Home > Challenge > Coffee and Cream

Coffee and Cream

I was recently reminded of an excellent math problem involving mixtures.

Imagine yourself sitting in front of a cup of coffee and a cup of cream. Suppose you take a spoonful of cream, pour it into the coffee, and stir it up. Now once that’s thoroughly mixed, you take a spoonful of the mixture and pour it back into the cream. Then you mix that up.

After all of this, is there more more coffee in the cream, more cream in the coffee, or equal amounts in both?


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  1. kIMbER-ly
    October 26, 2010 at 12:20 am

    Does anyone know the correct answer?
    I picked there’s more cream in the coffee…

    Hope I am right!!

  2. Jonathan Schwartz
    October 31, 2010 at 6:17 am

    I remember this problem!

  3. October 31, 2010 at 8:00 am

    Ah, but do you remember the solution?

  4. Amrisha
    November 12, 2010 at 2:35 pm

    I found a similar problem on the quantitative math section of the common admission test (CAT) in India. its a ratio problem, the mixture transfered back takes mostly coffee (since the mixture is mostly coffee) and leaves a large amount of cream. the cream cup get mostly coffee the way it works out, the amount of coffee transfered back is the same as the amount of cream left behind.

    the ratios are equal

  5. Prof. W.
    January 3, 2012 at 3:07 pm

    Imagine you have 2 units of coffee and 2 units of cream. Take 1 unit of cream and mix it into the coffee. So now the coffee cup is 2 units coffee, 1 unit cream. Now take 1 unit from the coffee cup (which is 1/3 units cream and 2/3 units coffee) and return it to the original cup. Do the math. The coffee cup now has 2/3 units cream and 4/3 units coffee. Therefore, the cream cup now has 4/3 units cream and 4/3 units coffee. The ratios are the same. Note that the result will be the same regardless of the original amounts of each liquid.

  6. January 3, 2012 at 5:26 pm

    Thanks for the response. Your solution is similar to the “Straightforward” solution below.

    Moreover, if instead of assuming there are 2 units of each, a remarkable think happens when you assume there is only 1 unit of each–see the “Make a Wish” solution.

    Lastly, I encourage you check out my colleague’s very elegant solution that doesn’t bother with actual amounts.

  1. November 2, 2010 at 7:05 am
  2. November 9, 2010 at 8:01 am
  3. November 22, 2010 at 8:40 am
  4. December 3, 2010 at 9:05 am

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