1 man and 4 women can complete a work in $\frac{65}{4}$ days, while 3 men and 4 women can complete it in $\frac{13}{2}$ days. In how many days will 13 women complete the same?

20

1M + 4W = \(\frac{65}{4}\) days,

\(\frac{65}{4}\)M + 65W =1 day   ..(1) (Multiplied by \(\frac{65}{4}\) to get it in 1 day)

3M + 4W = \(\frac{13}{2}\) days,

\(\frac{39}{2}\) + 26W = 1 day     ..(2)  (Multiplied by \(\frac{13}{2}\) to get it in 1 day)

NOW, (1) - (2), WE GET 

⇒ 13M = 156W

⇒ M : W = 156 : 13 = 12 : 1 (Efficiencies)

Putting efficiency in equation (1), and get the total work

\(\frac{65}{4}\) (12) + 65(1) = 260 units.

Hence, Time taken by 13W to complete the total work = \(\frac{260}{13(1)}\) = 20 days. ..(\(\frac{Work}{Efficiency}\) = Time)