Randall wants to mix 22 lb of nuts worth $1 per lb with some nuts worth $9 per lb to make a mixture worth $8 per lb. How many pounds of $9 nuts must he use?
1*22 + 9x = 8(x+22)
To solve this problem, we can use the concept of weighted averages.
Let's assume that Randall needs to use x pounds of $9 nuts to make the mixture.
The total weight of the mixture will be the sum of the weights of the $1 nuts and $9 nuts, which is 22 lb + x lb.
Now, let's calculate the weighted average of the mixture. The weighted average is calculated by multiplying the value of each component by its weight, summing them up, and dividing by the total weight of the mixture.
For the $1 nuts, the value is $1 per lb, and the weight is 22 lb. So, the contribution of the $1 nuts to the mixture is 1 * 22 = 22.
For the $9 nuts, the value is $9 per lb, and the weight is x lb. So, the contribution of the $9 nuts to the mixture is 9 * x = 9x.
The total contribution of the mixture is 22 + 9x.
The weighted average is given as $8 per lb, which means the total contribution of the mixture should be (22 + 9x) lb * $8.
Setting up the equation, we have:
(22 + 9x) lb * $8 = (22 lb * $1) + (9x lb * $9)
Simplifying the equation, we get:
176 + 72x = 22 + 81x
Rearranging the equation, we have:
72x - 81x = 22 - 176
-9x = -154
Dividing both sides by -9, we get:
x = 17
Therefore, Randall needs to use 17 pounds of $9 nuts to make a mixture worth $8 per lb.