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

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B

$24$ days

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C

$30$ days

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D

$36$ days

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Solution

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 $=24$ daysSo 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}$$\text{A+B=}\frac{1}{30}\text{,B+C=}\frac{1}{24}\text{,C+A=}\frac{1}{20}$$\text{A+B+B+C+C+A=}\frac{1}{30}\text{+}\frac{1}{24}\text{+}\frac{1}{20}\phantom{\rule{0ex}{0ex}}\text{2A+2B+2C=}\frac{1}{30}\text{+}\frac{1}{24}\text{+}\frac{1}{20}\phantom{\rule{0ex}{0ex}}\text{2(A+B+C)=}\frac{1×4+1×5+1×6}{120}\phantom{\rule{0ex}{0ex}}\text{2(A+B+C)=}\frac{4+5+6}{120}\phantom{\rule{0ex}{0ex}}2\left(\text{A+B+C)=}\frac{15}{120}\text{=}\frac{1}{8}\phantom{\rule{0ex}{0ex}}\left(\text{A+B+C)=}\frac{1}{2×8}\text{=}\frac{1}{16}\phantom{\rule{0ex}{0ex}}$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 $=\left(\text{A+B+C)-(B+C)(onedaywork)}\phantom{\rule{0ex}{0ex}}=\frac{1}{16}-\frac{1}{24}=\frac{1×3-1×2}{48}=\frac{3-2}{48}=\frac{1}{48}$Time taken by $\text{A}$ to do the work $=48$ daysBut $\text{A}$ done only $\frac{3}{8}$ work alone $=\frac{3}{8}×48=3×6=18$daysHence $\text{A}$ will take $18$ more days to complete the work .Therefore option(A) is the correct answer and all other options are incorrect answers .

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