Question

$A$ and $B$ can do a piece of work in $12$ days; $B$ and $C$ in $15$ days; $C$ and $A$ in $20$ days. How much time will $A$ alone take to finish the work?

Answer 1

Time taken by $A+B$ to finish the work $=12$days

Time taken by $B+C$ to finish the work $=15$ days

Time taken by $A+C$ to finish the work $=20$ days

Now,

Work done by $A+B$ in one day $=\frac{1}{12}$

Work done by $B+C$ in one day $=\frac{1}{15}$

Work done by $A+C$ in one day $=\frac{1}{20}$

∴ Work done together is,

$A+B+B+C+A+C=\frac{1}{12}+\frac{1}{15}+\frac{1}{20}$

$\Rightarrow 2(A+B+C)=\frac{5+4+3}{60}$

$\Rightarrow 2(A+B+C)=\frac{12}{60}$

$\Rightarrow 2(A+B+C)=\frac{1}{5}$

$\Rightarrow A+B+C=\frac{1}{10}$

∴ Work done together $=A+B+C=\frac{1}{10}$

∴ Work done by A alone $=(A+B+C)-(B+C)$

$=\frac{1}{10}-\frac{1}{15}$

$=\frac{3-2}{30}$

$=\frac{1}{30}$

Thus, $A$ alone can finish the work in $30$ days.