Question

A and B together can complete a work in 4 days. A alone can finish the work in 12 days. In how many days B alone can finish the work?
A. 5
B. 15
C. 6
D. 8

Answer 1

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., the 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 solution:
It is given that A and B can do a piece of work in 4 days.
Let us suppose the amount of work as X.
If A and B both can finish X amount of work in 4 days then, we get that the work done in one day is $\dfrac{1}{4}$ of the total amount of work,
$ \Rightarrow $ Work was done in one day by both A and B $ = \dfrac{X}{4}$
It is also given that A finishes the work in 12 days, hence we get that if A finishes work in 12 days then work done by A in 1 day is $\dfrac{1}{{12}}$ of the total work, so we get
$ \Rightarrow $Work is done by A alone in 1 day $ = \dfrac{X}{{12}}$
Now, we know that work done by B alone in one day,
$ \Rightarrow $Work is done by B alone in 1 day = (work done by both in 1 day) – (work is done by A alone in 1 day)
Substitute the values,
$ \Rightarrow $Work is done by B alone in 1 day $ = \dfrac{X}{4} - \dfrac{X}{{12}}$
By taking LCM we get,
$ \Rightarrow $Work is done by B alone in 1 day $ = \dfrac{{3X - X}}{{12}}$
Simplify the terms,
$ \Rightarrow $Work is done by B alone in 1 day $ = \dfrac{{2X}}{{12}}$
Cancel out the common factors,
$ \Rightarrow $Work is done by B alone in 1 day $ = \dfrac{X}{6}$
So, work done by B alone in 1 day is $\dfrac{1}{6}$ of the total amount of work.
Multiplying by 6 on both sides of the above expression we get,
$ \Rightarrow $ Work is done by B alone in 6 days = total amount of work
Thus, B alone can do the work in 6 days

Hence, option (C) is the correct answer.

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 the amount of work directly for 4 days instead of 1,
X amount of work is done by both A and B in 4 days,
$\dfrac{X}{{12}}$ amount of work is done by A alone in one day
So, $\dfrac{{4X}}{{12}}$ is the amount of work done by A alone in 4 days.
$ \Rightarrow $ Amount of work done by B in 4 days = work is done by both in 4 days – work is done by A alone in 4 days
Substitute the values,
$ \Rightarrow $ Amount of work done by B in 4 days $ = X - \dfrac{{4X}}{{12}}$
Cancel out the common factor,
$ \Rightarrow $ Amount of work done by B in 4 days $ = X - \dfrac{X}{3}$
Take LCM on the right side,
$ \Rightarrow $ Amount of work done by B in 4 days $ = \dfrac{{3X - X}}{3}$
Simplify the term,
$ \Rightarrow $ Amount of work done by B in 4 days $ = \dfrac{{2X}}{3}$
So, the amount of work done by B alone in 4 days is $\dfrac{2}{3}rd$ of total work.
Hence, total work done will be done by B alone in ($4 \times \dfrac{3}{2}$) days i.e., 6 days.