A student 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 $\frac{2}{3}$ 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.

Let the number cherry Danishes be x Let the number cheese Danishes be y   Given, The number of cherry Danishes the student brings is at least 3 more than 2/3 the number of cheese Danishes x ≥ 3 + (2/3)y   Given, The number of cherry Danishes the student brings is no more than twice the number of cheese Danishes. x ≤ 2y   Merging, the two inequalities above, 3 + (2/3)y ≤ x ≤ 2y 3 + (2/3)y ≤ 2y OR 2y ≥ 3 + (2/3)y 2y – (2/3)y ≥ 3   Multiply both sides by 3 (3 * 2)y – (3 * 2/3)y ≥ 3 * 3 6y – 2y ≥ 9 4y ≥ 9 y ≥ 9/4   Since x ≥ 3 + (2/3)y, and y ≥ 9/4 Then, x ≥ 3 + (2/3 * 9/4) x ≥ 3 + (2/3 * 9/4) x ≥ 3 + 3/2 x ≥ 9/2   Since y ≥ 9/4 and x ≥ 9/2 x + y ≥ 9/4 +9/2 x + y ≥ 27/4 x + y ≥ 6.75     x + y represents the total number of Danishes the student brings x + y ≥ 6.75 means that the total number of Danishes the student brings is 6.75   BUT since the total number of Danishes must be an integer, then that the total number of Danishes the student brings is 6.