Question

$\text{A}$ and $\text{B}$ can do a piece of work in $30$ days , while $\text{B}$ and $\text{C}$ can do the same work in $24$ days and $\text{C}$ and $\text{A}$ in $\text{20}$ days .They all work together for $10$ days when $\text{B}$ and $\text{C}$ leave . How many days more will $\text{A}$ take to finish the work ?

• A. $18$ days
• B. $24$ days
• C. $30$ days
• D. $36$ days

Answer 1

The correct option is A

$18$ days

Time taken by $\text{A}$ and $\text{B}$ to do the work $=30$ days

So work done by $\text{A}$ and $\text{B}$ in one day $=\frac{1}{30}$

Time taken by $\text{B}$ and $\text{C}$ to do the work $=24$ days

So work done by $\text{B}$ and $\text{C}$ in one day $=\frac{1}{24}$

Time taken by $\text{C}$ and $\text{A}$ to do the work $=20$ days

So work done by $\text{C}$ and $\text{A}$ in one day $=\frac{1}{20}$

So work done by $\text{A,B,C}$ in one day $=\frac{1}{30}+\frac{1}{24}+\frac{1}{20}$

$\text{A+B=}\frac{1}{30}\text{, B+C=}\frac{1}{24}\text{,C+A=}\frac{1}{20}$

$$\begin{gathered} \text{A+B+B+C+C+A=}\frac{1}{30}\text{+}\frac{1}{24}\text{+}\frac{1}{20} \\ \text{2A+2B+2C=}\frac{1}{30}\text{+}\frac{1}{24}\text{+}\frac{1}{20} \\ \text{2(A+B+C)=}\frac{1\times 4+1\times 5+1\times 6}{120} \\ \text{2(A+B+C)=}\frac{4+5+6}{120} \\ 2\left(\text{A+B+C)=}\frac{15}{120}\text{=}\frac{1}{8} \\ \right(\text{A+B+C)=}\frac{1}{2\times 8}\text{=}\frac{1}{16} \end{gathered}$$

Therefore one day work of $\text{A+B+C}$ $=\frac{1}{16}$

So work done in $10$ days of $\text{A+B+C= 10×}\frac{1}{16}\text{=}\frac{5}{8}$

So remaining work done by $\text{A}$ is $=1-\frac{5}{8}=\frac{3}{8}$

Now $\text{A}$ one day work is

$$\begin{gathered} =(\text{A+B+C)-(B+C) (one day work)} \\ =\frac{1}{16}-\frac{1}{24}=\frac{1\times 3-1\times 2}{48}=\frac{3-2}{48}=\frac{1}{48} \end{gathered}$$

Time taken by $\text{A}$ to do the work $=48$ days

But $\text{A}$ done only $\frac{3}{8}$ work alone $=\frac{3}{8}\times 48=3\times 6=18$days

Hence $\text{A}$ will take $18$ more days to complete the work .

Therefore option(A) is the correct answer and all other options are incorrect answers .