A man can drink a cask of wine in 20 days, but if his wife drinks with him it will take only 14 days—how long would it take for the wife alone?
120+1x=114
1x=20−1414∗20=614∗20
x=14∗206=46.6666666667
for the wife alone 46.6666666667 days
P.S.
1V20daysrate for 1 dayman1Vxdaysrate for 1 daywoman1V20days+1Vxdaysrate for 1 dayman and woman(1V20days+1Vxdays)× 14 days=1 V⇒120+1x=114
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.....