Time and Work Problems::

Question

A,B and C work at different speeds.when they slowest two work together they take x days to finish the Job.when the quickest two of the three work together they take y days to finish the Job.one of them working alone takes three times time taken when all of the three work together.Find the time taken to finish the job when all the three work together.

I have tried:

let the speeds of A,B,C be a,b,c

From this question which two i consider fastest , two as slowest and working alone.

Please guide me on how should one go about solving this and such other related problems involving time-work constraints.

Answer 1

We consider $A < B < C $ in terms of efficiency. Thus, $B $ and $C $ complete the work in $x $ days and $A $ and $B $ do it in $y $ days.

If the quickest one is only one-third as efficient as the entire team, the other two cannot add up to two-thirds. By a similar logic, the slowest one cannot be the person who is one-third as efficient.

So, $B $ is one third efficient giving us that if the three complete the work in $k $ days, $B $ alone will do it in $3k$ days.

We have $B $ and $C $ complete $\frac {1}{x}$ part of work in a day and $A $ and $B $ complete $\frac {1}{y} $ part in a day. We also have $B $ does $\frac {1}{3k} $ part of work in a day.

So, $A,B $ and $C $ do $\frac {1}{x}+\frac {1}{y}-\frac {1}{3k} $ work in one day. (Why?) This is equal to $\frac {1}{k} $ as all three take $k $ days to finish the work.

Hope you can take it from here.