Show all possible combinations of roses and carnations that Carlo could buy.?

Carlo needs to buy at least eight flowers and has $15 to spend.Roses cost $3 per stem, carnations cost $1 per stem.

Show all possible combinations of roses and carnations that Carlo could buy.?
R + C %26gt;= 8

3R + C %26lt;= 15



And obviously R %26gt; 0 and C %26gt; 0, so we can confine ourselves to first quadrant answers.



The shaded region (aka feasible region) created by the above two inequalities in QI is the triangular region with vertices (R,C) at (0,8) (0,15) and (3.5, 4.5). You can look at that triangle and any point in or on the triangle having integer coordinates to be a valid combination.



If he buys 0 roses, he can buy anywhere from 8 to 15 carnations.



If he buys 1 rose, he can buy up to 7 to 12 carnations.

If he buys 2 roses, he can buy 6 to 9 carnations.

The most roses he can buy is 3 because R = 4 is outside the triangle. And with 3 roses ($9) he can buy 5 or 6 carnations.
Reply:Let R be the number of roses and C be the number of carnations. Then

R+C %26gt;= 8

3*R + 1*C = 15



Solve this second one for R

R = (15 - C)/3.

R = 5 - (C/3)



Now try values of C from 15 (which is the most amount of carnations he could possibly afford) down to 0 so that R+C is at least 8. Since R has to be an integer, C has to be a multiple of 3. So just try C = 15, 12, 9, 6, 3 and 0. You get



15 carnations, no roses

12 carnations, one rose

9 carnations, two roses

6 carnations, three roses



Trying with 3 carnations gives 4 roses and 0 carnations gives 5 roses, but these sums don't add up to 8 or more flowers. So the only possible combinations are the above 4.
Reply:Okay, let's go through all of the possibilities using the most expensive flower, the rose, as the basis.



Rose: $3

Carnation: $1



0 roses, 15 carnations ($15.00)

1 rose, 12 carnations ($15.00)

2 roses, 9 carnations ($15.00)

3 roses, 6 carnations ($15.00)



These are the only ones that work and allow at least 8 flowers to be bought. Also note that these combos require Carlos to spend all $15 dollars exactly. If you are also looking for combos that don't require Carlos to spend all $15 but still buy at least 8 flowers, they are as follows:



0 roses, 8 carnations ($8.00)

0 roses, 9 carnations ($9.00)

0 roses, 10 carnations ($10.00)

0 roses, 11 carnations ($11.00)

0 roses, 12 carnations ($12.00)

0 roses, 13 carnations ($13.00)

0 roses, 14 carnations ($14.00)

1 rose, 7 carnations ($10.00)

1 rose, 8 carnations ($11.00)

1 rose, 9 carnations ($12.00)

1 rose, 10 carnations ($13.00)

1 rose, 11 carnations ($14.00)

2 roses, 6 carnations ($12.00)

2 roses, 7 carnations ($13.00)

2 roses, 8 carnations ($14.00)

3 roses, 5 carnations ($14.00)



Wow, that is a total of just 4 combos if you are going along with the first thing, but it's a total of 20 combos if you count all of them!
Reply:Make a list

he can buy 1 rose and 12 carnations (13 flowers)

2 roses and 9 carnations (11 flowers)

3 roses and 6 carnations (9 flowers)

rain roots