Answer:
at least 8
Step-by-step explanation:
Let the variable $r$ represent the number of cherry danishes and $s$ represent the number of cheese danishes. We have$2s \ge r \ge 3+\frac{2}{3}s.$The left side of the inequality must be greater than or equal to the right side, so we have$2s \ge 3+\frac{2}{3}s\qquad
\Rightarrow\qquad 6s \ge 9+2s \qquad
\Rightarrow\qquad 4s \ge 9 \qquad
\Rightarrow\qquad s \ge \frac{9}{4}.$The least possible integer that $s$ can be is 3. The least possible value of $r$ is$r \ge 3+\frac{2}{3}(3) \qquad \Rightarrow \qquad r \ge 5.$Therefore, the student brings at least 8 danishes
* credits: AoPS
