Question

2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

Hint: In this type of question where men and women together perform a work. The first thing is to assume the one day work of a woman and one day work of a man will be x and y respectively. After this form the two equations in two variables using the given data in question.

Complete step-by-step solution -
In the question, it is given that a certain amount of work is completed by 2 women and 5 men in 4 days, while 3 women and 6 men can finish the same work in 3 days. And we have to find time taken if 1 woman or 1 man alone does the work.
So first let us assume that one day work of one woman = x
And one day work of a man = y
It is given that time taken by 2 women and 5 men to complete a work= 4 days
$\therefore$ In one day work done by 2 women and 5 men= $\dfrac{1}{4}$
So using the above information, we can write:
${\text{2x + 5y = }}\dfrac{1}{4} \\ \Rightarrow 4\left( {{\text{2x + 5y}}} \right) = 1 \\$
$\Rightarrow 8{\text{x + 20y = 1}}$ _______________________ (1)
It is given that time taken by 3 women and 6 men to complete a work= 3 days
$\therefore$ In one day work done by 3 women and 6 men= $\dfrac{1}{3}$

So using the above information, we can write:
${\text{3x + 6y = }}\dfrac{1}{3} \\ \Rightarrow 3\left( {{\text{3x + 6y}}} \right) = 1 \\$
$\Rightarrow 9{\text{x + 18y = 1}}$ _________________ (2)
Multiplying equation 2 by 10 and equation 1 by 9, we get:
72x+180y = 9 _______________________ (3)
And
90x+180y = 10 ______________________ (4)
Now on subtracting equation 3 from equation 4, we get:
18x=1
$\Rightarrow {\text{x = }}\dfrac{1}{{18}}$
Putting the value of x in equation 2, we get:
y =$\dfrac{1}{{36}}$ .
So the work done by 1 woman in 1 day = $\dfrac{1}{{18}}$
$\therefore$ Time taken by 1 woman to complete the whole work =$\dfrac{1}{{\dfrac{1}{{18}}}} = 18$ days.
Similarly, the work done by 1 man in 1 day = $\dfrac{1}{{36}}$
$\therefore$ Time taken by 1 man to complete the whole work =$\dfrac{1}{{\dfrac{1}{{36}}}} = 36$ days.

Note: In this type of question, first you should know how to calculate work done in one day by 1 person. After this form the equations using the given information in terms of variable x and y. After getting the values of x and y just divide the values of x and y by 1 to get the time taken by 1 person to complete the whole work.