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 women alone to finish the work, also that taken by 1 man alone.​

Answer 1

$\huge\underline\mathbb{SOLUTION}$

$\bf{Assumption\::}$

The time taken by woman to complete the work be x days and by man be y days.

$\bf{Given\::}$

2 women and 5 men can together finish an embroidery work in 4 days.

3 women and 6 men can finish the same work in 3 days.

$\underline\mathfrak\purple{According\:to\:question,}$

⇒ 2/x + 5/y = 1/4

⇒ (2y + 5x)/xy = 1/4

⇒ 4(2y + 5x) = xy

⇒ 8y + 20x = xy...........(1)

On the other hand, 3 women and 6 men can finish it in 3 days.

$\underline\mathfrak\purple{According\:to\:question,}$

⇒ 3/x + 6/y = 1/3

⇒ (3y + 6x)/xy = 1/3

⇒ 3(3y + 6x) = xy

⇒ 9y + 18x = xy............(2)

On multiplying (eq 1) with 9 and (eq 2) with 8 we get,

⇒ 72y = 9xy - 180x...........(3)

⇒ 72y = 8xy - 144x...........(4)

On comparing we get,

⇒ 9xy - 180x = 8xy - 144x

⇒ 9xy - 8xy = - 144x + 180x

⇒ xy = 36x

⇒ y = 36 days

Substitute value of y in (eq 1)

⇒ 8(36) + 20x = 36x

⇒ 288 = 36x - 20x

⇒ 288 = 16x

⇒ x = 18 days

$\bf{Therefore,}$

Woman alone can finish the work in 18 days and man alone in 36 days.

Answer 2

Heya !!

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Let 1 woman can finish the embroidery work in 'x' days and 1 man can finish it in 'y' days.

A woman's one day work = 1/x

and a man's one day work = 1/y

A.T.Q.

2/x + 5/y = 1/4 and

3/x + 6/y = 1/3

Let 1/x = a and 1/y = b

Now, 2a + 5b = 1/4 _(1)

and 3a + 6b = 1/3 _(2)

Multiplying _(1) by 3 and _(2) by 2 and subtracting

6a + 15b = 1/4

6a + 18b = 1/3

___________

0 + 3b = 1/4 - 1/3

=> 3b = 1/12

=> b = 1/(12×3)

=> b = 1/36

Putting b = 1/36 in _(1)

2a + 5(1/36) = 1/4

=> 72a + 5 = 36/4

=> 72a + 5 = 9

=> 72a = 9 - 5

=> 72a = 4

=> a = 4/72

=> a = 1/18

Now, a = 1/x = 1/18

=> x = 18

And, b = 1/y = 1/36

=> y = 36

So, 1 woman can finish the embroidery work in 18 days and 1 man can finish the work in 36 days.

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Hope my ans.'s satisfactory.☺