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

is there a shortcut to tackle this?

I'm gonna go with 4/9

Tom's rate = 1/6
Peter's rate = 1/3 = 2/6
John's rate = 1/2 = 3/6

First hour:
John paints 1/6 of the room

Second hour:
John is joined by Peter and they work for an hour. John paints 1/6 of the room and Peter paints 2/6 of the room. After the 2nd hour, 4/6th of the room is already painted, with Peter having painted 2/6th of that.

Third hour:
The boys are now joined by John and things are getting interesting They still have 2/6th of the room to paint and their combined rate is 1 room per hour. So working together, they will finish that room in 20 minutes. Peter will contribute his usual rate (2/6th)

Peter contributed 2/6 of 1/3 of the remaining job in the 3rd hour or 2/6 * 1/3 = 1/9.

Alright so Peter painted 2/6th of the room during the 2nd hour and 1/9th of the room during the third hour. In total he painted 2/6 + 1/9 = 4/9 of the room.

What did everyone else get?