# Word Problem During a break, each pupil in a class made a full cup of coffee with milk for himself or herself. After having finished his cup, Brandon calculated that he consumed one-fifth of the...

**Word Problem**

During a break, each pupil in a class made a full cup of coffee with milk for himself or herself. After having finished his cup, Brandon calculated that he consumed one-fifth of the total amount of coffee and one-seventh of the total amount of milk used by all students in the class. Assuming that all students had cups of the same capacity, how many students were in the class? Find all possible (practical) solutions.

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### 1 Answer

This can either be worked out based on the proportions of coffee used or the proportions of milk used. Once one or other is defined there is no extra '*slack*' in the problem.

Suppose we think about the amount of coffee used. This is more practical in fact, because the amount of coffee is likely to be discrete (measured in spoons). The amount of milk, however, is likely to vary more continuously since it is liquid.

Let us assume that the amount of coffee used by students varies between 1/2 tsp and 2 tsps ( it is important to reach a practical solution).

This means that Brandon's 1/5 of the total amount of coffee might vary between 1/2 tsp and 2 tsps so that the total amount of coffee would vary between 2.5 tsps and 10 tsps.

**Scenario 1:**

If Brandon has 1/2 tsp of coffee, then there are 2 tsps left. Depending on their preferences for the amount of coffee they have this implies that there may be between 1 and 4 other students in this scenario. Result: there are between 2 and 5 students in the class.

**Scenario 2:**

If Brandon has 2 tsp of coffee, then this implies that there are 8 tsps left. Again depending on their preferences for the amount of coffee they have this further implies that there may be between 4 and 16 other students in this scenario. Result: there are between 5 and 17 students in the class.

Since these are the two '*extreme'* scenarios (where Brandon has the minimum amount of coffee and the maximum amount of coffee), all other solutions lie between these two.

Combining the two together there may be between 2 and 17 students in the class. The amount of milk each has may vary between a tiny amount (2 ml?) and a large amount (20 ml?). If Brandon has 2 ml, then the other student could have as much as 12 ml. If the other student has 20 ml then Brandon has 10/3 ml

**Answer: Assuming that each student has between 1/2 and 2 tsps of coffee, there may be between 2 and 17 students in the class. A sensible assumption for the amount of milk they have is that it varies (continously) between 2 and 20 ml.**