Question

2 Gents and 5 Ladies complete a work in 4 dyas. 4 Gents and 4 Ladies takes
days to complete the same work. Then how many days need to complete the
same work for a single Gents or Ladies?​

Answer 1

2 Gents and 5 Ladies complete a work in 4 dyas. 4 Gents and 4 Ladies takes  *3 days to complete the same work.

12 days are needed to complete the  same work for a single Gents and Ladies.

Let one gent take x days to complete the work.

So, work done by him in one day = 1/x

Let one lady take y days to complete the work.

So, work done by him in one day = 1/y.

Given, 2 gents and 5 ladies complete a work in 4 days.

Thus, 2/x + 5/y = 1/4

⇒ 2y + 5x = xy/4                 [Multiplying both sides by xy]

⇒ 8y + 20x = xy                  [Multiplying by 4 on both sides]

⇒ 96y + 240x = 12xy          [Multiplying by 12 on both sides]   ...(1)

Again 4 gents and 4 ladies takes 3 days to complete the work.

Thus, 4/x + 4/y = 1/3

⇒ 4y + 4x = xy/3                 [Multiplying by xy on both sides]

⇒ 12y + 12x = xy                 [Multiplying by 3 on both sides]

⇒ 96y + 96x = 8xy             [Multiplying by 8 on both sides]    ...(2)

Subtracting (2) from (1),

240x - 96x = 12xy - 8xy

⇒ 144x = 4xy

⇒ y = 144/4  = 36

As 8y + 20x = xy

So, 8(36) + 20x  = x(36)

⇒ 20x + 288 = 36x

⇒ 16x = 288

⇒ x = 288/16 = 18

Let one gent and one lady take 'd' days.

So, 1/x + 1/y = 1/d

⇒ 1/18 + 1/36 = 1/d

⇒ 3/36 = 1/d

⇒ 1/d = 1/12

⇒ d = 12

12 days is the answer.