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Difficulty: Sub 505 Level,   Work and Rate Problems,                              
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Re: Machines A and B always operate independently and at their respective [#permalink]
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Option A.
Let total units of work=10
A's rate=2 u/hr
B's rate=10/x u/hr.
Combined=10/(2+10/x)=2
Solving this equation we get x=10/3 or 3.33

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Re: Machines A and B always operate independently and at their respective [#permalink]
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As most of the times in such problems, I am creating the RTW chart:

_______R_____T___W
A_____1/5____5____1
B____3/10____x____1
Both___1/2____2___1

So, now let me explain:
From the stem we know that A is doing the job (1 job) in 5 hours. For under T we add 5. From R*T=W, we get R=W/T, so in this case R=1/5. So, we add this under R.

From the stem we know that both machines together are doing the job in 2 hours. We add 2 under T and 1/2 under R.

Now, since we have the conbined time of both of the machines and the time of machine A we can find the time for machine B:
1/2 - 1/5 = 3 /10 or even easier 0.5 - 0.2 = 0.3, which is 3/10. We add 3/10 under R for machine B.

Finally, we are asked to find x, which is the time machine B needs to complete the job. Using R*T=W --> (3/10)X=1 -->(3X)/10 = 1 --> 3X = 10 --> X = 10/3 --> X = 3+1/3.

*an easy way to calculate the mixed number (mixed fraction) is like this:
To turn 10/3 to a mixed number you are looking to find a number with which you can multiply the denominator, add sth to it and get the nominator.

So, you will always have the same denominator: in this case 3.
You are looking for a number lower than the nominator. You will multiply your denominator with this number and add sth to get 10 (your nominator).
For example, you have 3 in this case in the denominator, multiply 3 by 3 and you get 9, add 1 and you get 10. You are done. The number you multiplied your denominator with goes to the left of the fraction and what you added goes to the nomintor.
You now have 3 + 1/3.
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Re: Machines A and B always operate independently and at their respective [#permalink]
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Hi All,

This prompt is an example of a "Work Formula" question. Any time a question involves two entities (people, machines, etc.) working on a task together and there are no "twists" to the question (someone stops working, someone shows up late to the job, etc.), you can use the Work Formula:

(A)(B)/(A+B) where A and B are the "times" that it takes for each entity to finish the job on his/her/its own.

Here, we're told:
Machine A can do the job in 5 hours
Machine B can do the job in X hours
Working together, the two machines can do the job in 2 hours.

Using the Work Formula, we have:

(5)(X)/(5 + X) = 2

5X = 10 + 2X
3X = 10
X = 10/3 hours

So, Machine B can do the job on its own in 10/3 = 3 1/3 hours.

Final Answer:

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Re: Machines A and B always operate independently and at their respective [#permalink]
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Hi All,

This prompt can also be solved by using the Work Formula:

Work = (A)(B)/(A+B) where A and B are the individual times that it takes to complete the 'job'

We're told that 2 machines can complete a task in 5 hours and X hours, respectively and working together will take 2 hours to complete the task. Working together, it would take them...

(5)(X)/(5+X) = 2 hours to complete the task

Using a bit of algebra, we can now solve for X...

5X = 10 + 2X
3X = 10
X = 10/3

Final Answer:

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Re: Machines A and B always operate independently and at their respective [#permalink]
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niheil wrote:
Machines A and B always operate independently and at their respective constant rates. When working alone, machine A can fill a production lot in 5 hours, and machine B can fill the same lot in x hours. When the two machines operate simultaneously to fill the production lot, it takes them 2 hours to complete the job. What is the value of x?

