Trouble with Direct and Inverse Proportion

[Kyoma]

1. The problem statements, all variables and given/known data
a. 12 workers took 6 hours to complete 2/3 of a job. To speed up the work, another 6 workers were hired. How many more hours of work are needed before the job is completed?

b. It takes a painter 6 hours to paint 2 rooms. His assistant takes 4 hours to paint one room. If both of them were to work together, how many hours will it take for them to paint one room?

c. For a camp, food is brought to last for 15 days for a group of 40 students. If 4 students left the camp after 6 days, how many days can the food last for the remaining 36 students?

2. The attempt at a solution
a. I have no idea.

b. The speed of the painter is 1/3 room per hour while the assistant's speed is 1/4 room per hour, so if they work together, the speed will be 7/12 room per hour. Thus, 1 room will take 12/7 hours. I'm not sure with this.

c. I have no idea.

Thanks.

 

[HallsofIvy]

a) In problems like these, rates add. The first 12 workers completed 2/3 of the job in 6 hours so they were working at a rate of (2/3 job)/6 hours= 1/9 job per hour. Assuming all workers worked at the same rate, each worker then did (1/9)/12= 1/108 job per hour. Assuming each new person also works at that rate 6 new workers will work at 6(1/108)= 1/18 job per worker (more simply, that is 1/2 the rate of the original 12 workers because 6=(1/2)(12)). All 18 will work at 1/18+ 1/9= 1/18+ 2/18= 3/18= 1/6 job per hour. How long will it take them to finish the remaining 1/3 job?

b) Yes, 1/3+ 1/4= 4/12+ 3/12= 7/12 room per hour. They will do one room in (1 room)/(7/12 room per hour= (1 room)(12/7 hour per room)= 12/7 hours. You are correct.

c) There was enough to last 40 students 15 days and there 40 students so after 4 days there will be enough food to last 40 students the remaining 11 days. That is there will be food for 40/11 students per day. How long will that last 36 students. (Look at the units: students times (days per student)= (students)(days/students)= days.)