Final answer:
To solve the mixture problem, we set up an equation based on the total weight and another based on the total cost, then solve the system of equations algebraically to find out how many pounds of peanuts and raisins are needed.
Step-by-step explanation:
The question involves creating a cost-based blend of two items, peanuts and raisins, to produce a mixture with a target cost per pound. This is a typical algebraic mixture problem, commonly encountered in high school math. To solve this problem, we want to find out how many pounds of peanuts and raisins are needed to make a 50-pound mixture that costs $1.47 per pound when peanuts cost $1.20 per pound and raisins cost $2.10 per pound.
Let's denote the weight of peanuts as P pounds and the weight of raisins as R pounds. The total weight of the mixture is given as 50 pounds, which gives us the equation:
P + R = 50
Next, we need to consider the total cost of the mixture. The cost of the peanuts is P times $1.20, and the cost of the raisins is R times $2.10. Since the mixture should have an overall cost of $1.47 per pound, our cost equation becomes:
1.20P + 2.10R = 1.47 × 50
From our first equation, we can express R as 50 - P, and substitute it into our second equation to find the value of P. After solving these linear equations, we will obtain the exact quantities of peanuts and raisins required to make the desire mixture.