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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.

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.

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