a can do a piece of work in 14 days while B can do it in 21 days they begin together but three days before the completion of the work a leaves off find the total number of days taken to complete the work
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As you said in your solution, AA can do 1/141/14 of the job per day, and BB can do 1/211/21 of the job per day. On each day that they work together, then, they do
114+121=542114+121=542
of the job. Up to here you were doing fine; it’s at this point that you went astray. You know that for the last three days of the job BB will be working alone. In those 33 days he’ll do
3⋅121=173⋅121=17
of the job. That means that the two of them working together must have done 6767 of the job before AA left. This would have taken them
6/75/42=67⋅425=365 days.6/75/42=67⋅425=365 days.
Add that to the 33 days that BB worked alone, and you get the correct total:
365+3=515=10.2 days.365+3=515=10.2 days.
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114+121=542114+121=542
of the job. Up to here you were doing fine; it’s at this point that you went astray. You know that for the last three days of the job BB will be working alone. In those 33 days he’ll do
3⋅121=173⋅121=17
of the job. That means that the two of them working together must have done 6767 of the job before AA left. This would have taken them
6/75/42=67⋅425=365 days.6/75/42=67⋅425=365 days.
Add that to the 33 days that BB worked alone, and you get the correct total:
365+3=515=10.2 days.365+3=515=10.2 days.
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