Albert can do a piece of work in 9 hours . Brian can do the same work in 12…

Concept

When several people work on the same job for different lengths of time, each person's contribution to the job equals (time worked) multiplied by their individual work rate, where the individual rate is 1 divided by the time that person alone would take to finish the whole job. The fraction of work completed by the group at any instant is the sum of each person's contribution, so the fraction still left is 1 minus that sum; whoever finishes the job needs extra time equal to (fraction remaining) divided by (their own rate).

Application

1. Albert's rate of work is 1/9 of the job per hour. Albert works from 9am to 11am, that is 2 hours, so his contribution is 2 x 1/9, which is 2/9 of the job.

2. Brian's rate of work is 1/12 of the job per hour. Brian works from 9am to 12 noon, that is 3 hours, so his contribution is 3 x 1/12, which is 1/4 of the job.

3. Adding Albert's and Brian's contributions: 2/9 + 1/4. Using a common denominator of 36, this is 8/36 + 9/36, which equals 17/36 of the job.

4. The work remaining for Charles is 1 - 17/36, which is 19/36 of the job.

5. Charles's rate of work is 1/15 of the job per hour, so the time he needs is 19/36 divided by 1/15, which is 19/36 x 15 = 285/36 hours, that is 7 hours 55 minutes.

6. Charles started at 9am along with Albert and Brian, so the work finishes at 9:00 am + 7 hours 55 minutes = 4:55 pm.

Cross-check

As a check, add all three contributions to confirm they total the whole job: Albert's 2/9 (8/36) + Brian's 1/4 (9/36) + Charles's 19/36 = 36/36 = 1, confirming the job is exactly completed. Also, 285/36 hours = 7 and 33/36 hours = 7 and 11/12 hours, and 11/12 of 60 minutes is 55 minutes, confirming the time conversion.