In a village, $90\%$ of people drink Tea, $80\%$ Coffee, $70\%$ Whiskey, $60\%$ Gin. Nobody drinks all four. What percentage of people drinks alcohol?

In a small village $90\%$ of the people drink Tea, $80\%$ Coffee, $70\%$ Whiskey and $60\%$ Gin. Nobody drinks all four beverages. What percentage of people of this village drinks alcohol?

I got this riddle from a relative and first thought it can be solved with the inclusion-, exclusion principle. That the percentage of people who drink alcohol has to be in the range from $70\%$ to $100\%$ is obvious to me

When $T$, $C$, $W$, and $G$ are sets, and I assume a village with $100$ people, then what I am looking for is $$\lvert W\cup G\rvert = \lvert W\rvert+\lvert G\rvert-\lvert W\cap G\rvert$$ I know that $$\lvert T \cap C \cap W \cap G \rvert = 0$$ and also the absolute values of the singletons.

But I do not see how this brings me any closer, since I still need to figure out what $\lvert W\cap G\rvert$ is and that looks similar hard at this point

On the way there I also noticed that $\lvert T\cap C\rvert \ge 70$ and similar $\lvert W\cap G\rvert \ge 30$

By now I think there is too little information to solve it precisely.

Solution 1:

If you add up the percentages, they come out to $300\%$. This means that the average number of beverages per person is $3$. No one drinks more than that, so no one can drink less than that, either. Since everyone drinks exactly three beverages, everyone has exactly one beverage that they don't drink. So no one doesn't drink both whiskey and gin, i.e. everyone drinks alcohol.

Solution 2:

Yes, there is enough information. Clearly the overlap between whiskey drinkers and gin drinkers is at least $30\%$, being exactly $30\%$ iff $40\%$ drink whiskey but not gin, and $30\%$ drink gin but not whiskey. Similarly at least $70\%$ drink both tea and coffee. Since nobody drinks all four, the number who drink both whiskey and gin is exactly $30\%$, and the number who drink alcohol is $30\% + 30\% + 40\%$, i.e. everybody.

Solution 3:

Hint: consider what "Nobody drinks all four beverages" means in terms of the four groups

• People who don't drink tea
• People who don't drink coffee
• People who don't drink whiskey
• People who don't drink gin

and then see how big each of those groups are.

Solution 4:

From the problem we can say that the village is $90\%+80\%+70\%+60\% = 300\%$ saturated with liquids. From this we can say that every person should drink strictly $3$ beverages, no more no less.

Having drinking three beverages, while there are only two non-alcoholic beverages supplied to the village, you are guaranteed to get drunk. Thus, $100\%$ of the village drink alcohol.

Solution 5:

You can use inclusion/exclusion but you might not have enough information. Or then again you might.

The number of people who are in $A$ or $B$ is $A + B - (A\cap B)$ and so if $A+B > 100$ percent we can conclude $A+B - 100\le A\cap B \le \min (A,B)$

So $WHISKEY + GIN - 100 = 70+60 -100 = 30 \le(WHISKEY \cap GIN) \le \min WHISKEY, GIN = GIN = 60$.

Likewise $TEA + COFFEE - 100 = 90 + 80 100 = 70 \le(COFFEE \cap TEA)\le \min COFFEE, TEA = COFFEE = 80$.

Let $A = (COFFEE \cap TEA)$ and $B = (WHISKEY \cap GIN)$ and $70 + 30 = 100 \le A + B$ so $A+B -100 \le A\cap B$. But we know that $A \cap B = COFFEE \cap TEA \cap WHISKEY \cap GIN)=0$.

That can only happen if $A = 70$ and $B = 30$.

So the people drink alcohol $= WHISKEY + GIN - (WHISKEY \cap GIN) = 70 + 60 -30 = 100$ percent.

(We have everybody drinking alcohol and everbody drinking caffeine, and everybody drinks two of one and one of the other:

$30\%$ drink gin, tea, and coffee.

$40\%$ drink whiskey, tea, and coffee.

$10\%$ drink whiskey, gin, and coffee.

$20\%$ drink whiskey, gin, and tea.