Tom, working alone, can paint a room in 6 hours. Peter and John

Could you please explain after the 1/3 remaining. I understood until all 3 complete 1 work together,so from this point on wards what is the work remaining ?

Previous case tom and peter complete (1/6+1/3) in one hour so total work completed is (1/6+1/2) is 2/3 , now when peter,tom and jack together work (1/6+1/2+1/3) is 1.Does that mean work gets completed when peter comes in,should the work be added 2/3+1 .

1/3 work is to be completed so how do you proceed from here.
Thanks.

Tom, working alone, can paint a room in 6 hours. :- Tom is finishing 100/6 i.e. 16.66% of the work in 1 hour.

Peter, working independently, can paint the same room in 3 hours. :- Peter is finishing 100/3 i.e. 33.33% of the work in 1 hour.

John, working independently, can paint the same room in 2 hours. :- Tom is finishing 100/2 i.e. 50% of the work in 1 hour.

Tom starts painting the room and works on his own for one hour. :- Tom Completed 16.66 of the work. 83.34% work is balance

He is then joined by Peter and they work together for an hour. :- Tom + Peter Completed (16.66% + 33.33% = 50%) of the work. 33.33% work is balance
Finally, John joins them and the three of them work together to finish the room :- Together Tom + Peter + John Complete (16.66% + 33.33% + 50= 100%) work in 1 hour, So to finish balance 33.33% work it would take them \(\frac{33.33}{100} = \frac{1}{3} hour.\)

What fraction of the whole job was done by Peter? :- We know Peter worked for \(1 + \frac{1}{3} hour.\) He must have completed \(33.33 + \frac{1}{3}(33.33)\) work ------> He completed 44.44% work which equals to \(\frac{4}{9}\)

Let's use smart numbers here --> Work=18
Rate * Time = Work
Tom: 3 x 6 = 18
Peter: 6 x 3 = 18
John: 9 x 2 = 18

Before John joined Tom and Peter: Tom worked 2 Hours -> 2*3=6 and Peter 1*6=6 gives us 12. So we are left with 18-12=6 for all three of them --> (3+6+9)*t=6, thus t=1/3 this means that Peter worked 2+1/3 Hours = 6+2=8 --> 8/18=4/9
At least this approach helps me... Don't like dealind with fractions when you're tired.

T-1/6
P-2/6
J-3/6

1*1/6+1*(1/6+2/6)+x(1/6+2/6+3/6)=1
4/6+x*1=1
x=2/6=1/3

P=2/6+2/6*1/3=2/6+2/18=8/18=4/9

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Let`s first put some value to the area of the room to make this problem easier to solve. We are looking for a value that is advisable by the working rate of each one, Tom =6, Peter = 3, and John = 2. So the best value for the room area is 6x3x2 = 36 feet.
Tom starts painting the room and works on his own for one hour, so he paint 6 feet of the room, and 30 feet is the remaining.
He is then joined by Peter and they work together for an hour, so Tom paint another 6 feet, Peter paint 12 feet, and 30 – 6 -12 = 12 feet is the remaining.
Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. Here we need to divide the remaining 6 feet among Tom, Peter and John based on their working rate. Tom = 12/6 = 2 feet, Peter = 12/3 = 4 feet, and John = 12/2 =6feet.
What fraction of the whole job was done by Peter?
Peter fraction = (12+4)/36 =4/9 ---D

So i used different approach. I wasnt able to solve it, but brought it down to two options.
If peter worked for one hour thats 33% of work and he worked with tom n john after that so technically he finished more than 33%.
That eliminates A,B and C.

And then i guessed D. I know its a wrong answer but better chance at guessing out of 2 than out of 5.

Tom's individual rate is 1 job / 6 hours or 1/6.
During the hour that Tom works alone, he completes 1/6 of the job (using rt = w).

Peter's individual rate is 1 job / 3 hours.
Peter joins Tom and they work together for another hour; Peter and Tom's respective individual rates can be added together to calculate their combined rate: 1/6 + 1/3 = 1/2.
Working together then they will complete 1/2 of the job in the 1 hour they work together.

At this point, 2/3 of the job has been completed (1/6 by Peter alone + 1/2 by Peter and Tom), and 1/3 remains.

When John joins Tom and Peter, the new combined rate for all three is: 1/6 + 1/3 + 1/2 = 1.
The time that it will take them to finish the remaining 1/3 of the job can be solved:
rt = w (1)(t) = 1/3 t = 1/3.

The question asks us for the fraction of the job that Peter completed. In the hour that Peter worked with Tom he alone completed: rt = w w = (1/3)(1) = 1/3 of the job.
In the last 1/3 of an hour that all three worked together, Peter alone completed:
(1/3)(1/3) = 1/9 of the job.
Adding these two values together, we get 1/3 + 1/9 of the job = 4/9 of the job.

The correct answer is E.

can you explain this part please?

Peter worked for t+1 hours at the rate of 1/3 job hour, so in t+1 hour and has done 1/3*(t+1) part of the job. t = 1/3, so he's done \(time*rate=\frac{1}{3}*(\frac{1}{3}+1)=\frac{4}{9}\) part of the job.

Hope it's clear.

