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# A drinking problem.

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481
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+92623

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?

Melody  Jun 20, 2014

#1
+19480
+8

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

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

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

for the wife alone 46.6666666667 days

P.S.

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

heureka  Jun 20, 2014
#1
+19480
+8

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

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

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

for the wife alone 46.6666666667 days

P.S.

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

heureka  Jun 20, 2014
#2
+26718
+5

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.

Alan  Jun 20, 2014
#3
+86859
0

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.....

CPhill  Jun 20, 2014