17. 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?
Let the amount of cashews be x and the amount of peanuts be y.
We know that the total weight of the mixture is 1/2 pound, so:
x + y = 1/2
We also know that the mixture is worth $2.80 per pound, so the total cost of the mixture is:
0.5 * $2.80 = $1.40
The cost of the cashews is $5.50 per pound, so the cost of x pounds of cashews is:
$5.50x
Similarly, the cost of y pounds of peanuts is:
$2.30y
Since we want the total cost of the mixture to be $1.40, we can set up the following equation:
$5.50x + $2.30y = $1.40
Now we have two equations with two variables:
x + y = 1/2
$5.50x + $2.30y = $1.40
We can solve for one variable in terms of the other from the first equation:
y = 1/2 - x
Substituting this into the second equation, we get:
$5.50x + $2.30(1/2 - x) = $1.40
Simplifying, we get:
$5.50x + $1.15 - $2.30x = $1.40
Combining like terms, we get:
$3.20x = $0.25
Dividing both sides by $3.20, we get:
x ≈ 0.078
Substituting this value back into the equation y = 1/2 - x, we get:
y ≈ 0.422
Therefore, the owner should mix approximately 0.078 pounds of cashews with 0.422 pounds of peanuts to get a 1/2 pound mixture that is worth $2.80 per pound.