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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.
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