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?

A. 1/9
B. 1/6
C. 1/3
D. 7/18
E. 4/9

I am not able to answer this question. Please help.

saadishah 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?

A. 1/9
B. 1/6
C. 1/3
D. 7/18
E. 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.

saadishah 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?

A. 1/9
B. 1/6
C. 1/3
D. 7/18
E. 4/9

I am not able to answer this question. Please help.

Let's put in the following symbols:

Rate at which Tom works in an hour is T = 1/6 (Since Tom can paint the room in 6 hours, in one hour he can paint 1/6 of the room)
Rate at which Peter works in an hour is P = 1/3
Rate at which John works in an hour is J = 1/2

We also need two combined rates:
Hourly rate of work with Tom & Peter working together - TP = 1/6 + 1/3 = 3/6 = 1/2
Hourly rate of work with all 3 working together = TPJ = 1/6 + 1/3 + 1/2 = 6/6 = 1

Now, coming to the problem:
For the first one hour Tom works alone. This implies that in hour 1:
Room painted = 1/6 (hourly rate of work for Tom)
Room remaining to be painted = 1 - 1/6 = 5/6

In the second hour Peter joins Tom:
Room painted in hour 2 = 1/2 (Hourly rate of painting for TP)
Room painted at the end of hour 2 = 1/6 + 1/2 = 4/6 = 2/3
Room remaining to be painted = 1/3

Now, check how long will the 3 of them take to paint this remaining 1 room:
Time taken for the 3 of them to paint the remaining room = 1/3 / 1 (Task/Rate) = 1/3
Thus, the 3 of them work for 20 minutes more.

This gives you the total time that Peter has worked on the job ie the second hour and the last 20 minutes = 1 + 1/3 = 4/3
The proportion of work that Peter would have done would be calculated by multiplying the time with the rate:
4/3 x 1/3 = 4/9

Thus, Peter painted 4/9 of the room. The answer is E

Think of the room as being made of blocks. Let's say the room has 12 blocks. Tom can do this in 6 hours, so he can paint 2 blocks per hour. Likewise, Peter can paint 4 and John can paint 6.

When the guys work together, we get

First hour = 2, by Tom
Second hour = 6, by Tom and Peter

So we only have 4 blocks left. John + Peter + Tom can do 12 blocks in an hour, so 4 blocks will only take 20 minutes, or (1/3) of an hour.

That means that Peter does 4 blocks in hour two + (1/3)*4 blocks in "hour" three (the last 20 minutes).

Thus Peter does (4 + 4/3) out of the 12 total blocks, or 4/9 of the job.

An even easier (but less intuitive) way would be to see that our ratio is going to be

4 * Peter's time / 12

or Peter's time / 3.

Since Peter will work for MORE than an hour (he does hour two plus some of "hour" three), Peter's time / 3 must be > 1 / 3.

From there, the answer is D or E, but if Peter's time / 3 = 7/18, Peter's time = some hideous fraction. That's unlikely, so the answer "must" be E!