Ruby, Joey and sully are baking cookies together. they need 3/4 cup of flour and 1/3 cup of butter to make a dozen cookies. Ruby brought 2 cups of flour and 1/4 cup of butter, Joey brought 1 cup of flour and 1/2 cup of butter, sully brought 1 1/4 cups of flour and 3/4 cup of butter. if the students have plenty of other ingredients that they need. how many whole batches of a dozen cookies can they make.​​

4

Step-by-step explanation:

Amounts needed for 1 batch of 1 dozen cookies:

3/4 cup flour

1/3 cup butter

It's easier to create a table with all amounts to find the total amount of flour and total amount of butter available.

            Ruby         Joey            Sully         Total

Flour       2               1                  1 1/4          4 1/4

Butter     1/4           1/2                  3/4           1 1/2

The table above shows the total amount of flour is 4 1/4 cups, and the total amount of butter is 1 1/2 cups.

Flour:

Amount available: 4 1/4 cups

Amount needed for 1 batch: 3/4 cup

(4 1/4 cups)/(3/4 cup) = 17/4 * 4/3 = 17/3 = 5 2/3

The amount of flour is enough for 5 2/3 batches.

This makes 5 full batches.

Butter:

Amount available: 1 1/2 cups

Amount needed for 1 batch: 1/3 cup

(1 1/2 cups)/(1/3 cup) = 3/2 * 3/1 = 9/2 = 4 1/2

The amount of flour is enough for 4 1/2 batches.

This makes 4 full batches.

The amount of flour is enough for 5 batches, but the amount of butter is enough for only 4 batches, so the greatest number of whole batches they can make is 4.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

6 on both sides and twelve on the top and the bottom

step-by-step explanation: