Question

If A can do a piece of work in $4$ hours, B and C together in $3$ hours and A and C together in $2$ hours. How long will B be alone taken to do it?
(A)$10$ hours
(B)$12$ hours
(C)$8$ hours
(D)$24$ hours

Answer 1

Hint:
If a piece of work is done in $x$ hours, then the work done in one hour will be $\dfrac{1}{x}$. Use this approach to find the work done by A alone, B and C together & A and C together in one hour.

Complete step by step solution:
 Let $a,b\& c$be the number of hours taken by A, B & C to complete the work alone.
Given, A can do a piece of work in $4$ hours.
As we know that if a piece of work is done in $x$ hours, then the work done in one hour will be $\dfrac{1}{x}$.
$\therefore $Work done by A in $1$ hour$ = \dfrac{1}{4}$$ \Rightarrow \dfrac{1}{a}$$ = \dfrac{1}{4}$ ….. (1)
Similarly, B and C together can do a piece of work in $3$ hours.
$\therefore $Work done by B and C together in $1$ hour$ = \dfrac{1}{3}$$ \Rightarrow \dfrac{1}{b} + \dfrac{1}{c}$ $ = \dfrac{1}{3}$ ….. (2)
A and C together can do a piece of work in $2$ hours.
$\therefore $Work done by A and C together in $1$ hour$\dfrac{1}{2}$$ \Rightarrow \dfrac{1}{a} + \dfrac{1}{c} = $$\dfrac{1}{2}$ ….. (3)
Now, subtracting (1) from (3);
$\left( {\dfrac{1}{a} + \dfrac{1}{c}} \right) - \dfrac{1}{a} = \dfrac{1}{2} - \dfrac{1}{4}$
$ \Rightarrow \dfrac{1}{c} = \dfrac{{2 - 1}}{4}$
$ \Rightarrow \dfrac{1}{c} = \dfrac{1}{4}$ ….. (4)
Now, subtracting (4) from (2);
$\left( {\dfrac{1}{b} + \dfrac{1}{c}} \right) - \dfrac{1}{c} = \dfrac{1}{3} - \dfrac{1}{4}$
$ \Rightarrow \dfrac{1}{b} = \dfrac{{4 - 3}}{{12}}$
$ \Rightarrow \dfrac{1}{b} = \dfrac{1}{{12}}$
$ \Rightarrow b = 12$
Therefore, time taken by B to complete the work alone is $12$ hours.

Hence, option (B) is the correct answer.

Note:
An another method to solve this question is described below:
The formula of total work (in term of hours and efficiency) is given by,
Total work = Efficiency $ \times $ Time
Given, A can do a piece of work in $4$ hours, B and C together in $3$ hours and A and C together in $2$ hours. The total work can be obtained by taking the LCM of all these given times.
$\therefore $ Total work = LCM of $4$, $3$& $2$
$ \Rightarrow $Total work =$12$

Use formula, Total work = Efficiency $ \times $ Time to find the efficiencies of A,B and C.
A’s efficiency = $\dfrac{{12}}{4}$
$ \Rightarrow $ A’s efficiency $ = 3$work per hour …. (1)
Similarly, (C+A)’s efficiency $ = \dfrac{{12}}{2}$ [Using formula, Total work = Efficiency $ \times $ Time]
$ \Rightarrow $ (C+A)’s efficiency $ = 6$work per hour
$\therefore $C’s efficiency$ = 6 - $A’s efficiency
 $ \Rightarrow $ C’s efficiency$ = 6 - $ $3$ [Using (1)]
  $ \Rightarrow $ C’s efficiency$ = 3$work per hour ….. (2)
Similarly, (B+C)’s efficiency $ = \dfrac{{12}}{3}$ [Using formula, Total work = Efficiency $ \times $ Time]
$ \Rightarrow $ (B+C)’s efficiency $ = 4$work per hour
$\therefore $B’s efficiency$ = 4 - $C’s efficiency
 $ \Rightarrow $ B’s efficiency$ = 4 - $ $3$ [Using (2)]
  $ \Rightarrow $ B’s efficiency=$1$work per hour ….. (3)
Therefore, time taken by B alone to complete the work$ = \dfrac{{12}}{1}$$ = 12$hours [Using formula, Total work = Efficiency $ \times $ Time]
Therefore, time taken by B to complete the work alone is $12$ hours.
Hence, option (B) is the correct answer.