Formulate the following problems as a pair of equations, and hence find their solutions:

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 woman alone to finish the work, and also that taken by 1 man alone.

#### Solution

Let the number of days taken by a woman and a man be x and y respectively.

Therefore, work done by a woman in 1 day = 1/x

According to the question,

`4(2/x + 5/y) = 1`

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

`3(3/x + 6/y) = 1`

`3/x + 6/y = 1/3`

Putting `1/x = p ` in these equations, we get

2p + 5q = 1/4

By cross multiplication, we get

`p/(-20-(-18)) = q/(-9-(-18)) = 1/(144-180)`

`p/-2 = q/-1 = 1/-36`

`p/-2 = -1/36 `

`p = 1/18 `

`p = 1/x = 1/18 `

x = 18 and y = 36

Hence, number of days taken by a woman = 18 and number of days taken by a man = 36