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# 2 women and 5 men can together finish a piece of embroidery 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 embroidery and that taken by 1 man alone.

Given:

2 women and 5 men can together finish a piece of embroidery in 4 days, while 3 women and 6 men can finish it in 3 days.

To do:

We have to find the time taken by 1 woman alone to finish the embroidery and that taken by 1 man alone.

Solution:

Let the number of days taken by one man alone to finish a piece of embroidery be $x$.

This implies,

The amount of work done by one man in a day $=\frac{1}{x}$.

Let the number of days taken by one woman alone to finish the embroidery be $y$.

This implies,

The amount of work done by one woman in a day $=\frac{1}{y}$.

In the first case, 2 women and 5 men can together finish the piece of embroidery in 4 days.

The amount of work done by 5 men in 1 day $=5\times\frac{1}{x}=\frac{5}{x}$.

The amount of work done by 2 women in 1 day $=2\times\frac{1}{y}=\frac{2}{y}$.

According to the question,

$4(\frac{5}{x}+\frac{2}{y})=1$

$\frac{20}{x}+\frac{8}{y}=1$....(i)

In the second case, 6 men and 3 women finish the work in 3 days.

The amount of work done by 6 men in 1 day $=6\times\frac{1}{x}=\frac{6}{x}$.

The amount of work done by 3 women in 1 day $=3\times\frac{1}{y}=\frac{3}{y}$.

According to the question,

$3(\frac{6}{x}+\frac{3}{y})=1$

$\frac{18}{x}+\frac{9}{y}=1$....(ii)

Multiplying equation (i) by 9 and equation (ii) by 8, we get,

$9(\frac{20}{x}+\frac{8}{y})=9(1)$

$\frac{180}{x}+\frac{72}{y}=9$.....(iii)

$8(\frac{18}{x}+\frac{9}{y})=8(1)$

$\frac{144}{x}+\frac{72}{y}=8$.....(iv)

Subtracting (iv) from (iii), we get,

$\frac{180}{x}+\frac{72}{y}-(\frac{144}{x}+\frac{72}{y})=9-8$

$\frac{180-144}{x}=1$

$x=36$

Substituting $x=36$ in (i), we get,

$\frac{20}{36}+\frac{8}{y}=1$

$\frac{8}{y}=1-\frac{5}{9}$

$\frac{8}{y}=\frac{9-5}{9}$

$\frac{8}{y}=\frac{4}{9}$

$y=\frac{8\times9}{4}$

$y=18$

Therefore, one man alone takes 36 days to finish the work and one woman alone takes 18 days to finish the work.