Wild rice sells for $6 pound, and plain rice sells for $1 per pound. A grocer wants 10 pounds if the mixture to sell for $2 per pound. How much of each type should he used?
wild rice needed --- x lbs
plain rice needed -- 10-x lbs
6x + 1(10-x) = 2(10)
6x + 10-x = 20
5x = 10
x = 2
he needs 2 lbs of wild rice and 8 lbs of plain rice
Let's assume the grocer uses x pounds of wild rice and y pounds of plain rice.
The total weight of the mixture is x + y pounds.
The cost of the wild rice is $6 per pound, so the cost of x pounds of wild rice is 6x dollars.
The cost of the plain rice is $1 per pound, so the cost of y pounds of plain rice is y dollars.
The total cost of the mixture is 6x + y dollars.
The mixture should sell for $2 per pound, and the total weight of the mixture is x + y pounds. So, the total cost of the mixture should be (x + y) * 2 dollars.
Since the grocer wants the cost of the mixture to be $2 per pound, we can set up the equation:
6x + y = (x + y) * 2
Simplifying the equation:
6x + y = 2x + 2y
Rearranging the terms:
6x - 2x = 2y - y
4x = y
So, y = 4x.
Since the grocer wants 10 pounds of the mixture, we can set up another equation for the total weight:
x + y = 10
Substituting y = 4x:
x + 4x = 10
5x = 10
x = 10 / 5
x = 2
So, the grocer should use 2 pounds of wild rice.
Substituting x = 2 in y = 4x:
y = 4 * 2
y = 8
So, the grocer should use 2 pounds of wild rice and 8 pounds of plain rice.
To determine how much of each type of rice the grocer should use, we can set up a system of equations based on the given information.
Let's assume the grocer uses 'x' pounds of wild rice and 'y' pounds of plain rice:
1) The total weight of the mixture is 10 pounds, so we have the equation:
x + y = 10
2) The grocer wants the mixture to sell for $2 per pound, so the total cost of the mixture can be expressed as:
6x + 1y = 2 * 10
Now, we have a system of two equations:
1) x + y = 10
2) 6x + y = 20
We can solve this system of equations using the method of substitution or elimination.
Using the substitution method:
From equation 1, we can express x in terms of y by subtracting y from both sides:
x = 10 - y
Substituting this value of x into equation 2:
6(10 - y) + y = 20
60 - 6y + y = 20
-5y = 20 - 60
-5y = -40
y = (-40) / (-5)
y = 8
Now, substitute the value of y into equation 1 to find x:
x + 8 = 10
x = 2
Therefore, the grocer should use 2 pounds of wild rice and 8 pounds of plain rice to achieve the desired mixture.