = 1/4+1/3 = 7/12
But Tab=1/Rab = 12/7 hrs

Posted from my mobile device

TheUltimateWinner · Answer

dave13 
In  \frac{7}{12}  the  unit  is  \frac{job}{hour} . You need   hour   (time). So, just flip it. You'll get the   hour  .
Thanks__

TheUltimateWinner · Answer

Hello all,
We can solve this question logically that will save your time for sure!
 Choice D and E: 
One machine (x) takes 3 hrs to complete whole the job and another machine (y) takes 4 hrs to complete the  same   job. If if they do the same job simultaneously, then their total time will be less than 3 hours. So, the answer choices (D,E) which keeps more than 3 hours are crossed out. 
In real life example:
You do a job by 3 hrs; i do the  same   job by 4 hrs. If i help you to do the SAME job (with respective constant rate), you will need LESS THAN 3 hours to complete whole the job. 
 Choice A, B and C: 
Suppose, x takes 3 hours to take the job; and y takes 3 hours to take the same job (with respective constant rate). If they do simultaneously, then they need  1\frac{1}{2}  hours to complete whole the job, right? But, in the question prompt, y takes 4 hours (who is slower than x). So, they need more than  1\frac{1}{2}  hours if they do simultaneously. So, A and B are out. The correct choice is  C.

EMPOWERgmatRichC · Answer

Hi Asad,

That is a great series of deductions - and it shows how GMAT questions can almost always be approached in more than one way. 
 
GMAT assassins aren't born, they're made,
Rich

ArnauG · Answer

Let's assume that the production order is 1 unit in size.

Machine 1 can complete 1/4 of the order in 1 hour, while Machine 2 can complete 1/3 of the order in 1 hour. Working together, they will complete 1/4 + 1/3 = 3/12 + 4/12 = 7/12 of the order in 1 hour.

Therefore, we can set up the equation:

(7/12) * t = 1

where "t" is the time it takes both machines working together to complete the order.

Solving for "t", we get:

t = 12/7 hours

Therefore, the answer is approximately 1.71 hours or 1 hour and 42 minutes, and the correct choice is (C) 1 5/7.

totaltestprepNick · Answer

C https://www.youtube.com/watch?v=Jz7C1WRAoOc Nick Slavkovich, GMAT/GRE tutor with 20+ years of experience tutoring@totaltestprep.net

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