# 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.

Question

Answer:

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.
solved

general
9 months ago
1907