Question

2 women and 5 men can complete an embroidery work in 4 days, while 3 women and 6 men can do it in 3 days. Find the time taken by one woman and one man Alone to complete the work.

Answer 1

Let time taken by 1 woman alone to finish the work = x days

Let time taken by 1 man alone to finish the work = y days

So, 1 woman’s 1-day work = (1/x)th part of the work

And, 1 man’s 1-day work = (1/y)th part of the work

So, 2 women’s 1-day work = (2/x)th part of the work

And, 5 men’s 1-day work = (5/y)th part of the work

Therefore, 2 women and 5 men’s 1-day work = (2/x+5/y)th part of the work

                                                                                                              … (1)

It is given that 2 women and 5 men complete work in = 4 days

It means that in 1 day, they will be completing 1/4th part of the work   … (2)

Clearly, we can see that (1) = (2)

⇒  2/x+5/y=1/4                                                                                      … (3)

Similarly,3/x+6/y=1/3                                                                             … (4)

Let 

Putting this in (3) and (4), we get

2p + 5q =1/4  and 3p + 6q =1/3 

⇒ 8p + 20q = 1                                                                                        … (5) 

and 9p + 18q = 1                                                                                     … (6) 

Multiplying (5) by 9 and (6) by 8, we get

72p + 180q = 9                                                                                        … (7)

72p + 144q = 8                                                                                        … (8)

Subtracting (8) from (7), we get

36q = 1⇒ q =1/36 

Putting this in (6), we get

9p + 18 (1/36) = 1

⇒ 9p = 1/2⇒ p = 1/18 

Putting values of p and q in , we get x = 18 and y = 36

Therefore, 1 woman completes work in = 18 days

And, 1 man completes work in = 36 days

:) Hope this helps!!!

Answer 2

Hello Dear ✔️✔️↙️↙️

Q) 2 women and 5 men can complete an embroidery work in 4 days, while 3 women and 6 men can do it in 3 days. Find the time taken by one woman and one man Alone to complete the work.

Solution :

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

Then, 1 women's 1 days work $= \frac{1}{x}$

1 man's 1 days work $= \frac{1}{y}$

$\therefore \: \frac{2}{x} + \frac{5}{y} = \frac{1}{4} \:and \: \frac{3}{x} + \frac{6}{y} = \frac{1}{3} \\ \\Putting \: \frac{1}{x} = u \: and \: \frac{1}{y} = v \: these \: equation \\became \\ 2u + 5v = \frac{1}{4} \: \: \: \: \: \: \: \: ...(1) \\\\ 3u + 6v = \frac{1}{3} \: \: \: \: \: \: \: \: ...(2) \\ \\Multiply \: (1) \: by \: 3 \: and \: (2) \: by \: 2 \: and \\subtracting, \: we \: get \: \\ \\ 6u + 15v \: = \frac{3}{4} \: \: \: \: \: \: \: \: ...(3) \\\\ 6u + 12v = \frac{2}{3} \: \: \: \: \: \: \: \: ...(4)\\\\ 3v = \frac{3}{4} - \frac{2}{3} = \frac{9 - 8}{12} = \frac{1}{12} \\ \\ \implies \: v = \frac{1}{36} \\ \\Putting\:v\:=\frac{1}{36}\:in\:(1),\:we\:get\\\\2u\:=\frac{1}{4}-\frac{5}{36}=\frac{9-5}{36}=\frac{4}{36} \\ \\ \implies\:u=\frac{4}{36}\times\frac{1}{2}=\frac{1}{18}\\ \\Now, \: u = \frac{1}{18} \implies \: \frac{1}{x} = \frac{1}{18} \implies \: x = 18 \\\\ and \: v = \frac{1}{36} \implies \: = \frac{1}{y} = \frac{1}{36} \implies \: y = 36$

Thus, 1 woman alone can finish the embroidery work in 18 days and 1 man alone can finish it in 36 days.

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