Final answer:
To determine how many pounds of peanuts are in a 10-pound mixture that costs $84, with peanuts costing $7 per pound and cashews $9 per pound, we set up and solve a system of equations. With C representing cashews and P peanuts, we get P + C = 10 and 7P + 9C = 84. Solving this, we find there are 3 pounds of peanuts in the mixture.
Step-by-step explanation:
To solve the problem of determining the quantity of peanuts in a 10-pound mixture of peanuts and cashews that costs $84, we can set up a system of equations. Let P represent the number of pounds of peanuts and C represent the number of pounds of cashews.
First, we know that the total weight of the mixture is 10 pounds, which gives us the equation:
P + C = 10.
Second, we have the pricing for the peanuts and cashews. Peanuts cost $7 a pound and cashews cost $9 a pound. This gives us another equation based on the total cost: 7P + 9C = $84.
To find the value of P, we can solve the system of equations. Multiply the first equation by 7 to set it up for elimination with the second equation:
- 7P + 7C = 70 (first equation multiplied by 7)
- 7P + 9C = 84 (original cost equation)
Subtracting the first new equation from the second gives us:
Dividing both sides by 2 we find that C = 7. Plugging C = 7 into P + C = 10 gives us P = 3. Thus, there are 3 pounds of peanuts in the mixture.