The owner of an organic fruit stand also sells nuts. She wants to mix cashews worth $5.50 per pound with peanuts worth $2.30 per pound to get a 1/2 pound mixture that is worth $2.80 per pound. How much of each kind of nut should she include in the mixed bag?
amount of cashews ---- x lbs
amount of peanuts = (1/2 - x) lbs
5.5x + 2.3(1/2-x) = 2.8(1/2)
5.5x + 1.15 - 2.3x = 1.4
3.2x = .25
320x = 25
x = 25/320 lbs or 5/64 lbs of cashews
She needs 5/64 lbs of cashews and
27/64 lbs of peanuts.
Let's assume that x represents the number of pounds of cashews and y represents the number of pounds of peanuts needed for the mixture.
To determine the amount of cashews and peanuts needed, we can set up a system of equations based on the given information:
1) The total weight of the mixture is 1/2 pound:
x + y = 1/2
2) The cost per pound of the mixture is $2.80:
(5.50x + 2.30y) / (x + y) = 2.80
To solve this system of equations, we can use the method of substitution.
Let's solve Equation 1 for x:
x = 1/2 - y
Next, substitute this value of x into Equation 2:
(5.50(1/2 - y) + 2.30y) / (1/2 - y + y) = 2.80
Simplifying the equation:
2.75 - 5.50y + 2.30y = 2.80
Combine like terms:
-3.20y = 0.05
Dividing both sides by -3.20:
y ≈ 0.0156
Now, substitute the value of y back into the equation x = 1/2 - y:
x = 1/2 - 0.0156
x ≈ 0.4844
Therefore, the owner should include approximately 0.4844 pounds of cashews and 0.0156 pounds of peanuts in the mixture.
To find out how much cashews and peanuts the owner should include in the mixture, we can set up a system of equations based on the given information.
Let's assume:
Let's represent the weight of cashews in the mixture as "C" (in pounds)
Let's represent the weight of peanuts in the mixture as "P" (in pounds)
Now let's set up the equations:
1. Equation based on the total weight of the mixture:
C + P = 1/2
2. Equation based on the total value (price) of the mixture:
(5.50C + 2.30P) / (C + P) = 2.80
To solve this system of equations, we can use the substitution method.
Let's start by solving equation 1 for C:
C = 1/2 - P
Now substitute this value of C in equation 2:
(5.50(1/2 - P) + 2.30P) / (1/2 - P + P) = 2.80
Simplify the equation:
(2.75 - 5.50P + 2.30P) / (1/2) = 2.80
Multiply both sides of the equation by 1/2 to get rid of the denominator:
2.75 - 5.50P + 2.30P = 1.40
Combine like terms:
2.75 - 3.20P = 1.40
Subtract 2.75 from both sides:
-3.20P = -1.35
Divide both sides by -3.20:
P = 1.35 / 3.20
P ≈ 0.422
Now substitute the value of P back into equation 1 to find C:
C + 0.422 = 1/2
C ≈ 1/2 - 0.422
C ≈ 0.078
So, the owner should include approximately 0.078 pounds (or about 1.25 ounces) of cashews and approximately 0.422 pounds (or about 6.75 ounces) of peanuts in the mixed bag.