(A) \(3 \frac{1}{3}\)

(B) \(3\)

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

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

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


The rate of machine A is ⅕, and the rate of machine B is 1/x. Their combined rate is ½. Thus, we can create the equation:

1/5 + 1/x = 1/2

Multiplying by 10x, we have:

2x + 10 = 5x

10 = 3x

x = 10/3 = 3 1/3

Answer: A
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Re: Machines A and B always operate independently and at their respective [#permalink]
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VeritasKarishma wrote:

Time for a Teaser: A and B, working together, can finish a job in 10 days, B and C, working together, can finish the same job in 12 days and A and C, working together, can finish the same job in 15 days. If all three work together, how long will they take to finish the same job?


Is this answer 4 days?
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Re: Machines A and B always operate independently and at their respective [#permalink]
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VeritasKarishma wrote:
Work Rate problems are based on the concept that rates are additive. That is to say that if I paint half a wall in an hour and if you paint half a wall in an hour, if we both work together on a wall, we will finish the wall in an hour (Assuming that you are not repainting whatever I am painting to cover up my shoddy work!).
Remember, Rate of work = Work done per unit time
So, the proper way to express rate is 1/2 wall per hour and not 1 wall in 2 hours

If my rate of work is 1/2 wall/hour and yours is 1/2 wall/hour, our total rate of work is 1/2 + 1/2 = 1 wall/hour.

The basic questions of work rate are of the following form:
If A, working independently, completes a job in 10 hours and B, working independently, completes a job in 5 hours, how long will they take to complete the same job if they are working together?

Since A completes a job in 10 hours, his rate of work is 1/10th of the job per hour. B's rate of work is 1/5th of the job per hour.
Their combined rate of work would then be 1/10 + 1/5 = 3/10th of the job per hour.
As we said before, Rate of work = Work done/Time so 3/10 = 1/T (because 1 job has to be done)
or T = 10/3 hours.
This implies that A and B will together take 3.33 hours to do the job.
Note: Time taken when A and B work together will obviously be less than time taken by A or B when they are working independently.

Coming back to your question (finally! I know!), if A takes 5 hours to fill a lot and B takes x hours, and together they fill it in 2 hours, what is x?
Rate of work of A = 1/5th of the lot per hour
Rate of work of B = 1/xth of the lot per hour
Combined rate of work = 1/2 of the lot per hour
1/2 = 1/5 + 1/x
x = 10/3 hours
Note: Without solving, I know that E cannot be the answer since they both together take 2 hours to complete the work so one person alone can definitely not do the work in less than 2 hours.

Time for a Teaser: A and B, working together, can finish a job in 10 days, B and C, working together, can finish the same job in 12 days and A and C, working together, can finish the same job in 15 days. If all three work together, how long will they take to finish the same job?


Hi Karishma,

The explanation was wonderful. I tried to solve the teaser and my answer is 4 days. Is it correct?
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Re: Machines A and B always operate independently and at their respective [#permalink]
1/2 -1/5 = 3/10
Answer is 10/3
Option A
I m posting this if somebody has a problem in the conceptual understanding can ask otherwise it's a simple sum ...

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Re: Machines A and B always operate independently and at their respective [#permalink]
we can simplify:
5x/ (5+ x) = 2
then solve for x or substitute.. x = 10/3
hence A is correct
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Re: Machines A and B always operate independently and at their respective [#permalink]
5X/(5+X)=2
3X=10
X=10/3
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Re: Machines A and B always operate independently and at their respective [#permalink]
Let's say the capacity of the production lot is 1 unit. When Machine A works alone, it can fill the lot in 5 hours, so its rate is 1/5 units per hour. When Machine B works alone, it can fill the same lot in x hours, so its rate is 1/x units per hour.

When the two machines operate simultaneously, they can fill the lot in 2 hours, so their combined rate is 1/2 units per hour.

We can set up the equation:

1/5 + 1/x = 1/2

Multiplying both sides by 10x, we get:

2x + 10 = 5x

Solving for x, we get:

3x = 10

x = 10/3

Therefore, the answer is (A) 3 1/3.
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Re: Machines A and B always operate independently and at their respective [#permalink]
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