A can do a piece of work in 14 days and B in 21 days. They begin together, but 3 days before the completion of the work, A leaves off. In how many days is the work complete? (a) $10$ days (b) $10\dfrac{1}{5}$ days (c) $15$ days (d) $7$ days
ANSWER
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Hint: We will first find the fraction of work they do in one day, each alone and together. And, we will calculate the work done in the last 3 days. Then we will use this data to solve the question.
Complete step-by-step answer:
According to question,
A completes the work in 14 days if he works alone. Therefore, he must have completed ${{\left( \dfrac{1}{14} \right)}^{th}}$ of the total work in one day.
And, B completes the work in 21 days if he works alone. Therefore, he must have completed ${{\left( \dfrac{1}{21} \right)}^{th}}$ of the total work in one day.
Now, if they work together, then the fraction of total work done by them in one day must be$=\dfrac{1}{14}+\dfrac{1}{21}$
Now, B has worked alone in the last 3 days after A leaves, to complete the work. Also, fraction of total work done by B in 3 days = $3\times \dfrac{1}{21}$
=$\dfrac{1}{7}$.
Therefore, remaining fraction of work, that A and B have completed together = $1-\dfrac{1}{7}$
Taking LCM and then subtracting,
$\begin{align}
& =\dfrac{1\times 7}{1\times 7}-\dfrac{1}{7} \\
& =\dfrac{7-1}{7} \\
& =\dfrac{6}{7} \\
\end{align}$
Here, we have,
Number of days taken by A and B together to complete \[{{\left( \dfrac{5}{42} \right)}^{th}}\] of the total work = 1 day.
Therefore, the number of days taken by A and B together to complete the whole work = $\dfrac{42}{5}$ days.
Hence, number of days taken by A and B together to complete \[{{\left( \dfrac{6}{7} \right)}^{th}}\] of the total work,
$=\dfrac{42}{5}\times \dfrac{6}{7}$ days
Simplifying, we get,
$=\dfrac{6\times 6}{5}$ days
$=\dfrac{36}{5}$ days.
Therefore, time taken to complete the whole work,
$=\dfrac{36}{5}$ days + $3$ days
Taking LCM,
$=\left( \dfrac{36}{5}+\dfrac{3\times 5}{1\times 5} \right)$ days
$=\dfrac{36+15}{5}$ days
$=\dfrac{51}{5}$ days.
$=10\dfrac{1}{5}$ days
Hence, the correct answer is option (b).
Note: In such types of questions, do not confuse fraction of work with number of works. And keep in mind that if we are considering a fraction of work, then whole work will be represented by 1.