Just as a guess, since cashews cost so much more, you'd need to use a lot less of them. So I'd choose A or F. To be sure, solve
p+c = 1/2
5.50c + 2.30p = 2.80 * 1/2
ACashews: 0.10 lb.; peanuts: 0.40 1b.
BCashews: 0.42 lb.; peanuts: 0.08 1b.
CCashews: 0.40 lb.; peanuts: 0.10 1b
DCashews: 0.27 lb.; peanuts: 0.23 1b.
E Cashews: 0.23 lb.; peanuts: 0.27 1b.
F Cashews: 0.08 lb.; peanuts: 0.42 1b
p+c = 1/2
5.50c + 2.30p = 2.80 * 1/2
Let's use the variables C and P to represent the amount of cashews and peanuts, respectively, in pounds.
According to the problem, we want to mix cashews worth $5.50 per pound with peanuts worth $2.30 per pound to get a 1/2 pound mixture worth $2.80 per pound.
From this information, we can write the following equations:
1. The total weight equation:
C + P = 1/2
2. The value equation:
(5.50 * C + 2.30 * P) / (C + P) = 2.80
To solve the system of equations, we can use substitution or elimination.
Let's use substitution:
Rearrange the total weight equation to solve for C:
C = 1/2 - P
Substitute this value for C in the value equation:
(5.50 * (1/2 - P) + 2.30 * P) / ((1/2 - P) + P) = 2.80
Simplify and solve for P:
(2.75 - 5.50P + 2.30P) / (1/2 + 0.5P) = 2.80
(2.75 - 3.20P) / (1/2 + 0.5P) = 2.80
Cross-multiply:
2.75 - 3.20P = 2.80(1/2 + 0.5P)
2.75 - 3.20P = 1.40 + 1.40P
Combine like terms:
3.20P + 1.40P = 2.75 - 1.40
4.60P = 1.35
Divide by 4.60:
P = 1.35 / 4.60
P ≈ 0.2935
Now, substitute the value of P back into the total weight equation to find the value of C:
C + 0.2935 = 1/2
C ≈ 1/2 - 0.2935
C ≈ 0.2065
Therefore, the owner should include approximately 0.2065 pounds of cashews and 0.2935 pounds of peanuts in the mixed bag.
So, the correct answer is option D: Cashews: 0.27 lb.; peanuts: 0.23 lb.