Please help with finite word problem!

Question

There are three types of coffee, Kona, Latin Dark Toast and Moravia. Equal parts of these three coffees are put into the house blend which sells for $7 per pound. A 30 lb bag is made up of 1.5 lbs Kona, 8 lbs of Latin Dark and 7 lbs of Moravia and sells for $216.20. Another popular blend is made of 7 lbs Kona and 3 lbs Moravia and sells for $65.40. What is the coast per pound of the three coffees?

Answer 1

Hi Kenya.

 

There's a lot of info here, but let's just start by calling some variables. I'm going to say that:

 

K = cost of 1 pound of Kona coffee

L = cost of 1 pound of Dark Roast coffee

M = cost of 1 pound Moravia coffee

 

The house blend is equal parts of all three and sells for $7 per pound. That means 1/3 of a pound of K, plus 1/3 of a pound of L, plus 1/3 of a pound of M equals 7.

 

1/3K + 1/3L + 1/3M = 7

 

Let's move onto the 30 pound bag. 15 lbs of K plus 8lbs of L plus 7lbs of M costs $216.20.

 

15K + 8L + 7M = 216.20

 

We also know that 7lbs of K plus 3lbs of M is $65.40.

 

7K + 3M = 65.40

 

Now we have a system of equations.

 

1/3K + 1/3L + 1/3M = 7

15K + 8L + 7M = 216.20

7K + 0L + 3M = 65.40

 

This is going to get really messy if we do it by hand. Let's use a matrix solver. My personal favorite is Wolfram Alpha.

 

https://www.wolframalpha.com/

 

You want to use the Reduced Row Echelon Code to solve this matrix. The code

 

rref()

 

will indicate you want to solve the matrix using that form.

 

Plug in the coefficients of your system row by row, separated by commas, enclosed by brackets. For example, look at your first row.

 

1/3K + 1/3L + 1/3M = 7

 

In matrix form, this will translate to

 

[1/3,1/3,1/3,7]

 

In the end the code will look like this:

 

rref([1/3,1/3,1/3,7],[15,8,7,216.20],[7,0,3,65.40])

 

When you punch that in, it'll spit out something that looks like this:

 

1 0 0 7.5

0 1 0 9.2

0 0 1 4.3

 

This is the "reduced row echelon form" of the problem matrix.

 

If you look at the matrix we defined, K is the first column. That means K = 7.5.

 

L is the second column, so L = 9.2

 

M = 4.3

 

This means that a pound of Kona roast costs $7.50, a pound of Latin Dark roast costs $9.20, and a pound of Moravia costs $4.30.

 

Hope this helps.