Question

A completes half as much work as B and C completes half as much work as A and B
together, at the same time. If C alone can complete the work in 40 days, all of them can together
finish the work in
E. $13\dfrac{1}{3}\;{\rm{days}}$
F. $14\dfrac{1}{3}\;{\rm{days}}$
G. $20\;{\rm{days}}$
H. $30\;{\rm{days}}$

Answer 1

Hint: In this problem first we have to calculate how much work is done by $A$ and $B$ in one day. After that, from the given information we need to find the relation between $A, B$ and $C$.

Complete step by step solution:
With the relations of $A,B$ and $C$ we can calculate the required number of days needed to
finish the work when all of them work together.
Let $B$ can complete the work in $d$ days.
It is given that the number of days needed by $B$ to complete the work is twice the number of
day’s $A$ needs.
Therefore time needed by $A$ to complete the work $ = \dfrac{d}{2}$
Now we have to find out how much work $A$ and $B$ can do in 1 day.
In $d$ days $B$ can complete one work.
In 1 day $B$ can complete $\dfrac{1}{d}$ work.
Similarly,
In $\dfrac{d}{2}$ days $A$ can complete one work.
In 1 day $A$ can complete $\dfrac{2}{d}$ work.
Now, find the work $A$ and $B$ together can do.
$\begin{array}{c}\dfrac{1}{A} + \dfrac{1}{B} = \dfrac{1}{{\dfrac{2}{d}}} +
\dfrac{1}{{\dfrac{1}{d}}}\\ = \dfrac{d}{2} + \dfrac{d}{1}\\ = \dfrac{{3d}}{2}\end{array}$
Therefore $A$ and $B$ together can do in 1 day
 $\begin{array}{c} =
\dfrac{1}{{\dfrac{{3d}}{2}}}\\ = \dfrac{2}{{3d}}\end{array}$

Given that the time required to complete the work by $C$ is half of the time required when $A$
and $B$ work together.
$\therefore $ The number of days required by $C$ to complete the work $ = 2 \times
\dfrac{2}{{3d}}$
Given $C$ can complete the work in 40 days.
$\begin{array}{l}\therefore \dfrac{4}{{3d}} = 40\\ \Rightarrow 3d = \dfrac{1}{{10}}\\
\Rightarrow d = \dfrac{1}{{30}}\end{array}$
Therefore, time needed by $A$ to complete the work
 $\begin{array}{c} = \dfrac{d}{2}\\ =
\dfrac{1}{{60}}\end{array}$

Time needed by$B$to complete the work
$\begin{array}{c} = d\\ =
\dfrac{1}{{30}}\end{array}$

In 40 days, $C$ can complete one work.
$\therefore $ In one day $C$ can complete $\dfrac{1}{{40}}$ work.
$A$, $B$ and $C$ can work together in one day
$\begin{array}{c} = \dfrac{1}{{60}} +
\dfrac{1}{{30}} + \dfrac{1}{{40}}\\ = \dfrac{3}{{40}}\end{array}$

Therefore, $A$, $B$ and $C$ can work together
$\begin{array}{c} =
\dfrac{1}{{\dfrac{3}{{40}}}}\\ = \dfrac{{40}}{3}\\ = 13\dfrac{1}{3}\end{array}$

Hence, the correct option is A.

Note: Here, we have to determine the number of days required to complete the work when $A$, $B$ and $C$ work together. From the given information we can find the relations between $A$, $B$ and $C$. From those expressions we can find the number of days required by each of them to complete the work. After getting the time required to complete the work individually. Thus, it is easier to find the total days required to complete the work when $A$, $B$ and $C$ work together.