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If A and B can do a piece of work in 6 days and A alone can do it in 9 days. Then what is the time taken by B alone to do the work?
(a) 18 days
(b) 15 days
(c) 12 days
(d) $ 7\dfrac{1}{2} $ days

Answer Verified Verified
Hint: First try to find the fraction of work done by both A and B in one day and then find the fraction of work done by A alone in a day after that find the fraction of work done by B alone in one day. Using the information i.e. fraction of work done by B alone in a day, we can easily find out the amount of time B will take to finish the work alone.

Complete step-by-step answer:
It is given that A and B can do a piece of work in 6 days. Let us suppose the amount of work as X.
If A and B both can finish X amount work in 6 days then, we get that the work done in one day is $ \dfrac{1}{6}th $ of the total amount of work, i.e.
Work done in one day by both A and B = $ \dfrac{X}{6} $
It is also given that A finishes the work in 9 days, hence we get that if A finishes work in 9 days then work done by A in 1 day is $ \dfrac{1}{9}th $ of the total work, so we get
Work done by A alone in 1 day = $ \dfrac{X}{9} $
Now, we know that work done by B alone in one day would be equal to
 = (work done by both in 1 day) – (work done by A alone in 1 day)
= $ \dfrac{X}{6} $ - $ \dfrac{X}{9} $
By taking LCM we get,
= $ \dfrac{9X-6X}{9\times 6} $
= $ \dfrac{3X}{54} $
= $ \dfrac{X}{18} $
So, work done by B alone in 1 day is = $ \dfrac{1}{18}th $ of the total amount of work.
Multiplying by 18 on both sides of above expression we get,
Work done by B alone in 18 days = total amount of work
Hence B alone can do the work in 18 days
So, the correct answer is “Option A”.

Note: Be careful while solving this question and don’t try to use any shortcuts here as it can lead to wrong answers.
This question can also be solved by considering amount of work directly for 6 days instead of 1,
For example,
X amount of work is done by both A and B in 6 days,
And $ \dfrac{X}{9} $ amount of work is done by A alone in one day
So, $ \dfrac{6X}{9} $ is the amount of work done by A alone in 6 days
Amount of work done by B in 6 days = work done by both in 6 days – work done by A alone in 6 days
= $ X-\dfrac{6X}{9} $
 $ \begin{align}
  & =\dfrac{9X-6X}{9} \\
 & =\dfrac{3X}{9} \\
 & =\dfrac{X}{3} \\
\end{align} $
Hence amount of work done by B alone in 6 days is $ \dfrac{1}{3}rd $ of total work,
So, total work done will be done by B alone in (6 $ \times $ 3) days i.e. 18 days.
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