A grocer wishes to blend two different coffee brands in order to make a blend of 400 pounds to sell at $3.00 a pound. If he uses a brand of coffee worth $2.70 a pound with another brand worth $3.20 a pound, how many pounds of each does he use?
To find the number of pounds of each coffee brand the grocer needs to use, we can set up a system of equations.
Let's assume he uses x pounds of the $2.70 coffee brand, and y pounds of the $3.20 coffee brand.
We can then set up the following equations:
Equation 1: x + y = 400 (since the total blend is 400 pounds)
Equation 2: (2.70 * x) + (3.20 * y) = (3.00 * 400) (since the total value of the blend is equal to the total weight multiplied by the selling price)
To solve this system of equations, we can use either substitution or elimination method. Let's use the elimination method in this case.
Multiply Equation 1 by 2.70 to simplify it:
2.70 * (x + y) = 2.70 * 400
2.70x + 2.70y = 1080
Now we have the following system of equations:
Equation 1: 2.70x + 2.70y = 1080
Equation 2: 2.70x + 3.20y = 1200
Next, subtract Equation 1 from Equation 2 to eliminate the x term:
(2.70x + 3.20y) - (2.70x + 2.70y) = 1200 - 1080
0.50y = 120
y = 120 / 0.50
y = 240
Now we know that the grocer needs to use 240 pounds of the $3.20 coffee brand.
Substitute the value of y back into Equation 1:
x + 240 = 400
x = 400 - 240
x = 160
Therefore, the grocer needs to use 160 pounds of the $2.70 coffee brand and 240 pounds of the $3.20 coffee brand to make a blend of 400 pounds.