A coffee distributor needs to mix a(n) House coffee blend that normally sells for $8.40 per pound with a Mexican Shade Grown coffee blend that normally sells for $14.70 per pound to create 70 pounds of a coffee that can sell for $14.61 per pound. How many pounds of each kind of coffee should they mix?
To solve this problem, we need to determine the quantities of House coffee blend and Mexican Shade Grown coffee blend required to create the desired coffee blend.
Let's assume the number of pounds of House coffee blend needed is 'x', and the number of pounds of Mexican Shade Grown coffee blend needed is 'y'.
According to the given information, the total weight of the coffee blend needed is 70 pounds.
Therefore, we can write the following equation based on the weight:
x + y = 70 --(Equation 1)
The desired coffee blend is expected to sell for $14.61 per pound. To find the price equation,we can calculate the weighted average of the prices of the two coffee blends.
The total cost of the House coffee blend (x pounds) is 8.40x dollars.
The total cost of the Mexican Shade Grown coffee blend (y pounds) is 14.70y dollars.
The total cost of the coffee blend (70 pounds) is 14.61 * 70 = 1022.70 dollars.
Therefore, we can write the following equation based on the price:
8.40x + 14.70y = 1022.70 --(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of 'x' and 'y'.
Using any method of solving systems of equations (substitution, elimination, or matrix), we can find the solution.
One way to solve this is by substitution:
From Equation 1, we can express 'x' in terms of 'y':
x = 70 - y
Substituting this expression into Equation 2, we get:
8.40(70 - y) + 14.70y = 1022.70
Simplifying the equation:
588 - 8.40y + 14.70y = 1022.70
Combining 'y' terms:
6.30y = 434.70
Dividing both sides by 6.30:
y = 69
Now, substituting the value of 'y' back into Equation 1:
x + 69 = 70
x = 1
So, the coffee distributor should mix 1 pound of House coffee blend with 69 pounds of Mexican Shade Grown coffee blend to create 70 pounds of coffee that can sell for $14.61 per pound.
Let's assume the coffee distributor needs to mix x pounds of the House coffee blend and (70 - x) pounds of Mexican Shade Grown coffee blend to create 70 pounds of a coffee blend that can sell for $14.61 per pound.
The cost of the House coffee blend is $8.40 per pound, so the total cost of x pounds of the House coffee blend is 8.40x dollars.
The cost of the Mexican Shade Grown coffee blend is $14.70 per pound, so the total cost of (70 - x) pounds of the Mexican Shade Grown coffee blend is 14.70(70 - x) dollars.
The total cost of the coffee blend is the sum of the costs of the two blends, which should be equal to the selling price of $14.61 per pound multiplied by 70 pounds:
8.40x + 14.70(70 - x) = 14.61 * 70
Simplifying the equation:
8.40x + 1029 - 14.70x = 1022.7
Combining like terms:
-6.30x + 1029 = 1022.7
Subtracting 1029 from both sides:
-6.30x = -6.3
Dividing both sides by -6.30:
x = 1
So, the coffee distributor needs to mix 1 pound of the House coffee blend with (70 - 1) = 69 pounds of the Mexican Shade Grown coffee blend.
add up the value of the parts:
8.40x + 14.70(70-x) = 14.61*70
Expect a value of nearly zero for x, since the price of the mix is very close to the price of the Mexican coffee.