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# $A$ and $B$ can do a piece of work in $15$ days. If $A$’s one day work is $1\dfrac{1}{2}$ times the one day work of $B$. Find in how many days can each do the work?

Last updated date: 20th Jun 2024
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Hint: Let us assume that $A$ alone can do the work in $x$ days and similarly $B$ alone can do the same work in $y$ days.
Then one day work of $A$$= \dfrac{1}{x} One day work of B$$ = \dfrac{1}{y}$
Now we have to do the rest calculation.

In this question it is given that $A$ and $B$ can do a piece of work in $15$ days. It is also given that $A$’s one day work is $1\dfrac{1}{2}$ times the one day work of $B$.
Now we have to calculate the total time taken by $A$ alone and $B$ alone to do this work.
So first of all let us assume that:
$A$ alone can do the work in $x$ days.
$B$ alone can do the same work in $y$ days.
So if $A$ alone can do the work in $x$ days, then one day work of $A$$= \dfrac{1}{x} And if B alone can do the same work in y days, then one day work of B$$ = \dfrac{1}{y}$
And in this question we know the relation between both one day’s work.
$A$’s one day work is $1\dfrac{1}{2}$ times the one day work of $B$.
Here we know that
One day work of $A$$= \dfrac{1}{x} One day work of B$$ = \dfrac{1}{y}$
And $\dfrac{1}{x} = 1\dfrac{1}{2}.\dfrac{1}{y}$
$\dfrac{1}{x} = \dfrac{3}{{2y}}$
$y = \dfrac{{3x}}{2}$ $- - - - - \left( 1 \right)$
Now it is also given that they can complete work together in $15$ days. So one day work of together is:
$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{{15}}$ $- - - - - \left( 2 \right)$
Now putting the value of $y = \dfrac{{3x}}{2}$ in equation $\left( 2 \right)$
$\dfrac{1}{x} + \dfrac{2}{{3x}} = \dfrac{1}{{15}}$
$\dfrac{{3 + 2}}{{3x}} = \dfrac{1}{{15}}$
$\dfrac{5}{{3x}} = \dfrac{1}{{15}}$
$x = 25$days
Similarly $y = \dfrac{{3x}}{2}$$= \dfrac{{75}}{2}$ days
So $A$ alone can do the work in $25$ days.
$B$ alone can do it in $\dfrac{{75}}{2}$ days.

Note: In this type of question you need to find the two questions. First equation you will get from the statement that they can complete work together in $15$ days. And the second equation, you will get from the statement that $A$’s one day work is $1\dfrac{1}{2}$ times the one day work of $B$.