Question

$4$ men and $9$ women finish a job in$12$ days, while$3$ men and $10$ women finish it in $14$ days. In how many days will \[13\] women finish it?
(A) 24 days
(B) 26 days
(C) 28 days
(D) 30 days
(E) None of these

Answer 1

Hint: First we have to find the job per day by men and women. Then we can calculate it for 13 women.

Complete Step by Step solution:
Let us consider$'m'$is the days it takes for a man to do the job and$'w'$is the days for women to do the job.
So, $4$ men and $9$ women complete $\dfrac{1}{{12}}$ of the job per day.
Therefore,$\dfrac{4}{m} + \dfrac{9}{w} = \dfrac{1}{{12}}$ . . . . . . (1)
And, similarly $3$ men and $10$ women complete $\dfrac{1}{{14}}$ of the job per day.
Then,
$\dfrac{3}{m} + \dfrac{{10}}{w} = \dfrac{1}{{14}}$ . . . . . . . (2)
Now solving (1) and (2) we get,
$\dfrac{4}{m} + \dfrac{9}{w} = \dfrac{1}{{12}}$,
$\dfrac{3}{m} + \dfrac{{10}}{w} = \dfrac{1}{{14}}$
Multiply Equation (1) by equation$3$ and
Multiply Equation (2) by equation$4$, then
$3\left( {\dfrac{4}{m} + \dfrac{9}{w}} \right) = \left( {\dfrac{1}{{12}}} \right)3$
$\dfrac{{12}}{m} + \dfrac{{27}}{w} = \dfrac{1}{4}$ . . . . . . (3)
$4\left( {\dfrac{3}{m} + \dfrac{{10}}{w}} \right) = \left( {\dfrac{1}{{14}}} \right)4$
$\dfrac{{12}}{m} + \dfrac{{40}}{w} = \dfrac{2}{7}$ . . . . . . (4)
From (3) and (4)
$\dfrac{{27}}{w} - \dfrac{{40}}{w} = \dfrac{1}{4} - \dfrac{2}{7}.$
$\dfrac{{27 - 40}}{w} = \dfrac{{7 - 8}}{{28}}$
$ - \dfrac{{13}}{w} = - \dfrac{1}{{28}}$
$\dfrac{{13}}{w} = \dfrac{1}{{28}}$
$w = 364$
Solving the system, $w = 364$. Thirteen women can do the job in$\dfrac{{364}}{{13}} = 28$days.
Therefore, from the above explanation the correct option is [C] $28$days.

Note: Consider complete work as 1. Then calculate the number of days. It makes the calculation easier.
Number of days =Total work units $/$No. of units completed per day.