Question

A is 50% as efficient as $B$. $C$ does half of the work done by $A$ and $B$ together. If $C$ alone does the work in $40$ days then, $A$,$B$ and $C$ together can do the work in
A) $13\dfrac{1}{3}days$
B) $15days$
C) $20days$
D) $30days$

Answer 1

Hint: To solve this question we need to know the concept of Time and Work and fraction. In this question we have changed the unlike fraction into mixed fraction.

Complete step by step solution:
The question asks us to find the total time taken by $A,B$ and $C$ to complete a certain amount of work when conditions are given. Let us start with the first part of the question which says $A$ works $50\%$ as $B$ work.
Let the amount of time taken by $B$ to complete the work be $'x'$ days. Hence, the time taken by $A$ to complete the work will be $'2x'$ days
Now amount of work done by $A$ in one day =$\dfrac{1}{2x}$
Amount of work done by $B$ in one day is equal =$\dfrac{1}{x}$
Amount of work done by both $A$ and $B$ =$\dfrac{1}{2x}+\dfrac{1}{x}=\dfrac{3}{2x}$
$C$ does half the work of $A$ and $B$ combined.
So work done by $C$ in one day = $\dfrac{1}{2}\left( \dfrac{3}{2x} \right)=\dfrac{3}{4x}$
Time taken by C to complete the work is equal to $\dfrac{4x}{3}$
According to the question, time taken by $C$ to complete the work is $40$ days
So we will equate the time taken by $C$ to complete the work. On doing this we get:
$\dfrac{4x}{3}=40days$
$\Rightarrow x=\dfrac{40\times 3}{4}days$
$\Rightarrow x=30days$
So time taken to complete the work by $B$ is $30days$, while the time taken to complete the work by $A$ is $60days$.
Now work done by $A,B$ and $C$ in one day is $=\dfrac{1}{60}+\dfrac{1}{30}+\dfrac{1}{40}$
On calculating the value we get:
$\Rightarrow \dfrac{1\times 2+1\times 4+1\times 3}{120}$
$\Rightarrow \dfrac{9}{120}$
Time taken by $A,B,C$ to complete the work will be =$\dfrac{120}{9}$
On converting the fraction in lowest term and changing it into mixed fraction, we get:
$\Rightarrow \dfrac{40}{3}days$
$\Rightarrow 13\dfrac{1}{3}days$

So, the correct answer is “Option A”.

Note: While solving this question always keep in mind that \[Time\text{ }=\text{ }\dfrac{Work\text{ }done}{Efficiency}\].To solve these type of question we need to find the work done in one day. To recheck the answer we can see that the time taken by $A,B,C$ together to complete a work is less than the time taken by any of these $AB,BC,CA$ to complete the work which is further less than time taken by any one of $A,B,C$ to complete the same work. So through the answer it is clear that time taken by all $A,B,C$ is $13\dfrac{1}{3}days$ which is the least time taken to complete the given work.