An apple farmer is deciding how to use each day’s harvest. she can use the harvest to produce apple cider or apple juice for the apple festival in two weeks. the information she uses to make the decision is listed below.
1 bushel of apples will make 20 quarts of apple cider.
1 bushel of apples will make 15 quarts of apple juice.
the apple farmer collect 18 bushels of apples.
today the apple farmer needs to produce a total of 330 quarts.

a. the information given can be modeled with a system of equations. define your x and y variables represent in this situation.

b. the information given can be modeled with a system of equations. write two different equations that can be used to model the situation. explain what each equation represents.

c. use the equations in part b to determine the number of bushels that were used to make apple cider and were used to make apple juice. show or explain your work.

A)

There are 18 bushels of apples.

1 bushel of apples will make 20 quarts of apple cider.  

1 bushel of apples will make 15 quarts of apple juice.

x = number of bushels to make cider out of.

y = number of bushels to make juice out of.

B)

As there are 18 bushels

x + y = 18

The apple farmer needs to produce a total of 330 quarts.

1 bushel of apples will make 20 quarts of apple cider.  

1 bushel of apples will make 15 quarts of apple juice.

20x + 15y = 330

C)  

x + y = 18. Thus we can express y in terms of x:

y = 18 - x

We can substitute the y in 20x + 15y = 330 and solve for x:

20x + 15 (18 - x) = 330

20x + 270 - 15x = 330

5x = 60

x = 12

And as y = 18 - x

y = 18 - 12  

y = 6

[ C_1 ]: We could have done the same with x, expressing x in terms of y:

x = 18 - y, substitute the x in 20x + 15y = 330 and solve for y:

20 (18 - y) + 15y = 330

360 - 20y + 15y = 330

-5y = -30

y = 6

And as x = 18 - y

x = 18 - 6

x = 12

D)

As we had defined in A)  

x = number of bushels to make cider out of

y = number of bushels to make juice out of,

there will 12 bushels of apples be processed into cider and

there will 6 bushels of apples be processed into juice

in order to produce

20 quarts of cider per bushel * 12 bushels = 240 quarts of cider, and

15 quarts of juice per bushel * 6 bushels = 90 quarts of juice,

which sum up to a total of

240 + 90 = 330 quarts of apple beverages, as required.