Cofffe and Cream — A Straightforward Solution

Published by patrick honner on

I offered a classic mixture problem–Coffee and Cream–and I suppose it is appropriate that the poll results show a mixture of opinions.  Although one answer clearly seems to be the favorite, there is still strong support for the other two options.  I offer here one approach to solving this classic conundrum.

We begin by adopting a powerful problem solving strategy, one that when used incorrectly can lead to disaster, but when used properly, can turn hard problems into simple ones:  we make a wish.

In the statement of the Coffee and Cream problem, no real mention of measurements were made, and, since all the results are ratios anyway, we can wish for whatever starting amount we want.  So, let us wish that each cup contains 10 spoonfuls, and proceed from and cream solution 1So there are 10 spoonfuls of coffee on the left, and 10 spoonfuls of cream on the right.  We take one spoonful of cream, and add it to the and cream solution 2There are now 11 spoonfuls of liquid in the cup on the left:  10 spoonfuls of coffee and 1 spoonful of cream.  Thus, the cup on the left is 1/11 cream and 10/11 coffee.  Now we and cream solution 3Mixing doesn’t change the amounts of coffee and cream in the cup:  the cup on the left is still 1/11 cream and 10/11 coffee.  So we take one spoonful of that, and add it to the and cream solution 4The cup on the left once again has 10 spoonfuls of liquid, and that liquid is 1/11 cream and 10/11 coffee.  So of the liquid on the left,  1/11 of 10 spoonfuls, or 10/11 of a spoonful, is cream.  In other words, there is 10/11 of a spoonful of cream in the coffee.

Now the cup on the right also again has 10 spoonfuls of liquid in it.  How much of that is coffee?  Well, the only coffee in the cup on the right came from the one spoonful of mixture we just added.  And that mixture is 10/11 coffee! coffee and cream solution 5Whether we mix it up or not, there is now 10/11 of a spoonful of coffee in the cream.

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Categories: Challenge

patrick honner

Math teacher in Brooklyn, New York


Scott Matthews · November 2, 2010 at 9:29 pm

The simplest explanation of this problem that I have heard involves no fractions: In the end, both cups have the same amount of liquid. Therefore, by the simple law of displacement, the amount of cream in the coffee is the the same as the amount of coffee in the cream.

Hilary · November 4, 2010 at 11:16 am

Scott, woah! Woah.

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