Astudent brings whole cherry and cheese danishes to his class for his birthday. the number of cherry danishes he brings is at least 3 more than the number of cheese danishes, but no more than twice the number of cheese danishes. find the smallest possible value for the total number of danishes he brings.

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

you can first factor out the n to get n(1+5). we know that 1+5 is 6, so we just have 6n.