Final answer:
To achieve a 25% mixture worth $5.73 per pound by blending chocolate and trail mix, a proportional relationship shows that we need to blend more trail mix than chocolate. Using the approximate ratio and rounding to the nearest pound, the answer that comes closest to the desired outcome is 4 pounds of chocolate and 6 pounds of trail mix. Option a is the correct answer.
Step-by-step explanation:
To create a new 25% mixture worth $5.73 per pound by mixing chocolate costing $7.80 per pound with trail mix costing $3.10 per pound, we can set up a system of equations to reflect the total cost of the mixture and solve for the amount of each ingredient. Let's denote c as the pounds of chocolate and t as the pounds of trail mix.
The total cost of chocolate would be $7.80c and the total cost of trail mix would be $3.10t. When mixed, the total weight is c + t and the mixture's total cost is $5.73(c + t).
Now we can set up two equations:
- $7.80c + $3.10t = $5.73(c + t)
- c + t = c + t (the total weight of the mixture, this does not give us new information, so we actually only need the first equation)
The first equation can be simplified and organized to $7.80c + $3.10t = $5.73c + $5.73t. Subtracting $5.73c and $3.10t from both sides leaves us with $2.07c = $2.63t. Dividing both sides by $2.07 gives us c = (2.63/2.07)t.
Since the mixture must total 100%, and we want a 25% mixture, we can assume that 25% of the mixture should be chocolate and 75% should be trail mix, meaning that there should be three times as much trail mix as chocolate. Hence, t = 3c. Using our earlier equation, c = (2.63/2.07)(3c) simplifies to c = (2.63/2.07)(3)c, which can be solved to find that c ≈ 4 pounds (rounding to the nearest pound) and t = 3c ≈ 12 pounds (rounding to the nearest pound). However, this does not match the given options.
Since none of the options provided match the exact answer, let's review the options and approximate proportionally. Option B provides the nearest ratio of three times as much trail mix as chocolate, which suggests that the correct answer is 4 pounds of chocolate and 6 pounds of trail mix (rounded to the nearest whole numbers and fitting the 3-to-1 ratio).