An herbalist has 40 oz of herbs costing $4 per ounce. How many ounces of herbs costing $1 per ounce should be mixed with these 40 oz of herbs to produce a mixture costing $2.20 per ounce?
oz
number of ounces of $1 stuff --- x
4(40) + 1(x) = 2.2(40+x)
solve for x, you should get x = 60
To solve this problem, we can use the concept of weighted averages. Let x be the number of ounces of herbs costing $1 per ounce that need to be mixed with the 40 oz of herbs costing $4 per ounce.
The total cost of the mixture can be calculated by multiplying the cost per ounce by the total number of ounces in the mixture. We want the total cost of the mixture to be $2.20 per ounce.
The total cost of the herbs costing $4 per ounce is 40 oz * $4/oz = $160.
The total cost of the herbs costing $1 per ounce is x oz * $1/oz = $x.
The total cost of the mixture is ($160 + $x).
The total number of ounces in the mixture is (40 oz + x oz).
According to the given condition, the cost per ounce of the mixture should be $2.20. We can set up the equation:
($160 + $x) / (40 oz + x oz) = $2.20/oz.
Now we can solve for x:
$160 + $x = $2.20(40 oz + x oz).
$160 + $x = $88 + $2.20x.
Simplifying, we get:
$x - $2.20x = $88 - $160.
$0.80x = -$72.
Dividing both sides by $0.80, we have:
x = -$72 / $0.80.
x = -90.
Since x represents the number of ounces, it cannot be negative. Therefore, there is no solution to this problem.
In conclusion, there is no amount of herbs costing $1 per ounce that can be mixed with the 40 oz of herbs costing $4 per ounce to produce a mixture costing $2.20 per ounce.
To solve this problem, we'll use the concept of mixture problems. Let's break down the information given:
1. The herbalist has 40 ounces of herbs costing $4 per ounce. This means the total cost of these 40 ounces is (40 * $4 = $160).
2. We need to find out how many ounces of herbs costing $1 per ounce should be mixed with the 40 ounces of herbs to produce a mixture costing $2.20 per ounce.
3. Let's assume that x ounces of herbs costing $1 per ounce are mixed with the initial 40 ounces.
Now, we can set up an equation to solve for x:
Total cost of the mixture = (cost of the herbs costing $4 per ounce) + (cost of the herbs costing $1 per ounce)
(total ounces of the mixture) * (cost per ounce) = (40 + x) * $2.20
From here, we can set up the equation:
(40 + x) * $2.20 = $160 + x * $1
Simplifying the equation:
88 + 2.20x = 160 + x
Rearranging the equation:
2.20x - x = 160 - 88
1.20x = 72
Now, solve for x:
x = 72 / 1.20
x ≈ 60
So, you would need to mix approximately 60 ounces of herbs costing $1 per ounce with the initial 40 ounces to produce a mixture costing $2.20 per ounce.