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A coffee merchant has two types of coffee beans, one selling for $9 per pound and the other for $15 per pound. The beans are to be mixed to provide 100 lb of a mixture selling for $13.50 per pound. How much of each type of coffee bean should be used to form 100 lb of the mixture?

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Answer:

To form 100 lb of the mixture we need 25 lbs of $9 coffee beans and 75 lbs of $15 coffee beans.

Step-by-step explanation:

Let x be the number of pounds of $9 coffee beans and y be the number of pounds of $15 coffee beans.

We know that the mixture must weigh 100 lb

[tex]x+y=100[/tex]

and the total cost per pound is given by

[tex]9x+15y=13.50\cdot 100\\9x+15y=1350[/tex]

Now, we can solve the system of equations

[tex]\begin{bmatrix}x+y=100\\ 9x+15y=1350\end{bmatrix}[/tex]

Isolate x for [tex]x+y=100[/tex]

[tex]x=100-y[/tex]

Substitute [tex]x=100-y[/tex] into the second equation

[tex]9\left(100-y\right)+15y=1350[/tex]

Isolate y

[tex]900-9y+15y=1350\\900+6y=1350\\900+6y-900=1350-900\\6y=450\\\frac{6y}{6}=\frac{450}{6}\\y=75[/tex]

For [tex]x=100-y[/tex] substitute y = 75

[tex]x=100-75\\x=25[/tex]

To form 100 lb of the mixture we need 25 lbs of $9 coffee beans and 75 lbs of $15 coffee beans.

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