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# A can do $\dfrac{2}{3}$ of a certain work in $16$ days and B can do $\dfrac{1}{4}$ of the same work in $3$ days. In how many days can both finish the work working together?

Last updated date: 16th Jun 2024
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Hint: Let us first take the whole work as $1$ unit.
We need to find first the work done by $A$ and $B$ in $1$ day. Then we can calculate the work done by $A$ and $B$ in $1$ day.
Then applying the unitary method on the work done by $A$ and $B$ in $1$ day, we can find out the number of days they both need to finish the work, working together.
(The unitary method is a technique for solving a problem by first finding the value of a single unit and then finding the necessary value by multiplying the single unit value.)

It is given that, $A$ can do $\dfrac{2}{3}$ of a certain work in $16$ days and $B$ can do $\dfrac{1}{4}$ of the same work in $3$ days.
We need to find out the number of days they both need to finish the work, working together.
Let us first take the whole work as a $1$ unit.
Now, $A$ can be done in $16$ days $\dfrac{2}{3}$ units of a certain work.
In $1$ day $A$ can do:
$\Rightarrow \dfrac{{\dfrac{2}{3}}}{{16}} = \dfrac{2}{3} \times \dfrac{1}{{16}} = \dfrac{1}{{24}}$ units of work.
$B$ can do in $3$ days, $\dfrac{1}{4}$ units of the work.
In $1$ day $B$ can do:
$\Rightarrow \dfrac{{\dfrac{1}{4}}}{3} = \dfrac{1}{4} \times \dfrac{1}{3} = \dfrac{1}{{12}}$ units of the work.
Now, $A$ and $B$ together can do in $1$ day:
$\Rightarrow \dfrac{1}{{24}} + \dfrac{1}{{12}} = \dfrac{{1 + 2}}{{24}} = \dfrac{3}{{24}} = \dfrac{1}{8}$ unit of work.
Thus $A$ and $B$ together can do $\dfrac{1}{8}$ unit of work in $1$ day.
$A$ and $B$ together can do the whole work that is, $1$unit of work in $\dfrac{1}{{\dfrac{1}{8}}} = 1 \times \dfrac{8}{1} = 8$ days.

Hence, $A$ and $B$ together can finish the work in $8$ days.

Note:
We have solved the problem with the unitary method.
Now we need to know what is the unitary method?
In short, this method is used to find the value of a unit from the value of a multiple, and hence the value of a multiple.