Let total work be a multiple of 6, 3 and 2 i.e. 36 units
So, Tom alone does 6 units per hour, Peter alone does 12 units per hour and John alone does 18 units per hour
T:P:J = 1:2:3

1st hour
John works alone and does 6 units. Remaining work 36-6=30 units

2nd hour
John does 6 units and Peter does 12 units of work. Remaining 30-18=12 units

In the subsequent hour, Tom, Peter and John work together in the ratio of 1:2:3 to complete 12 units
John does (1/6)*12 = 2 units, Peter does (2/12)*12 = 4 units and John does 6 units

Peter's total = 12+14 = 16 units
Therefore, Peter's share = 16/36 = 4/9
Answer E

Lets try to solve this question without complicated algebra, with just very easy to compute numbers.

If the question permits, we should always try to use multiples of 10, to substitute any numbers to establish the relations in the question.

Lets consider the total work to paint the room = 60 units.

Given Tom alone can paint the room in 6 hours, Work Rate of Tom = 60/6 = 10 units/hour

Peter alone can paint the room in 3 hours, Work Rate of Peter = 60/3 = 20 units/hour

John Alone can paint the room in 2 hours, Work Rate of John = 60/2 = 30 units/hour


Tom starts the work & works alone for 1 hour, hence Tom finishes 10 units of the work.

Work left to finish = 60 - 10 = 50 units of work

Now Peter joins Tom & they together Work for an hour, hence Tom & Peter finish (10 + 20) = 30 units of work

Work left to finish now is = 50 - 30 = 20 units of work

Lastly, John joins Tom & Peter & the three work together to finish the job.

The three together can finish (10 + 20 + 30) = 60 units of work in one hour or 60 minutes.

So working together the rate at which the work is done is (60 units/60 minutes) = 1 unit/min.

Hence to finish the 20 units of work that is left, the three together will take 20 mins.

The question asks for the fraction of total work, as done by Peter.

Peter worked for a total of 1 hour & 20 mins. Hence total units of work done by peter = Work done in 1 hour + Work Done in 20 mins

Work Done by Peter in 60 mins = 20 units

Work Done by Peter in 20 mins = (20 * 20)/60 = 20/3 units of work

Total work done by Peter = 20 + 20/3 = 80/3 units of work.

Hence Peters work, as a fraction of Total Work = (80/3) / 60 = 80/(3*60) = 8/18 = 4/9

Answer E.

Tom’s rate = 1/6, Peter’s rate = 1/3, and John’s rate = 1/2.

We can let x = the number of hours John works, so x + 1 = the number of hours Peter works and x + 2 = the number of hours Tom works. We can create the following equation:

(1/6)(x + 2) + (1/3)(x + 1) + (1/2)x = 1

Multiplying the equation by 6, we have:

x + 2 + 2(x + 1) + 3x = 6

x + 2 + 2x + 2 + 3x = 6

6x = 2

x = 1/3

So Peter works for 1/3 + 1 = 4/3 hours, and the fraction of the whole job he does is (1/3)(4/3) = 4/9.

Answer: E

--->
\(\frac{1}{6}*(t+2)+\frac{1}{3}*(t+1)+\frac{1}{2}*t=1\)

Why did you equate this equal to 1?

I'm supposing that is because each individual's effort synergized to finish ONE (1) work. Although this doesn't seem authentic enough to me.

I am having a confusion here. Not sure whether this approach is good to proceed with.

Initially, Tom works for completes 1/6 of the work. So, remaining work is 1-1/6=5/6. Now, Tom and Peter works on this remaining work, (1/3 + 1/6)*5/6= 15/36.

Remaining work on which all of them will work together = 5/6-15/36 = 15/36.

Please let me know whether this approach is correct.

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This is a multi leg problem. You need a w/r/t chart for all three legs. The first leg is Tom workingg alone and for 1 hour and 1/6 room per hour 1/6 of the room is painted. Then for leg 2, Tom is joined by peter. They both work for an hour at their combined rates of 1/6 room + 1/3 room per hour = 1/2 of the room Ad this to the previous total and 1/6 + 1/2 = 4/6 = 2/3. So 1/3 of the room remains for leg three, in which all three work together at a combined rate of 1/6 + 1/3 + 1/2 = 1 room per hour. since they have 1/3 of the work left, it takes them 1/3/1 = 1/3 Hour. Thus Peter Works for 1/3+ 1 =4/3 hour. Multiply this by his rate of 1/3 room per hour, and we see that peter completed 4/9 of the room

Hi, thank you for this answer. I use this approach as well! I had a small doubt which I would really appreciate if you could clarify.
When the 3rd person joins in, they have 6 units left. And they complete the 6 units in 1/3 of the time. Now when we have to calculate Peters contribution in that 1/3 of the time, wouldn't we calculate that as 1/3*6/18 .. here the 6 is Peters rate and 18 is the combined rate. Please tell me where I am going wrong in calculating this? I'm not able to wrap my head around it.

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Hello Bunuel, I came across a different explanation for this question - basically, if you take LCM of all the answer choices you get 18. So assume that the total work is 18 units. Now the respective speeds for Tom, Peter, and John are 3, 6, 9 units per hour. Substituting these numbers in the situation given in the question we get: For the first hour - Tom has worked for 3 units + for the second hour Tom and Peter have worked for 3 + 6 units + for the last 6 units of work the rate of work for Tom, Peter, and John will be 1:2:3. In total Peter would have worked for 6+2 = 8 units out 18 that is 4/9.

While avoiding fractions does make things easier, I am not sure if this is the correct way to solve this question. Please could you clarify if such an approach could be used where we can assume the total work done from the answer choices?