## Saturday, January 16, 2010

### The Math of Cocktail

At one of the fancy holiday parties you tasted an amazing cocktail that you decided to attempt to reproduce later at home. When days later two of your friends stopped by, you offered to make it for them. You were sure it contains two ingredients (say liqueur and champagne), but not certain of their proportions. When you made the first cocktail, you filled half of the glass with champagne, then poured 1 teaspoon of liqueur and tasted it a bit. You realized that you need to add much more liqueur to match the taste from the party. When making the second glass, you poured three times more liqueur than you originally put in the first glass. However, when you tasted it, you saw that a bit of liqueur is still needed. Perhaps a third of the amount you had to add to the first glass. How much liqueur do you need to pour in the glass the next time to make the perfect cocktail?

Enter your answer on our Family Puzzle Marathon page. Solve three and get a prize!

Kim said...

Let x=the amount you added the first time.

You added x the first time, 3x the second time.

If you need to add 1/3 of what you added to the first glass, that's x/3. So we need to add 3x + x/3 = 3.33x

Maria said...

Clarification: both glasses needed more liqueur. You added some (unknown) amount to the first glass. You added only a third of this amount to the second glass. It was not a third of the original amount in the first glass, but rather a third of the amount you added after tasting.

Kim said...

Maria, I'm still confused. It sounded like you added more to the second glass than the first, but above you're saying you added less. Why would you add less to the second glass if you knew the first glass was too weak?

But OK, we have x in the first glass and 1/3 x to the second glass. Can you describe again what you're trying to achieve with the 3rd glass?

Maria said...

I wanted to say that "a picture is worth a thousand words" and draw something explaining this puzzle, but then I realized that I spent half of today trying to convince my kids of the exact opposite and pull them away from various screens...

So, we are stuck with words.
You made only two glasses of cocktail, one after another.
In each glass you poured half a glass of champagne. For each glass you poured some liqueur, then mixed the cocktail, then tasted, realized that more liqueur is needed and added it as necessary.
For the first glass you initially added 1 teaspoon of liqueur. After mixing and tasting you added more liqueur, say X.
For the second glass you initially added three times as much liqueur as you originally put into the first glass = 3 teaspoons. When you mixed and tasted the cocktail in the second glass, you realized that you still need to add more. But only one-third of what you had to add to the first. That would be 1/3 of the X.

The question is, how much liqueur should be mixed with half a glass of champaign next time you make this cocktail, so that you won't need to taste and add more.

Alin Grin said...

7 teaspoon of liqueur.

3 + 1/3X = 1 + x
4 = 2/3X
x = 6

Maria said...

There is just one small mistake hiding somewhere in this equation...

Kim said...

I get the same equation as Alin, but...

1+ x = 3 + 1/3x
2/3 x = 2
x = 2 * 3/2 = 3 tsp.

Kim said...

Sorry, that wasn't a complete answer... we have to add 4 tsp to get it right the first time (1+3 or 3+1 being what we did with the first two glasses).

Maria said...

And the cocktail goes to Kim, or the puzzle point.