the choir is selling snacks after the concert. cupcakes are $4 and hot dogs are $3 each you want to buy at least 5 items you want to spend no more than $36 total

Answer 1

Hello Daniel!

First of all, lets call the number of cupcakes by the letter c and the number of hotdogs by the letter h. Based on the informations we have:

h + c ≥ 5

4c + 3h ≤ 36

Realize that h and c must be natural numbers (you cannot buy half cupcake or negative hotdog).

Multiplying the first expression by 3, we have

3h + 3c ≥ 15 ⇒ 3h ≥ 15 - 3c

From the second expression we have:

4c + 3h ≤ 36 ⇒ 3h ≤ 36 - 4c

Putting the two expressions together, we have:

36 - 4c ≥ 15 - 3c

c ≤ 21

However, the expression 4c + 3h ≤ 36 is more restrictive (because c and h are natural numbers), if we considers the lowest value h can admit (zero), we have:

4c ≤ 36

c ≤ 9

and, by the same reasoning, considering c equals zero, we have

3h ≤ 36

h ≤ 12

So to find the solution, we must consider all the possible values of c

if c = 0, then 12 ≥ h ≥ 5

if c = 1, then 10 ≥ h ≥ 4

if c = 2, then 12 ≥ h ≥ 3

if c = 3, then 8 ≥ h ≥ 2

if c = 4, then 6 ≥ h ≥ 1

if c = 5, then 5 ≥ h ≥ 0

if c = 6, then 4 ≥ h ≥ 0

if c = 7, then 2 ≥ h ≥ 0

if c = 8, then 1 ≥ h ≥ 0

if c = 9, then h = 0

Answer 2

Graph c + d ≥ 5 and 4c + 3d ≤ 36

The intersection shall provide all possible combinations