A drinking problem.

$$\dfrac{1}{20}+\dfrac{1}{x}=\dfrac{1}{14}$$

$$\dfrac{1}{x}=\dfrac{20-14}{14*20}=\dfrac{6}{14*20}$$

$$x=\dfrac{14*20}{6}=46.6666666667$$  

for the wife alone 46.6666666667 days

P.S.

$$\begin{array}{lcr} \frac{1\;V}{20\;days} & \mbox{rate for 1 day} & \mbox{man}\\\\ \frac{1\;V}{x\;days} & \mbox{rate for 1 day} & \mbox{woman}\\\\ \frac{1\;V}{20\;days} + \frac{1\;V}{x\;days} & \mbox{rate for 1 day} & \mbox{man and woman}\\\\ \left(\frac{1\;V}{20\;days} + \frac{1\;V}{x\;days}\right) & \times \mbox{ 14 days} &=1\mbox{ V} \end{array}\\\\\\ \Rightarrow \dfrac{1}{20}+\dfrac{1}{x}=\dfrac{1}{14}$$

For those who don't understand why heureka manipulated reciprocal times (which is the correct thing to do) here's a more long-winded approach.

Here's the way I like to think about this sort of problem. Each day the man drinks 1/20 of the cask. When his wife helps him, they drink 1/14 of the cask each day. So 1/14 - 1/20 must be the portion that she drinks everyday.

1/14 - 1/20   = 6/280  .......  now, just take the reciprocal of the fraction on the right

280/6 = 46+2/3 days

And there you go.....