An apple a day problem

Question

A farmer is deciding how to use each day’s harvest. She can use the harvest to produce apple cider or juice for festival in 2 weeks. The information she uses to make decision is listed.

*1 bushel apples make 20 qts cider

*1 bushel apples make 15 qts juice

* the farmer collects 18 bushels apples

* today farmer needs to produce a total of 330 qts.......

A. The information given can be modeled with a system of equations. Define your X and Y variables represent in this situation.

X variable: ?

Y variable: ?

Write 2 different equations that can be used to model the situation, explain what each equation represents....

Answer 1

Hi Eric R.,

Let x equal the number of bushels needed to make cider.

Let y equal the number of bushels needed to make juice.

The total number of bushels needed to make cider, plus, the total number of bushels needed to make juice, has to equal, the total bushels of apples the farmer collects, x + y = 18 bushels.

If 1 bushel makes 20 qts. cider, then x bushels makes 20x qts.

If 1 bushel makes 15 qts. juice, then y bushels makes 15y qts.

And qts. of cider plus qts. of juice needs to equal a total of 330 qts.

Therefore: 20x + 15y = 330 qts.

I hope this helps, Joe.

P.S. The solution can be found by substituting y = 18 - x into 20x + 15(18 - x) = 330.

Answer 2

For the easiest modeling of this, it makes sense to have x=bushels used for cider and y=bushels used for juice. You can also reverse this. It doesn't really matter, as long as you know which is which. This will work well because you know the total bushels, 18. Using that info, you can say:

x+y=18

To incorporate the quarts of juice and cider as well as the total quarts needed, just multiply! One bushel of apples yielding 20 quarts of cider is just another way to describe multiplication.

20x+15y=330