Problem Solving

Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

Given a room to paint together, Tom, Peter and John will finish
the work in the ratio 1:2:3 (, because the time taken is in the
ratio 6:3:2 => work done is in the ratio 1/6:1/3:1/2).

We know that Peter has already done 1/3 of the room while Tom has
finished 1/6 + 1/6 = 1/3 when John joins. So, together Tom and Peter
completes 2/3 of the room when John joins. 1/3 of the room is left
which will be split in the ratio 1:2:3 => 1/18:1/9:1/6 among
Tom:Peter:John => Peter does 1/3 + 1/9 of the room => 4/9 => E,
if I'm not wrong.

HTH

alex.gellatly wrote:Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

gmat_and_me wrote:Given a room to paint together, Tom, Peter and John will finish
the work in the ratio 1:2:3 (, because the time taken is in the
ratio 6:3:2 => work done is in the ratio 1/6:1/3:1/2).

We know that Peter has already done 1/3 of the room while Tom has
finished 1/6 + 1/6 = 1/3 when John joins. So, together Tom and Peter
completes 2/3 of the room when John joins. 1/3 of the room is left
which will be split in the ratio 1:2:3 => 1/18:1/9:1/6 among
Tom:Peter:John => Peter does 1/3 + 1/9 of the room => 4/9 => E,
if I'm not wrong.

HTH

alex.gellatly wrote:Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

OA is E

alex.gellatly wrote:Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

Let the room = 36 units.
Rate for T = 36/6 = 6 units per hour.
Rate for P = 36/3 = 12 units per hour.
Rate for J = 36/2 = 18 units per hour.

Work produced by T in one hour = 6 units.
Remaining work = 36-6 = 30 units.

Combined rate for T+P = 6+12 = 18 units per hour.
Work produced by T+P in one hour = 18 units.
Of these 18 units, the number produced by P = 12.
Remaining work = 30-18 = 12 units.

Combined rate for T+P+J = 6+12+18 = 36 units per hour.
Of these 36 units, the fraction produced by P = 12/36 = 1/3.
Thus, of the remaining 12 units, the number produced by P = (1/3)12 = 4.

(Total for P)/(total work) = (12+4)/36 = 4/9.

The correct answer is E.

GMATGuruNY wrote:

alex.gellatly wrote:Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

Let the room = 36 units.
Rate for T = 36/6 = 6 units per hour.
Rate for P = 36/3 = 12 units per hour.
Rate for J = 36/2 = 18 units per hour.

Work produced by T in one hour = 6 units.
Remaining work = 36-6 = 30 units.

Combined rate for T+P = 6+12 = 18 units per hour.
Work produced by T+P in one hour = 18 units.
Of these 18 units, the number produced by P = 12.
Remaining work = 30-18 = 12 units.

Combined rate for T+P+J = 6+12+18 = 36 units per hour.
Of these 36 units, the fraction produced by P = 12/36 = 1/3.
Thus, of the remaining 12 units, the number produced by P = (1/3)12 = 4.

(Total for P)/(total work) = (12+4)/36 = 4/9.

The correct answer is E.

Could you please clarify one thing, if the whole work is 36, Tom worked alone for an hour so 6 done, then joined by Peter for 1 hour so 6+12=18 units done, 24 so far done. Then Tom joined by John and worked for 1 hour (6+18=24), so overall 48. And then three of them together worked to finish the room, BUT the room has alredy finished, since it was 36! I am sure i am missing something i just cannot catch that something.
Thanks in advance!

ziko wrote:

GMATGuruNY wrote:

alex.gellatly wrote:Tom, working alone, can paint a room in 6 hours. Peter and John, working independently, can paint the same room in 3 hours and 2 hours, respectively. Tom starts painting the room and works on his own for one hour. He is then joined by Peter and they work together for an hour. Finally, John joins them and the three of them work together to finish the room, each one working at his respective rate. What fraction of the whole job was done by Peter?

1/9
1/6
1/3
7/18
4/9

Could you please clarify one thing, if the whole work is 36, Tom worked alone for an hour so 6 done, then joined by Peter for 1 hour so 6+12=18 units done, 24 so far done. Then Tom joined by John and worked for 1 hour (6+18=24), so overall 48. And then three of them together worked to finish the room, BUT the room has alredy finished, since it was 36! I am sure i am missing something i just cannot catch that something.
Thanks in advance!

The portion in red is not discussed in the problem. Only 3 stages of work are described:
1. Tom starts painting the room and works on his own for one hour.
2. He is then joined by Peter and they work together for an hour.
3. Finally, John joins THEM and THE THREE OF THEM work together to finish the room.
At no point do John and Tom work together without Peter.