Question 1159402
r * t = q
r = rate
t = time
q = quantity of work


man rate is 1/8 of the job in 1 hour.
boy rate is 1/12 of the job in 1 hour.
girl rate is 1/16 of the job in 1 hour.


when they work together, their rates are additive.
for the man and the boy, the combined rate becomes 1/8 + 1/12 = 3/24 + 2/24 = 5/24.
for the boy and the girl, the combined rate becomes 1/12 + 1/16 = 4/48 + 3/48 = 7/48.


the combined rate for the man and the boy is 5/24 = 10/48.
the combined rate for the boy and the girl is 7/48.


the man and the boy work together for 1 hour.
the formula becomes 10/48 * 1 = q
solve for q to get q = 10/48


since the total job is equal to 1, you get 1 - 10/48 = 48/48 - 10/48 = 38/48 of the job still needs to be done.


the boy and the girl work for t hours to finish the job.
the formula becomes 7/48 * t = 38/48
solve for t to get t = 38/48 * 48/7 = 38/7 hours.


the man and the boy work for 1 hour.
the boy and the girl work for 38/7 hours.
the total time spent on the job is 1 + 38/7 = 7/7 + 38/7 = 45/7 hours.


to confirm, add the two formulas together to get:
r * t for man and boy equals 10/48 * 1 = 10/48 of the job.
r * t for boy and girl equals 7/48 * 38/7 = 38/48 of the job.
10/48 + 38/48 = 48/48 = the whole job.


your solution is that the total time to complete the job was 1 + 38/7 = 7/7 + 38/7 = 45/7 hours.


the decimal equivalent of that is 6.428571429 hours.


i'll be available to answer any questions you might have about this.