Question

Tom has twelve slips of paper which he wants to put into five cups labeled $A,$ $B,$ $C,$ $D,$ $E.$

He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E.$ The numbers on the papers are: $\begin{align*} &2, 2, 2, 2.5, 2.5, 3,\\& 3, 3, 3, 3.5, 4, 4.5.\end{align*}$ If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B,$ then the slip with $3.5$ must go into what cup?

$A$

$B$

$C$

$D$

$E$

Video solution:

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Written solution:

The sum of all the slips is $35$ , so the five consecutive integer cup sums must average $7$ . Therefore cups $A,B,C,D,E$ must have sums $5,6,7,8,9$ , respectively.

Cup $B$ already contains a $3$ and must sum to $6$ , so it must contain another $3$ . Cup $E$ already contains a $2$ , so the other slips in $E$ must sum to $7$ .

The $3.5$ slip cannot go in $A$ , because cup $A$ would need another $1.5$ . It cannot go in $B$ , which is already full. It cannot go in $C$ or $E$ , because either would then need another $3.5$ , and no remaining slips can make that total. Cup $D$ works, for example with $3.5+4.5=8$ .

Thus, D is the correct answer.

2015 AMC 8 Problem 23

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Written solution:

Problem 23 in Other Years