**Given Information:**

- 2 women and 5 men can finish the embroidery work in 4 days.
- 3 women and 6 men can finish the embroidery work in 3 days.

**Let's assume:**

- Let the work done by 1 woman in 1 day = x.
- Let the work done by 1 man in 1 day = y.

**Calculating the work done:**

- From the given information, we can determine the following equations:
- (2x + 5y) * 4 = 1 (work done by 2 women and 5 men in 4 days).
- (3x + 6y) * 3 = 1 (work done by 3 women and 6 men in 3 days).

**Solving the equations:**

- Simplifying the equations:
- 8x + 20y = 1
- 9x + 18y = 1

- Multiplying the second equation by 2 to make the coefficients of x equal:
- 18x + 36y = 2

- Subtracting the first equation from the second equation:
- (18x + 36y) - (8x + 20y) = 2 - 1
- 10x + 16y = 1

- Simplifying the equation:
- 5x + 8y = 0.5

- Multiplying the equation by 2 to eliminate the decimals:
- 10x + 16y = 1

- Solving the equations:
- 10x + 16y = 1
- 9x + 18y = 1

- Subtracting the second equation from the first equation:
- (10x + 16y) - (9x + 18y) = 1 - 1
- x - 2y = 0

- Solving for x:
- x = 2y

- Substituting the value of x in the equation 5x + 8y = 0.5:
- 5(2y) + 8y = 0.5
- 10y + 8y = 0.5
- 18y = 0.5
- y = 0.5/18
- y = 1/36

**Calculating the time taken by 1 woman alone:**

- Substituting the value of y in the equation x = 2y:
- x = 2(1/36)
- x = 1/18

- Therefore, 1 woman alone can finish the embroidery work in 1/18 days.

**Calculating the time taken by 1 man alone:**

- Since y = 1/36, 1 man alone can finish the embroidery work in 1/(1/36) = 36 days.

Let the work done by man and woman per day be x and y respectively.

When the work is completed in 4 days
Since 5 men and 2 women complete the work in 4 days
therefore work done by 5 men and 2 women in 1 day =
1/4

∴5x+2y= 1/4 ( equation -i)


When the work is completed in 3 days
Since 6 men and 3 women complete the work in 3 days
therefore work done by 6 men and 3 women in 1 day =
1/3


∴6x+3y= 1/3( equation ii)

Multiplying by 3 in eq i
we get
⇒15x+6y= 3/4 (equation iii)

Multiplying by 2 in equation ii
, we get
⇒12x+6y= 2/3 (equation iv)

On subtracting eq iv from eq iii
we get
⇒15x+6y−12x−6y= 3/4-2/3


⇒3x= 1/12
⇒x= 1/36


On substituting the value of x in eq ii we get

⇒6×1/36 + 3y = 1/3

3y = 1/3-1/6


⇒y= 1/18



Thus,
work done by 1 man in 1 day = 1/36

days
∴ Time taken by 1 man alone to finish the work =36 days

work done by 1 woman in 1 day = 1/18


days
∴ Time taken by 1 woman alone to finish the work =18 days