Question

Suppose $A$ takes twice as much time as $B$ and thrice as much time as $C$ to complete a work. If all of them work together, they can finish the work in $2$ days. How much time $B$ and $C$ working together to finish it?

Answer 1

Let the work done by $A$ be $x$

$A$ takes twice as much as $B$ Therefore $B$ take half of what time $A$ takes.

$\therefore B=\frac{x}{2}$

$A$ takes thrice as much as $C$ Therefore $C$ take one third of what time $A$ takes.

$\therefore C=\frac{x}{3}$

All have them finish the work in $2$ days

$\therefore \frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{2}$

$\Rightarrow \frac{1}{x}+\frac{2}{x}+\frac{3}{x}=\frac{1}{2}$

$\Rightarrow \frac{6}{x}=\frac{1}{2}$ Cross multiply

$\Rightarrow x=12$

$\therefore B=\frac{x}{2}=\frac{12}{2}=6$

$\therefore C=\frac{x}{3}=\frac{12}{3}=4$

If $B$ and $C$ work together

$=\frac{1}{B}+\frac{1}{C}$

$=\frac{1}{6}+\frac{1}{4}$

$=\frac{2+3}{12}$ (Take LCM)

$=\frac{5}{12}$

If $B$ and $C$ work togethere they will take $\frac{12}{5}$ days